THE U N I V E R S I T Y OF M I C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 20 AN EXISTENCE ANALYSIS FOR NONLINEAR NON-SELF-ADJOINT BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS John S. Locker ORA Project 07100 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP- 3920 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1965

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1965. ii

TABLE OF CONTENTS Page ABSTRACT iv CHAPTER 1. INTRODUCTION 1 2. A GENERAL THEORY FOR THE EQUATION Lx = Nx 6 2.1 The Right Inverse Operator H 6 2.2 The Projection Operators P and Q 10 2.3 Solving the Auxiliary Equation 19 2.4 Solving the Bifurcation Equation 25 3. DIFFERENTIAL OPERATORS 37 3.1 The Definition of the Differential Operator L 37 3.2 The Adjoint Operator L* 42 3.3 An Integral Representation for the Operator H 53 4. THE NONLINEA.R DIFFERENTIAL EQUATION Lx = Nx 70 4.1 The Existence Theorems 70 4.2 The Self-adjoint Case 74 5. THE BASIC CONDITIONS OF THE EXISTENCE THEOREMS 78 6. AN EXAMPLE 87 APPENDIX I 105 APPENDIX II 112 BIBLI OGRAPHIY 12 iii

ABSTRACT In this dissertation we study the equation Lx = Nx under the assumptions that L is a linear operator with domain and range in a real Hilbert space S., and N is an operator (not necessarily linear) whose domain and range also lie in S. Our purpose is to present an existence theory which guarantees that this equation has at least one solution. We apply this theory to the study of nonlinear boundary value problems of ordinary differential equations. The existence theory presented is closely related to an existence theory recently developed by L. Cesari for boundary value problems of ordinary and partial differential equations. Our theory reduces to the one of Cesari when L is a self-adjoint differential operator. Cesari chooses a system of axioms concerning the existence of certain linear operators H and P possessing special properties. Using these two operators, the problem of determining a solution to the equation Lx = Nx is reduced to solving a finite system of equations in finitely many unknowns. For our existence theory we assume that L is a closed linear operator satisfying the properties: (a) the domain of L is dense in S; (b) the range of L is closed in S; (c) the null spaces for L and the adjoint L* have finite dimensions p and q, respectively. It follows that there exist linear operators H, P, and Q with properties analogous to the properties of Cesari's operators H and P. The operator H is a continuous right inverse for L, while the operators P and Q are projection operators which project into the domains of L* and L, respectively. These three operators depend only on the linear operator L and are independent of the operator N. Using these three operators, we establish the existence of at least one solution xcS to the equation Lx = Nx provided a certain set of inequalities are satisfied and provided a particular finite system of equations in finitely many unknowns is solvable. At the same time we obtain estimates on the norm of such a solution x. If p > q, then the finite system of equations is a system having more unknowns than equations or the same number. For this case we present two existence theorems for the equation Lx = Nx. These existence theorems are used to study the equation Lx = Nx when L is a differential operator on a finite interval [a,b], S is the real Hilbert space L2[a,b], and N is a nonlinear operator in S. We prove that L is a closed linear operator which satisfies conditions iv

(a), (b), and (c), and hence, we obtain the existence of the operators H, P, and Q. The operator H is shown to have an integral representation. Our existence theorems then yield existence theorems for a solution of the differential equation Lx = Nx in the real Hilbert space S = L2[a,b]. In case L is a self-adjoint differential operator, we can construct H. P, and Q in such a way that P = Q, and such that H and P satisfy the axioms of Cesari. Thus, we show that the Cesari theory is applicable to the study of self-adjoint differential operators. Our theory is a generalization of the Cesari theory which is especially useful for the study of non-self-adjoint differential operators. If the differential equation Lx = Nx has a solution x, then we prove that all the hypotheses of our existence theorems are satisfied, and that x is one of the solutions guaranteed by the existence theorems. As an application of our theory we show that the nonlinear boundary value problem 2 x + X + =X = It, 0 < t < 25c, ix(O) = 0 has a solution provided the constants a,t satisfy the conditions lal < 1, JIP <.001, and we obtain estimates on the norm of such a solution. v

CHAPTER 1 INTRODUCTI ON In this thesis we discuss the existence of solutions to the equation Lx - Nx,lo 1) where L is a linear operator with domain and range in a real. Hi:l.ber-t space S, and N is an operator (not necessarily linear) whose domain and range also lie in So We present an existence theory wh.ich. is applicable when L is an unbounded operator defined on a proper subset of So This theory has applications to the study of nonlinear boundary value problems in ordinary differential equations. Our theory is closely related to an existence theory recently developed by Cesari [3] When L is a self-adjoint differential operator9 our theory reduces to the one of Cesari. In his study of Equation (1. 1) Cesari presents a system of axioms concerning the existence of linear operators H. and P with convenient properties. Using these two operators., the problem of determining a solution to Equation (1 1.) is reduced to the problem of solving a finite system of equ.ations in finitely many unknownso We examine the Cesari axioms and their -impii.cat+ions in Appendix I. Bartle [21, Cronin [71, and Nirenberg [191 have dev-eloped ex:i.stence theories for an equation which is similar to Equation (.lo. 1)o They con1

2 sider the equation b: + F(xy) 0 (1 2) in Banach spaces S and Y. In this equation L~S + S is a bounded linear operator and F(x9y) is a function which maps a neighborhood of the origin in S x Y into S with F(O,0) = 0. For y near the origin in Y they seek solutions xcS of Equation (1.2). They also reduce the problem to one of solving a finite system of equations in finitely many unknowns. In Appendix II we relate these three theories to the theory presented in this thesis, and we determine the relationships which exist between the three of them. We present an existence theory for Equation (1.1) in Chapter 2. By assuming that the linear operator L satisfies reasonable conditions, we show that there exist linear operators H, P, and Q with properties analogous to the properties of Cesari' s operators H and P. Using these three operators, the study of Equation (1.1) is reduced to solving a finite system of equations in finitely many unknowns. The conditions on L are satisfied by all differential operators on a finite interval [ab], so our existence theory may be used to study nonlinear boundary value problems of ordinary differential equations. If the operator L is a self-adjoint differential operator, then we can construct H, P, and Q in such a way that P Q, and such that H and P satisfy the axioms of Cesari. Thus, we show that the Cesari

theory is applicable to the study of self-adjoint differential operators. Our theory is a generalization of the Cesari theory and is especially useful for the study of non-self-adjoint differential operators. Let us summarize our existence theory as it appears in Chapter 2. We consider Equation (1.1), Lx = Nx, in a real Hilbert space S. The operator L is a closed linear operator with domain o(L) and range /i(L) in S, and the operator N is an operator in S with domain /(N) such that`(L)nq(N) f ~. We assume that L satisfies the following conditions: (a) The domain (YL) is dense in S; (b) The range ~(L) is closed in S; (c) The null space for L has dimension p < o, and the null space for the adjoint operator L* has dimension q < c. When these conditions are satisfied, there exist operators H, P, and Q with convenient properties (see Theorem 2.4). The operator H is a continuous right inverse for L, while the operators P and Q are projection operators which project into~,&(L*) and XL), respectively. These three operators depend only on the linear operator L and are independent of the operator N. Using these three operators, we obtain the existence of at least one solution HAS to Eguation (1.1) provided a certain set of inequalities are satisfied and provided a particular finite system of equations

4 in finitely many unknowns is solvable (see Theorem 2.7). At the same time we obtain estimates on the norm of such a solution x. If p > q, then the finite system of equations is a system having more unknowns than equations or the same number. For this case we present two existence theorems for Equation (1o 1) (see Theorems2.9 and 2.10). In Chapter 3 we introduce the notion of a differential operator L in the real Hilbert space S -= L2[a,b] and summarize some of the familiar properties of these operators. In particular, we show that each differential operator in S is a closed linear operator satisfying the above conditions (a), (b), and (c). For a given differential operator L we determine its.adjoint L* using classical methods and construct a continuous right inverse H using the results of Chapter 2. The operator H is represented as an integral operator. The results of Chapters 2 and 3 are combined in Chapter 4 to yield existence theorems for nonlinear ordinary differential equations of the form Lx = Nx (see Theorems 4.1 and 4.2). As a special case we examine self-adjoint differential operators L, showing that our existence theory reduces to the Cesari theory when L is a self-adjoint differential operator. In Chapter 5 we assume that there exists an exact solution x' to the ordinary differential equation Lx = Nx, and then show that all the hypotheses of our existence theorems are satisfied. It follows that x is one of the solutions guaranteed by our existence theorems.

5 As an application of our theory we study in Chapter 6 the nonlinear boundary value problem x" + x+ Cx2 -= t, 0 <t < 2t,9 x(O) = 0, where a and a are real constants. The linear part L of this equation is a non-self-adjoint differential operator with p > q (p = 1, q = 0). We show that this equation has a solution provided the constants c, 9 satisfy the conditions Icl<l, IP_<.001, and we obtain estimates on the norm of such a solution.

CHAPTER 2 A GENERAL THEORY FOR THE EQUATION Lx = Nx 2. 1 THE RIGHT INVERSE OPERATOR H In this chapter we study the functional equation Lx = Nx where L and N are operators defined in a real Hilbert space S. The symbols 4(jL) and LI,) will denote the domain and range, respectively, of any operator L defined in So In case L is a linear operator, the symbol?L(L) will denote the null space or kernel of L. Most of the linear operators we shall study will be closed operators. Definition 2o 1 An operator L is closed if the relations xneiL), xn + x, and Lxn + y imply that xcs B)L) and Lx = y. This definition is equivalent to the statement that the graph of L is a closed subset of the Hilbert space SMSo If L is a linear operator in S, then the adjoint operator L* is defined iff i(~L) is dense in S. When L* is defined, it is a closed operator whether L is closed or not. If L is a closed linear operator with A(L) dense in S, then XL*) is also dense in S and L** = Lo In the next chapter we shall show that all ordinary differential operators are closed linear operators with dense domainso Let S be a real Hi lbert space with inner product (xy) and norm Ix11K) and let L be a closed linear operator in S with the following 6

properties: (Ia) The domain 6(L) is dense in S, (Ib) The range'(L) is closed in S, (Ic) The null spaces 7/(L) and 7Z(L*) are finite-dimensional linear subspaces in S. Choose elements 81..- Bp in'QL) to form an orthonormal base for,7/(L), and choose elements l1,..., ~q in,/(L*) to form an orthonormal base for /_(L*). Then p = dim (L), q = dimIZ(L*). (2.1) Let'(L)I denote the orthogonal complement of j(L) in S, i.e., (L)]- = (xeSj(x,~i) = 0 for i = 1,...,p). We introduce a new linear operator L1 by taking the restriction of L to the linear subspace o6L)n7(L)' L L1 = LI &(L)N7L(L). (2.2) Lemma 201 The linear operator L1 is 1-1 and its range is precisely the range of L. Proof. First, suppose xc~(LI) and Llx = 0. Then x~ LL) N L)L and Lx = O, so x~c7(L) nI,(L). Hence, x = 0 and it follows that Lj is 1-1o

Next, we note that AJ(L1>) 7(L). Take ye i(L). Then there exists xe AL) such that Lx = y. Let p z = x - i=l We see that z~XAL) X7L(L)L and L1 z = Lz = Lx = y. Thus, ye f(L1) and ~(L)C Sc (L1). Q.E.D. Definition 2.2 We define a linear operator H in S by H = L1 = [L|i (L)072(L)%] H Ll >0 Theorem 2. 1 The linear operator H has the following properties:. (a) The domain of H is t(L) and the range of H is a/( L) K( L)J. (b) H is a 1-1 continuous linear operator. (c) LH3y = y for all ye7(L). p (d) HLx = x -Z( x,ji)~i for all x~ L). i=l Proof. (a) This follows from the last lemma. (b) Because L1 is 1-1, we get that H is 1-1. Now >(L) and /(L)[ are both closed linear subspaces in S., so we can consider them as Hi.lbert spaces under the induced inner products. We consider H as a linear operator defined on all of the Hilbert space'41(L) and having values in the Hilbert space 7(L). Take xne~(L), xe~(L), and

9 yc7(L)1 with xn + x and Hxn + y. Assert that xe4 9(L) and Hx = y. Let Yn = HXn~ Then ync:(L)n(V(L)~ and Lyn = Llyn = L1Hxn = Xn, so YnaC lL), Yn + Y, and Lyn + x. Since L is a closed operator, we conclude that yc4(~L) and Ly - x. Thus, yc/4(L)q(L)l and Lly = x. This implies that xc/(LL) =YIj(L) and Hx = HLLy = y which establishes the assertion. By the Closed Graph Theorem [9, p. 571 we conclude that H is continuous. (c) This follows from the definition of H. p (d) Take x~O(L) and let y = x - (xi)i. Then i=l ye f L) 0(L )^, so HLLy = y = HLy. Thus, p HLx = HL (y + (x>,i)i) i=l - HLy - y. Q, E. D. In this theorem we have constructed a continuous right inverse operator H for the given operator Lo This operator will be very imr

10 portant in our subsequent work. 2. 2 THE PROJECTION OPERATORS P AND Q Let N be an operator in S with (L)G)'NN)f. In general, the operator N will be nonlinear, but later in this chapter we shall assume certain continuity conditions on N. We want to examine the equation Lx = Nx, (2.3) establishing existence theorems for it. We shall introduce two more operators P and Q which, when utilized with H, will yield an.existence theory for Equation (2.3). Let m be an integer with m > q where q = dim/Z(L*). Choose elements w+,'.., c in,4(L*) such that the elements l,... Wm form an orthonormal set in S. Thus, wic~JL*) for i = 1,..o,m and ({ip %wj) = bij for ij = l1...,m (2. 4) Clearly (?L*X) c < i,.-, h.. > where the bracket symbol is used to denote the linear subspace spanned by cl,,..., no. We need a.lemma. Lemma 2. 2 (L) = ~(L*)' and'(L) -= (L*).

11 Proof. First, take ye k(L). Then there exists xe(L) such that Lx = y. For each zC /(L*) we have (yZ) = (Lx, z) = (x,L*z) = 0, or ye &(L*)v Thus, j(L) C ~(L*)/ Next, we take ye'J(L*)-. Then (y,z) 0 for all ze j(L*). Becausei (L) is closed, we can write Y = Y1 + Y2 where ylc (L) and Y2 is orthogonal tot(L), i.e., (Lx,y2) = 0 = (x,O) for all xc1 L). From the definition of L* we see that y2EC(/L*) and L*y2 = O. Hence, Y2e/%(L*) and (y,Y2) = O. But IIY2112 = (Y2,Y2-y) = -(2,Y) 0 or Y2 = O. Thus, y = Y1Ec(L) and 7L(L*)'Cj(L). This proves that;~( )= i(L*), and from this we get that.(L)i = / (L*)' -(L*) Q.E.D. From this lemma we note that the elements (q+l-,'-vGm belong to ~ (L), and hence, we can form the elements H0q,+l~,*..H(.. Clearly Hcui cS~L)nO(L) for i = q+l,...,m. Let So be the linear subspace in S spanned by the elements Ol,..*Op, which were chosen to form an orthonormal base for;(L), and by the elements Hgq+l,...,pHDm~ So = < 51e,**, 5Zp9,Hwq+l,'.,HJm> (2.5) We claim these elements are linearly independent. Suppose

12 P m ~ bi ~i + ~ Ci HLi = 0. i=l i=q+l Since Hcni <c(L) for i = q+l,...,m, we have (Bj,Hwoi) = 0 for j = l,...,p and i q+l...e m. Thus, bj = 0 for j =1,..,p and H( cict) = 0 i=q+l m But H is 1-1, so jcini = 0 and ci = 0 for i = q+l,...,m. This esi=q+l tablishes the independence. We shall assume throughout the rest of this chapter that SO is a subset of0~'N). Lemma 2.3 The linear subspace So has dimension p+m-q, So is a subset of,/((L)n (N), and the elements l),...,sp*,yo +q1...yHwm form a base for SoThe proof of the lemma is clear. We are now ready to define the operators P and Q. Definition 2.3 The operator P: S - S is defined by m Px = (x,cOi)0Wi for all xcS. i=l The operator Q: S - S is defined by P m Qx = (Xi)i + (x,L*1i)H Hi for all xcS. i=l i=q+l Clearly P and Q are continuous linear operators in S. The next few

13 theorems will establish their properties. Theorem 2.2 The operator P is a continuous linear operator in S with the following properties: (a) PLi = wi for i = l,...,m. (b) The range of P is the linear subspace <l,>>... ca> which is a subset of f(L*); the range of the operator I - P is a subset of (c) P2 p. (d) Px = x for all xc <di,.,.,vcOm>. Proof. (a) This follows from (2.4) and the definition of P. (b) The first part follows from (a). For the second part take xcS and let y = x -: Px. We want to show that ye~(L). For 1 < j < q we have m (ywo) =. )(xoj) (xwi)(ai,Lj) = 0 i=l or YCt *)- P(L). (c) For xcS we ha-ve from (a) that P x = PL (xsWi)Wi \1 =.l m = Px.

14 (d) This istrivial to check using (a). Q.EoD. Theorem 2.3 The operator Q is a continuous linear operator in S with the following properties: (a) Q/i = ~i for i = 1,...,p and QHwi = Hci for i = q+l,...,m. (b) The range of Q is the linear subspace So which is a subset of (L). (c) Q2 =Q. (d) Qx = x for all xeSo. Proof. (a) Since ~i ~&/(L) and wj (e/ L*), we have (i'L*,Lj) = (hi'j) = 0 for j = l,...,m, so p gQi = C (i'Oj)ij + 0 for i = lj) j0=p. Also we have H i L) L) and Hi = i for i=m Also, we have Hcnie~1( /L(L) and LH~ai = Wi for i=g+l,..,m, and hence, (HoiJj) = 0 for j = 1,...,p and (Hic, L*jj) =(LH- ij) = (uisj) for i,j = q+l...,m. Thus, QHwoi = Hri for i = q+l,...,m.

15 (b) and (d) follow from (a). To show (c) take xeS. Using (a) we have =1 i —+Q x = i~ vi Q +(x L cLi)H 1j P m = (x,5i)vi + (xL*wi)Hwi i=l i=q+l = Qx. Q.E.D. Note that P and Q are projection operators. The operator P is an orthogonal projection. In general, the operator Q is not an orthogonal projection. The next theorem relates the four operators L, H, P, and Q. Theorem 2.4 The following properties are valid: (a) H(I - P)LX = (I - Q)x for all xeg(L). (b) LH(I - P)x = (I - P)x for all xeS. (c) Qx = PLx for all xe IL). (d) QH(I - P)x = 0 for all xcS. Proof. The proof of (b) is clear since I - P has its range in ~(L) and H is a right inverse for L. To show (a) and (c), take xc L). Then m (I - P)Lx = x - (lxi)wi i=l

= Lx - (XL*~~i)owi i=l m = Lx - (xL*wi)wi, i=q+l and hence, by Theorem 2.1 (d) we have H(I - P)Lx = HL I (XL*i)Hwi i=q+l p m x Z(Xi)$i' (xL*wi)Hwi i=l i=q+l (I - Q)x. Also, we have p m LQx (=Ix,/ )Li + (x, L*c )u ) i-=l i=q+l m =E (X L*ei)~i i=q+l and m Pbx = Z (Lx',1i)w i=1 m - I (x,L*wi)Wi i=l m =; (x,'iL*Wi)~ai' i=q+l

17 Finally, for each xcS we have QH(I - P)x = (H(I - P)x,/ti)i + (H(I - P)xL*ci) i =.l i =q +1 m = (LH(I -P)xoli)Hi i=q+.l m C= (x-Px,1i )hi i =q + where m (xPx,ci) = (x,ui) ( x,cuj) (joji) j=l 0 for i = q+l,...,m. Thus, QH(I - P)x = 0. Q. ED. The properties listed in the last theorem are analogous to the properties satisfied by Cesari's operators in [3]. We shall use these properties to develop an existence theory for Equation (2.3): Lx = Nx. Suppose there exists an element xe o(L)faoN) with Lx = Nxo Using part (a) of the last theorem, we have H(I - P)Nx = H(I - P)Lx - x - Qxo Thus, there exists an element x*eSo such that

(2.6) x = x* + H(I - P)Nx. Let us try to reverse this argument. Take x*ESo and suppose there exists xC (N) such that x = x* + H(I - P)Nx. (2.7) Clearly xe,.(L). By part (d) of the last theorem we have Qx = Qx* or Qx = x*. (2.8) Thus, x = Qx + H(I - P)Nx and Lx = LQx + LE(T - P)Nx. Using parts (b) and (c) of the last theorem, we get Lx = PLx + Nx - PNx or Lx - Nx = P(Lx-Nx). (2.9) Therefore, x is a solution of Equation (2.3)- provided P(Lx-Nx) = 0. (2.10) We have shown that if xeO6(N) is a solution of Equation (2.7) corresponding to x*cSo and if x is also a solution of Equation (2.10), then x is a solution of the original Equation (2,3). Equation (2,.7) will be called the auxiliary equation.

19 In the next section we shall introduce sufficient conditions for the existence of a unique solution x to the auxiliary Equation (2.7) corresponding to each x* belonging to a subset V of So. In the last section of this chapter we will give sufficient conditions that there exist x*~V, such that the corresponding element x also satisfies Equation (2.10), and hence, yields a solution to the original Equation (2.3). 2.3 SOLVING THE AUXILIARY EQUATION Let S' be a linear subspace in S and let L be a seminorm defined in S'. In most applications t is actually a norm on S'. We assume the following condition is satisfied: (IIa) The linear subspace 4(L) is a subset of S'. In our applications S is the Hilbert space of square-integrable functions f(t) on a finite interval [a,b ], L is a differential operator in S whose domain 4(L) consists of functions which are at least continuous, and S' is the set of functions in S which are bounded almost everywhere; for this case condition (IIa) will certainly be satisfied. Note that SoC_ 4(L)C S'. We assume that the following condition is satisfied: (IIb) There exist constants'k> 0 and#' > 0 such that J1H(I: P)xJJ < Afor all xcS. (2.11)

20 Choose an element xoSo0. Noting that Xoc L)AL N), we let 7 = H(I - P)Nxo. (2.12) Choose constants e and e' such that JJy|j < e, i(-y) < e'. (2.13) Let c, d, r, and Ro be real numbers with 0 < c < d and 0 < r < Ro. We define sets V and So in S by V = (xeSoIJJx-xojI < c, ji(x-Xo) < r} (2.14) and so = XCS xcsllx-xolj < d, pL(x-xo) < Ro}. (2.15) Clearly xoeVC So, so these two sets are nonempty. For each x*eV we define S(X*) = (CxCS' Qx = x*, Ix-xoll < d, Vl(x-xo) < Rol. (2.16) Clearly x*cS(x*), so each of the sets S(x*) is nonempty. Note that x* c S(x*)C'(So for all x*eV. Assume the following condition is satisfied: (IIc) The set So is a subset ofZ (N), and there exists a constant R > 0 such that IINx-Nylj <'l x-ylj for all x,yeS0. (2.17) This is the continuity condition on N which we mentioned earlier. Definition 2.4 The operator T:' N) - S is defined by Tx = Qx + H(I - P)Nx for all xel N).

21 Observe that Tx~~(L) for each xcXiN), and hence, T ( N) + S'. For each x*cV let T(x*) denote the restriction of T to S(x*)' T(x*) = TIS(x*). (2.18) Then for each x*cV we have T(x*) S(x*) + S' with T(x*)-x = Qx + H(I - P)Nx or T(x*)-x = x* + H(I - P)Nx for all xcS(x*). (2.19) Theorem 2.5 If conditions (IIabc) are satisfied and if k <.1 c+e < (1-k )d, r+e' < Ro-.'d, (2.20) then for each x*cV the operator T(x*) is a contraction and maps S(x*) into S(x*). Proof. Fix an element x*cV, Take xcS(x*) and let y = T(x*)ox. We want to show that ycS(x*). Now y = x* + H(I - P)Nx, so ycS' and Qy - Qx* + QH(I - P)Nx = Qx* A..lso, we have II-xoll Ilx* + H(I - P)Nx. xoll

22 ||x*-xo + H(I - P)Nx - H(I - P)Nxo + yll ~ C + IINx-Nxoii +e < c + g|x-xo| + e < c + id + e < d and (Y-xo) = I(x*-xo + H(I - P)Nx - H(I - P)Nxo + y) < r +('llNx-NxolI + e' < r + +'id + e' < R0 Thus, yeS(x*) and T(x*) S(x*) -+ S(x*). For xl, x2 ~ S(x*) we have IIT(x*)x T(x*)(x*) x211 = IIH(I P)Nxj - H(I - P)Nxk11 < IINX1 - NX211 < illx - X211. Hence, T(x*) is a contraction. Q.E.D. We assume that one more condition is satisfied: (IId) For each x*cV the set S(x*) is closed in S. Theorem 2.6 If conditions (IIabcd) are satisfied and if relations (2.20) are valid, then for each x*~V there exists a unique element xES(x*) which is a solution to the auxiliary Equation (2.7) corresponding to x*:

23 x = x* + (I - P)Nx. Furthermore, xC d(L)(4N(N), Qx = x*, and Lx - Nx = P(Lx-Nx). Also, the solutions x vary continuously with the x*: llx-yll < (l-a)- lx* —y*11. (2.21) Proof. The first part of the theorem follows from the:Banach Fixed Point Theorem applied to the contraction T(x*) in the complete metric space S(x*). All the other parts of the theorem have been shown except the continuity (2.21). Take x*eV and y*cV, and let xeS(x*) and yCS(y*) be the elements with x = x* + H(I - P)Nx and y = y* + H(I - P)Ny. Then IIx-yll < IIx*-y*I + ||H(I - P)(Nx-Ny)1| < jx*-y*11l + 11x-Yll or (l-&a)lx-yll 1 ix*Y*i. QoE.D, This last theorem guarantees that the auxiliary Equation (2.7) can be solved for each x*cV. In fact', it permits us to set up a correspondence between each x*CV and the solution x.S(x*) of the auxiliary equatione

24 Definition 2.5 Let conditions (IIabcd) be satisfied and let relations (2.20) be valid. The continuous operator A: V + (L)FSo is defined by 2(x*) = x for x*eV where x is the unique element in S(x*) which is a solution to the auxiliary Equation (2.7) corresponding to x*. Note that for each x*eV we have kxk* ) C /(L) and _2 (x*) e S(x*)C So C,(N). Hence, the expression P(L2 x*-N2x*) defines an operator mapping V into the linear subspace < u1,..,cn 0>. The next theorem is really a corollary of Theorem 2.6. Theorem 2.7 Let conditions (IIabcd) be satisfied and let relations (2.20) be valid. If there exists an element x*eV such that P(L.x*-Nax*) = 0, (2.22) then the element x = Z(x*) is a solution of the original Equation (2.3), Qx = x*, and Ilx-xo[l < d, i(x-xo) < Ro. (2.23) In Theorem 2.7 the problem of solving the original Equation (2.3) has been reduced to the problem of solving Equation (2.22). This is quite a simplification since Equation (2.22) is really a system of m equations in p+m-q unknowns. Equation (2.22) is called the bifurcation equation or the determining equation. We examine it in more de

25 tail in the next section. 2.4 SOLVING THE BIFURCATION EQUATION In this section we introduce sufficient conditions for the existence of a solution x*EV to the bifurcation Equation (2.22). We begin by writing Equation (2.22) in a simpler form and showing that it can be defined by means of a continuous operator. Definition 2.6 The operator 9: (L)nj/(N) -+ <l,...,Wm> is defined by Ajx = P(Lx-Nx) for all x-~ (L) n(N). Note that if conditions (IIac) are satisfied, then the sets V and j (L) So are both subsets of,$'(L)n A(N), and hence, we can form the operators / I V and i 4 <X( L)n~Soe Lemma 2.4 Let conditions (IIac) be satisfied. Then' is a continuous operator when restricted to the set ~Y(L)nSo. Proof. Take x,y~S(L)nSO. Then fix Ad = P(Lx-Ly) - P(Nx-Ny) i=.l i=l so m m IYx Y|II <E I(x.y,L*i) I l (Nx-Ny,c) i=l i-l

26 < I IKL*ai l + ej IiX-yII Q.E.D. i=q+.l Throughout the remainder of this section we assume that conditions (IIabcd) are satisfied and relations (2.20) are valid. Thus, the continuous operators B,'IX(L)ONo, and IV exist with V- 2 L)nSd' <, — -rn and Note that llx*y = P(Li7x* - NLx*) for all x*eV. (2.24) Therefore, the continuous operator tiZmaps the "ball" V, which is a subset of the p+m-q, dimensional Euclidean space So, into the m dimensional Euclidean space <l,...,cu m>. The bifurcation Equation (2.22) can be rewritten as -x* -- = 0. According to Theorem 2.7 if 0 belongs to the range of the operator Y, then there exists a solution x to the original Equation (2.3) with IIx-xo ll d, t(x-xo) < Roo In other words, if we can establish sufficient conditions for 0 to belong to the range of the operator (Y,~ then we will obtain an existence theory for the equation Lx = Nxo

27 The operator j~f is a difficult operator to work with since is defined using the Banach Fixed Point Theorem, i.e., Zis defined by an iteration process. On the other hand, the continuous operator -IjV is an operator which is easily obtained. These two operators can be compared. Theorem 2.8 Let conditions (IIabcd) be satisfied and let relations (2.20) be valid. Then j/dx*-px*J <_< (id+ee)g for all x*EV. (2.25) Proof. Take x*cV and let x = X x*. Then xeS(x*), Qx = x* = Qx*, and PLx = LQx = LQx* PLx*, so /4x.x*.,&x* - P(Lx-Nx) - P(LI*-Nx*) P(Nx*-Nx). By Bessel's Inequality we have || J Lx* j-X*1 < IINx-Nx*JI. Now x*eVC.So and xcSo since g maps V into, L)nSo, and hence, I IJX* A. WX* || < aj||x x* e But x x* = H(I - P)Nx H(I - P)Nx - H(I - P)Nxo + Y

28 so |II x- _ x*I < O -kI|x-xoj + e] Q.E.D. Using this theorem we shall determine conditions on %Iv which will guarantee that 0 belongs to the range of the operator y'Z. Apply the Gram-Schmidt process to the linearly independent elements Hwq+l,.g..,Hn to obtain orthonormal elements -q+l'''"~m' Let m i = aijHj for i = q+l,...,m. (2.26) j=q+.l Then each ri is an element in oO(L) n(L)-, SO = <O ~ np-) Nq+15 n e * n and each xeSo can be written as p m x = bi~i + cirji (2.27) i=.l i-=q+l where bi = (xOi), ci = (xtli), and p m xiiX = Lb.2 +~ c.2 (2.28) i-=l i=q+l Let M = p+m-q; let E be a copy of Euclidean M-space where we represent M each point ecE as an M-tuple: = - (bl,... bpCq+l,..., cm); also, let Em be a copy of Euclidean m-space where we represent each

29 point ucEm as an m-tuple: u = (u,..*u We define two operators r1: EM + So and F2 ~ Wl,..., cm> Em by p m Fl(b,.. bpbpq+l,..,c) = bii +7 ciTi (2.29) i=l i=q+l and F2( ui(,..,um) = (..30) i=l / Clearly CF and r2 are continuous linear operators with Ilri()!ll H1I11 for all 5~EM and IIr2(x)fl = iIxil for all xEc<l,...,c>. Thus these two maps are isomorphisms. Let P m xo = E boii + Coi'i ~ Soy (2.31) i=l i=q+l and then we define M 50 = (bo,...,bopcoq+l,..,com) E E (2.32) Clearly Fl(0o) = xo. Lemma 2.5 Let conditions (IIabcd) be satisfied and let relations (2.20) be valid. Then there exists a number c > 0 such that the set is f EM1II- ito h < ci is mapped by Cl into the set V.

3o0 Proof. Choose a constant M1 > 0 such that t(~i) < M1 for i = 1,...,p and (i(i) < M1 for i -= q+l,...,m. Let = min ( c, - MM1 If = (bl,...,bp,cq+l-..o,Cm) C EM with < e, then we have r'(W) c So with and (rl( )-xo) ) = (j (bi-boi)~i + (ci- coi)ri) i=l i=q+l p m < C lbi-boi M, +E ICi-COIlm i-l i=q+l < r. Thus, rP(k)eV. Q.E.D. Remark. In applications the numbers c and r are often related in such a way that xeSo0, 7Jjx < c implies pi(x) < ro For this case we have v = (xESolxllx K c<,< and in the last lemma we can take E - c. Choose a number e > 0 such that the set is = IeapMi by i tolj e (2e. ) is mapped by TL into the set V. We have

31 c7 rl >-?t VV < >O2h -1f@nC>' E and %2 ~ —--- v _ /(L) nSo <1... o>. r2 > Em Definition 2.7 The operator V: EM + Em is defined by,(S) - r2 (lrj(( ) for all eEM. The operator'IV | is a continuous mapping from the ball 2, which is a subset of the Euclidean space EM, into the Euclidean space Em. If we write out 4 in coordinate form, then we have = (b,...,bpqcq+l,...,cm) p m r1(W) = j bjaj+ cjrj, j=l j=q+l 4rfj() PLr() PNr() = P L PNr1(O) P ~( cj aijci) PNr'(t) j=q+l i=q+l m m m orq (E aij C (~i - (Nrl( );(i)wi: or

32 P m -(N bja j +( cjb j]wi) for i=l,, q *i (bljv bp'cq+lq ",cm =i j=l j=+l qj =q + m p m =q+l j=1 j=q +l for i = q+l...,m. (2.34) The next theorem is an existence theorem for Equation (2.3). Theorem 2.9 Let m = 1, let conditions (IIabcd) be satisfied, and let relations (2.20) be valid. If there exists a number 5 > 0 such that the interval [-6,6] is a subset of *(If) and if (aed+e)e < 6, (2.35) then there exists x*eV such that the element x = 7(x*) is a solution of the original Equation (2.3), Qx = x*, and IIx-xoll < d, i(x-xo) _< Ro. Proof. Choose 1,S2e7fssuch that 4f(S1) = 5 and r(g2) = -6. Let X1* = rjl(l), rxl(). Clearly x9* and x2* are elements of V, and by Theorem 2.8 IIY4 SLXI* - %Xxj*lj <6 and il~/xxZ* - /x*il I< 6. Thus,

33 r2 rl( ) r2$rl(5 1) - r2 r rl(' ()l = IY" Ixi*.-'xi*ll < or r2 %, rj(jJ) > o. Similarly, we get r2grl(62)'< 0 Since r2%.IFj rj(l 2 is connected, there exists ~S l such that rL' rl((~) = 0. If we set x* = r'(O), then x*eV and P(L X x* - N Z x*) = o. Applying Theorem 2.7, we obtain the desired result. Q.E.D. To conclude this chapter we give another existence theorem which relaxes the condition m = 1. Let conditions (IIabcd) be satisfied and let relations (2.20) be valid. In addition we assume that the following conditions are satisfied: (IIIa) p > q, (IIIb) v(to) = 0, (IIIc) The first order partial derivatives of 4 exist and are continuous on A, (IIId) The Jacobian matrix for V has rank m at 5oo The first condition is equivalent to the condition M > m, which means that' maps from a high dimensional space into a lower dimensional

34 space. The second condition says that P(Lxo-Nxo) = 0, which means that xo can be thoughtof as an approximate solution to the equation Lx = Nx. In applications this suggests how one should choose xo, The third condition corresponds to putting certain differentiabi.lity conditions on the operator N. The last condition guarantees that the range of V covers a neighborhood of the origin in Etm Lemma 2.6 Let conditions (IIabcd) be satisfied, let conditions (IIIabcd) be satisfied, and let relations (2.20) be valid. Then there exists a number 5 > 0 such that the set W = (ueEmiIu|uh < ) is a subset of xv(2o. Furthermore, there exists a continuous mapping A: W + such that V[A(u)] = u for all ueW. Proof. This follows immediately from the Inverse Function Theorem of advanced calculus, the proof of which suggests how one can determine the number 6. Q.E.D. Theorem 2.10 Let conditions (IIabcd) be satisfied, let conditions (IIIabcd) be satisfied, and let relations (2.20) be valid. If 6 > O is a number chosen as in Lenma 2.6 and if

35 (&ed+e)e < 5, then there exists x*CV such that the element x = x* is a solution of the original Equation (2.3), Qx = x*, and I|x-xoll < d, C((x-xo) < Ro. Proof. Consider the two continuous maps I: W + E and rFr-l'A: W + Em where I(u)- u. Take uEW and let x* = FlA(u). Then x*eV and 1r12,.Z P A(u) - I(u) I = IlrSL (x*) - r2yrl A(u) -Ii~Z(x*) W-)(x*)ll < (i~d+e)e, or llrF2%lFA(u) - I(u)II < 6 for all uEW. This inequality implies that for each u6W,, the line segment joining I(u) and r'2%t. rlA(u) does not contain the origin of Em. By the Poincar4'-Bohl Theorem [7,p.32] we conclude that the local degree of 2(//lF1A at O relative to W is equal to the local degree of I at O relative to W: d(rGl4 fiA,WO) = d(I,w,O). But d(I,WO) = 1, and hence, d(rffr~Aw,o ) X o. Therefore, there exists an element ucW such that rS2 x rflA(u) = 0e Setting x* lA(u) we have x*CV and

36 P(Lx* d - N x*) = 0. The proof is completed using Theorem 2.7. Q.E.D.

CHAPTER 3 DIFFERENTIAL OPERATORS 3.1 THE DEFINITION OF THE DIFFERENTIAL OPERATOR L In the last chapter we developed a general theory for solving the functional equation Lx = Nx in a real Hi.lbert space S. This theory was designed for applications to nonlinear differential equations. We must define the notion of a differential operator in a Hilbert space and show that these operators have the properties of the operator L in Chapter 2. This is the purpose of Chapter 3. Our main reference for this work is Dunford and Schwartz [9, ch.13]. Throughout this chapter we let I denote the finite interval [a,b] of the real axis. Consider the real Hilbert space S = L2(I) consisting of all real-valued functions f(t) which are square-integrable on I. We take the usual inner product and norm in S: b (f,g) = f(t)g(t)dt a and 1/2 Ilfll = (frf) / Definition 3.1 A formal differential operator of order n on the interval I is an expression n T = ai(t) () i=0 where the real-valued functions ai(t) belong to C (I), and the function 37

38 an(t) is not zero at any point of I. The Hilbert space S = L2(I) contains many linear subspaces which could serve as a domain for a formal differential operator T. Corresponding to each such subspace is a different linear operator in S. The next two definitions give two of these linear subspaces which will be very important in our work. Definition 3.2 By Hn(I) we denote the linear subspace of S consisting of all functions f(t) with the following properties: f is continuous on I, the derivatives f',f",...,f(n ) exist and are continuous on i, and f(n-) is absolutely continuous on I with f(n)eS. Definition 3.3 Let H (I) denote the set of all functions in Hn(I) each of which vanishes outside some compact subset of the interior of I. (The compact subset may vary with the function). Before examining the linear operators which we get by applying T to the subspaces H (I) and Ho(I), let us give a well known existence and uniqueness theorem. Theorem 3.1 Let T be a formal differential operator of order n on the interval I, let g be a function in S., let toeI, and let co0,c,...cn-l be an arbitrary set of n real numbers. Then there exists a unique function fHn (I) such that (a) Tf = go (b) (-) f(to) =c for i =l 0,,nc.l

39 Proof. The classical proofs can be altered to prove this theorem. See [9,p.1281]1 Corollary 1 If g has k continuous derivatives in I, then f has n+k continuous derivativesin I and 7f(t) - g(t) for all teI. Corollary 2 The set of all functions fcH n(I) with Tf = 0 forms an ndimensional linear subspace of S. Now we define the linear operators mentioned above. Definition 3.4 Let T be a formal differential operator of order n on the interval I. The operators To(T) and Tl(T) are defined in S by (a) >(To(T)) = Ho(I), To(T)f = Tf for all fce To(T)), (b) Q'(T1(T)) = Hn(I), T1(T)f = Tf for all fe,~,Tl(T))The linear operators To0() and T1(T) both have domains which are dense in S, and hence, both operators have adjoints. Clearly To ( ) C T1(T). (35.1) and by the last corollary the null space for Tj(T) has dimension n. We assert that T].(T) is a closed operator. To show this we muost introduce the notion of the formal adjoint. Definition 3.5 Let T be a formal differential operator of order n on the interval I. The formal differential operator n T* X Ebj(t) (O ) j=o

40 where n bj(t) = X (-l) (T) kt j ak(t), k=j is called the formal adjoint of T. Each of the functions bj(t) belongs to Cd(I) and bn(t) = (-1) an(t), so T* is indeed a formal differential operator. It can be shown that T = (T*)*. Theorem 3.2 If T is a formal differential operator of order n on the interval I, then TI(T) = To(T* )*. Proof. See L9,p.1294]. The differential operators which we are going to study are obtained from the operator T1(T) by imposing a set of boundary conditions. The definition of a boundary value will be given in a modern abstract form, which permits us to study differential operators using the methods of functional analysis. Definition 3.6 Let T be a formal differential operator of order n on the interval I. Since Ti(T) is a closed operator by the last theorem, it is easily seen that its domainli(T,(T)) becomes a Hilbert space under the inner product: [f,g] = (f,g) + (T1(T)f, T1(m)g). (3.2) A boundary value ~or T is a continuous linear functional B on the Hi.lbert space ('Ti(T)) which vanishes on o&T0(T)). In addition, if B(f) = O

41 for each function f in1i(Tl(T)) which vanishes in a neighborhood of a, then B is called a boundary value at a. A boundary value at b is defined similarly. An equation B(f) = O, where B is a boundary value for T, is called a boundary condition for T. This concept of a boundary value can be related to the classical notion of a boundary value. Theorem 3.3 Let T be a formal differential operator of order n on the interval I. (a) The space of boundary values for T is a linear space of dimension 2n. (b) The functionals A(i)(f) - f(i)(a) i = 0,1,...,n-1., (353) and (i) (i) B (f) = (b), i = Oi,..,n, (354) are boundary values for T at a and b, respectively. They form a base for the space of boundary values for T. (c) If B is a boundary value for Tr then there exist real numbers o;lC *.. — 1C- n_ P Do1e, -en-1 such that n-1 n-1 B(f) aiA(i)() + iB(i)(f) (55) i=O i=-O for all fc(Tl1(T)). Conversely, each expression (3 5) defines a boundary value for 0. Proof. See [9, p.1298 and p.1301].

42 With these preliminaries out of the way, we can now introduce the notion of a differential operator. Let T be a formal differential- operator.of order n on the interval I and let Bi(f) = O, i = l,...,k be a set of k linearly independent boundary conditions for T (the set may be vacuous). Definition 3.7 The linear operator L, which is the restriction of Tl(T) to the linear subspace of 4(T1(T)) determined by the boundary conditions Bi(f) = O, i = 1,...,k, is called a differential operator of order n on the interval I: L = Tj(T)I I AL) (3.6) where C(aL) = (fEcI'Tl(T)) IBi(f) = 0 for i = l k...,k (3-7) n From this definition we observe thatO'(To(T)) = Ho(I) is a subset of4XiL), and hence,o(L) is a dense linear subspace in S. Thus, the differential operator L has an adjoint. In the study of differential equations the adjoint operator always plays an important role. We shall show how to determine the adjoint operator L* corresponding to a differential operator L in the Hilbert space S = L2(I). 3.2 THE ADJOINT OPERATOR L* Let r be a formal differential operator of order n on the interval I and let L be the differential operator obtained from v by the imposition of a set of k linearly independent boundary conditions Bi(f) = O0

i = l,....k. We shall determine the adjoint operator L*, showing that L* is a differential operator obtained from T* by the imposition of a set of (2n-k) linearly independent boundary conditions. The main reference for this section is the paper of Schwartz L211. For each pair of integers I,j with 0 < Q,j < n-1 we define functions Ft (T) on I by n-l i t T =\ X 1 dt ae+i+z (t), j+i < n-l Rj ~=j t () (3.8) 0, j+Q > n-l. These functions appear in the fundamental formula relating T and T*, the well known Green's formula. Lemma 3.1 (Green's formula) If f(t) and g(t) are functions in H (I), then n-l (Tf,g)-(f,T*g) = j Fbj()f(T)(b)g(J)(b) Q,j=o n-, (3.9) F- (T)f() ()(a)). i,j=o0 Proof. See L9,pp.1285-1288]. Let F1 and F2 be the nxn square matrices given by F1 = [F j(T)1 and F2 [F- j(T)]. (5.10) a b These two matrices are known to be non-singular [9,p.1287]. Using Theorem 3.3 we write out the boundary values B1,...,Bk which determine

44 L: n-1l n-l Bi AU ~ jPiBjA ( j) i = +l ijBk) (3.11) j=o j=o Let B be the kx2n matrix given by C1lo'''cl n-1 lioo — l n-1 B =... (3.12) ckoo,..0 n-1 Pko'''k n-1 Because the boundary values B1,...,Bk were assumed to be linearly independent, the matrix B has rank k. Consider the system of equations ni-] n-1 ijj + ij = 0 i = l,...,k. (3513) j=o j=o This is a system of k-equations in the 2n-unknowns xOn,..Sxr..SYO.a** Yn-lo Since B has rank k, the space of solutions to (3.13) has dimension 2n-ko Let Xi, i = 1,...,2n-k be a set of 2n-tuples of real num.bers which form a base for the set of solutions of (3.13). We write i i i Xi = (Xio =,x~n- Yo,',n-L) i =,...,2n-k. (3.14) Finally, let X be the (2n-k)x2n matrix whose rows are the Xi. Since the Xi are independent, the matrix X has rank 2n-k. We define a (2n-k) x2n matrix 2nako *2nknl1 %nko o-knl 1

45 by n-l.i.j -= A FaTj ~ a' =0 O for i = 1,...,2n-k j,...,n-l, n-l Pij X Ya FbI(T) (3.16) 9=0 and let B*, i = 1,...,2n-k be the boundary values for T* defined by i n-1 n-1 B* = * A(j) + * B(i) i = 1,...,2n-k. i ij ij j=o j=O (3.17) Since -F1 0 B* = X (3.18) L0 F2_ we conclude that B* has rank 2n-k, and hence, the boundary values B*,..., B* are linearly independent. 2n-k Theorem 3.4 The adjoint operator L* is precisely the differential operator obtained from the formal adjoint T* by imposing the 2n-k linearly independent boundary conditions B*(f) = 0O i = 1,...,2n-k. 1 Proof. Let L be the differential operator obtained from T* by the imposition of the set of (2n-k) linearly independent boundary conditions B*.(f) = 0, i = 1,...,2n-k. We shall show that L = L*. (a) Take ge (L) and let g* = Lg = T*g. Then for each fc J~L) the 2n-tuple

46 X = (f(a),f (a),..,f(n-) (a),f(b),f'(b),...f(n- (b)) is a solution of the system (3.13), and hence, there exist real numnbers c1...,c 2n-k such that 2n-k X - ciXig i=l that is, 2n-k }f )(a) = c;X i=l 2n-k f(l)(b) = ZciY'.iy Thus, using Green's formula we have (Lf,g)-(f,g*) = (Tf,g) (f,T*g) n-1 2n-k Yj ~(j) - j X Fj b (+')ciyg (b), j=O i=l n-1 2n-k C- Fa J(T) cixig (j)(a) 2,j=O i=l 2n-k n- =j ci j ~* B( )(g) i-=l j=o 2n-k n-i + C A (g) i=l j=o ciBi(g) i=.L

47 0 or (Lfg) = (fg*) for all fE L). This implies that g (L*) and L*g = g* = Lg. Therefore, LCL*. (b) To complete the proof it is sufficient to show that /i(L*) C/(L) Take any function g~/(L*). Now TO(T) C L, so by Theorem 3.2 we have L*C To(T)* = T1(T*), and hence, ge(Til(T*)) = Hn(I). Choose functions ai(t), i = 1,..., 2n-k in C (I) such that (a) = x' i (b) = Y for 0 = o,1,...,n-1 Clearly ai(t)cEAL) for i = 1,..o,2n-k. Again using Green's formula we get o = (Loig)_(oiL*g) ( (ig)-( i CT'g) n-1 n-i Fi(T)y g()(b) F@ (T)Xlg (a) Qg jO =, j=0 n- 1 n-1 = B Pij B (gg) + ij A (g) = B*(g) for i = l0...,2n-k. i

48 Hence, ge~ L). Q. E. D. Let us consider several examples. Example 1 Let I = [0,1], let Tf = f" + f, and let L be the differential operator obtained from T by imposing the two boundary conditions Bl(f) = f(O) = 0 I B2(f) = f(1) + f'(.1) Then n = 2, k = 2, a2(t) -1, al(t) - 0, and ao(t) - 1. By Definition 3.5 the formal adjoint T* is given by r*f = b2(t)f" + bl(t)f' + b0(t)f where b2(t) = (l)a ( )a2(t) = 1, bl(t) = (1) ()al(t) + () 2 ()a(t) = 0 bo(t) = (-I1) (~)ao(t) = 1, or T*f = f" + f. Thus, T is formally self-adjoint. From (3.8) we have Ft (T) = (-1) (o)al(t) + (-1) (o)a2(t) = 0, F1o(T) = (-1l)1()a2(t) = -1, Ft (T) = (-1) (O)a2(t) = 1, Ft (T) = 0, SO

49 F1 = F2 = L1- i Now 1 0 0 0 B = O 0 1.1 and from this we see that we can take 0 -1 0 0 X = x O.0 -.1 Therefore, by (3.18) we have B* 0 1 0 0.1 0 [ 0 00 0 0 o -1 1 0 0 0 0 1 1g O 0 0 -.1 0 0 1 o or B*(f) = f(o) = 0 B*(f) = f(l) + f'(1) = 0. 2 Hence, L and L* are determined from T = T* by the same set of'boundary conditions, that is, L.'= L*. This self-adjoint differential operator was studied by Cesari in [31]. Example 2 Let I = [0o,2c], let Tf = f" + f, and let L be the differentia.l operator obtained from T by imposing the single boundary condition B1(f) - f(o) = 0. In this examp.le n = 2 and k = 1. As in. the last example the formal adP joint is given by T*f = f" + f g

and also F1 = F2 = 1 0 Now B = 0 0 J so we can take 0-.1 0 0 X = 0 -.1 0 O O O 1. Thus, 0 -.1 0 0 1 0 0 1 0 0 0 B* =0 0 -.10 -.1 0 0 0 0 0 1 L0o 01 0 0 0-1 0 0 L O 0 0 1 Q or B*(f) = f(o) = O 1' B*(f) = f'(2g) = O LBW(f) = f(27T) = 0. The adjoint operator L* is obtained from T = T* by imposing the three boundary conditions B*(f) = 0, i = 1,2,3. We use the differential 1 operator L of this example in Chapter 6. The next theorem gives a characterization of the self-adjoint differential operators.

51 Theorem 3.5 Let T be a formal differential operator of order n on the interval I and let L be the differential operator obtained from T by the imposition of a set of k linearly independent boundary conditions Bi(f) = O, i = l,.,k. The operator L is self-adjoint, L = L*, iff the following conditions are satisfied: (a) k = n; (b) = r, i.e., T is formally self-adjoint; (c) The matrices B and B* are row equivalent. Proof. Suppose L is self-adjoint. Consider the two systems of equations n-1 n-1 (*) > ij X+ ~ ijY = 0, i = l,...,k j=O j=O and ni1il n-1 (**) ZC Xj +C i yj = O, i = In....2n-k, =0 =0 j-o j=o the coefficients are just the elements in the matrices B and B*. (a) Let W1 denote the solution space of (*) and let W2 denote the solution space of (**). We know that dim W1 = 2n-k and dim W2 = 2n - (2n-k) - k. Assert W1 = W2. Take a 2n-tuple (xo,.oo. nxnlyoo...,ynml)cWJl Then we can choose a function f(t) in Ca(I) such that f(J)(a) = xj f(J)(b) = yj for j = O,l,..o,n-l. Thus,

n-1 n-1 Bi(f) = f()(a) + ijf (b) j=o j=o = 0 for i = l1,...k, or fe~(L) = en*)e, n-L n-1 B*(f) = O= C Xj y i =,...,2n-k, j=0 j=0 or (xO,..., Xn-lyo... Yn-_l)EW2 and WlC W2. Similarly we can show that W2Q W1. We conclude that W1 = W2 and 2n-k = k or k = n. (b) We have n-1 T ai(t) (t) i-O and n-1 = bIi(t) i=O We must show that ai(t) - bi(t) for i = O,l,...,n-l. Fix the integer i with 0 < i < n and take any to with a < to < b. Choose a function >(t) in C (I) which is identically 1 in a neighborhood of to and is identically zero in neighborhoods of a and b. Consider the function ft! (t) (t -to) on I. Clearly fc4L) =AiL*), so Lf = L*f or Tf = T*f. But in a neighborhood of to we have f(t) = 1 (t-to)i and hence,

53 Tf(to) = ai(to) = T*f(to) = bi(to). Thus, ai(t) = bi(t) for a < t < b, which shows that ai(t) - bi(t) on I. Since i was arbitrary, we have T = T*. (c) We showed that the matrices B and B* were row equivalent in part (a) of the proof. The reverse implication is trivial to prove. Q.E.D. 3.3 AN INTEGRAL REPRESENTATION FOR THE OPERATOR H To conclude this chapter we show that each differential operator L in the real Hilbert space S - L2(I) is a closed linear operator and satisfies conditions (Iabc) of Chapter 2. This establishes the existence of the right inverse operator H, which can be represented as an integral operator in S = L2(I). n Let T= ai(t) ) be a formal differential operator of order i=O n on the interval I and let L be the differential operator in S = L2(I) obtained from T by the imposition of a set of k linearly independent boundary conditions Bi(f) = 0O i = l,...,k. We assert that L is a closed operator. Theorem 3.6 The differential operator L is a closed linear operator in S.

54 Proof. Take a sequence of functions fie4L) with fi jf and Lfi -g. We must show that feC4'L) and Lf = g. Since LCTl(T), we have fieC5(Tl(T)) and Lfi = T1(T)fi. But Tj(T) is a closed operator by Theorem 3.2, so fc4OT1(T)) and Tl(T)f = g. Now in the Hilbert space 4/T1(T)) we have [fi-f,fi-f] = Iifi-fll2 + IIT1(T)fi-T1(T)fl12 _ fi-fi + IlLfi-ggl, and hence, the functions fi converge to the function f in the Hilbert space 4GT1(T)). Since Bj(fi) = O by the continuity of Bj on.4/{Tj(Y)) -we get Bj(f) = limBj(fi) = 0 for j = 1,...,k. Thus, fe4pL) and Lf = Tl(T)f = g. Q.E.D. We observed earlier that the domain AL) is dense in S. Since TO(T) CL CT1(T) (3519) and To(T*) C L* C T1(*), (3.20) from Corollary 2 of Theorem 3.1 it follows that the null spaces 9(L) and k(L*) are finite-dimensional linear subspaces in S of dimension < n. We assert that the range of L is closed in S. Choose functions l,.,-n in nTi(T)) to form an orthonormal base for the null space 2(T1(T)). With no loss of generality we can

55 assume that the Xi are chosen so that the functions,-...,~p are in' L) and form a base for the null space7l(L). Clearly each Xi belongs to Cw(I). Consider the nxn matrix X2('(tt) n (t) =.. (3.21) in-) (t) (n-) (t It is well known that this matrix is non-singular for each teI [6,ch.3]. If we compute - 1 by forming the adjoint matrix of h,.then we find that the element in the j-th row, n-th column of -1 is just wj(t) det( (t) where W.(t) is the determinant of the matrix obtained from 0(t) by replacing the j-th column by (0,...,0,1). Thus, n 7 W (i)(t Wj(t) _ ~for i = 0,1...,n-2 L'~j~~~ det(t)~(t)(3.22) det~(t) for i = n-1. j=l for i = n-l. We define a function G(t,s) on the square IxI by j(t) Wj(s) G(t,s) L an(s)detO(s), a t, s b. (3.23) j=l Clearly G(t,s) is continuous on IxI. This function is frequently used in the study of differential equations, e.g., see [6,pp.87-881. We examine its properties in the next two lemmas.

Lemma 3.2 If u(t) is an absolutely continuous function on I and if u'(t) = g(t) a.e. where g(t) is a continuous function on I, then ucC'(I) and u'(t) = g(t) for all teI. Proof. We have t t u(t) = u(a) + u'(s)ds = u(a) + g(s)ds. a a The fundamental theorem of calculus tells us that u(t) is differentiable on I and u'(t) = g(t) for all teI. Q.E.D. Lemma 3.3 Let f(t) be any function in S = L2(I) and let u(t) = G(t,s)f(s)ds for teI. Then the function u(t) belongs to4(Tl(T)) and T(T)u = f. Proof. By definition we have u(t) = j(t) j Wj(s) i f(s)ds j=l a for a < t < b. This representation shows that u(t) is continuous on I and, in fact, is absolutely continuous on I. Using (3.22) we get n t n ul(t) = (t) Wjf ds + ) $(t) andetO an(t) j detO(t) j=l a j=1l = B j(t) L a(s)det (s) f(s)ds a e,, j=i a

57 By Lemma 3.2 we conclude that u'(t) exists everywhere on I and is a continuous function on I. By repeated use of (3.22) and Lemma 3.2 we conclude that the derivatives u',u',...u(n 1) exist and are continuous on I with n (i) \7(i) Wa(s) u (t) = > j (t) f(s)ds, a < t < b, a an(s)det()(s) (n-j) for i = 0.1,...,n-l. Thus, u is absolutely continuous on I, and again by (3.22) we have n t n t)j (t ) a (t - Jds + _,In (t( ) (t) andet~ an(t) J det~(t) j=l a j=l n 7-(n) j(s) f(t) La n (s) S )det(s) f(s)ds + a(t ae. j=l ( Hence, u cS and we have shown that u(t) belongs to /T.T1(Tr)). Finally we note that n-1 T1(T)u = a (t )u()(t) (t) i=O n n t f(t) C+ ai(t) (i(t) (s)'et J (s) f(s)ds i=O j=l a n t f(t) + Tl(T)O j(t). F Ws) (s 1 f(s)ds j=l Lan s)det (sJ f f QoE.D.

58 Theorem 3-7 The range of the differential operator L is closed in S. Proof. Let fi(t) be a sequence of functions in 49(L), let f(t)eS, and let fi + f. We must show that f (( L). Let t ui(t) = G(t,s)fi(s)ds for teI a and t u(t) = G(t,s)f(s)ds for t-eI. a By Lemma 3.3 the functions ui(t), i = 1,2,... and the function u(t) belong to4T1(T)) and T1(T)ui = fi, T1(T)u = f. Assert ui + u. Choose Mi > O0 such that IG(t,s)I < M1 on th square IxI. Then t ui(t)-u(t)l < JG(t,s) |fi(s)-f(s) ds a / < M1 ls Ifi(s)-f(s)jds a <M1 (b-a)1/2 Ilfi-fll, and hence, Ilui-ull2 M1(ba )211fi-fr|. Thus, uiea/T1T(T)), ucTl(T)), ui + u, and Ti(T)ui + T1(T)u. This means that the ui converge to u in the Hilbert space4(Tl(T)). By the continuity of Bj on nT1(YT)) we have lim Bj(ui) = Bj(u) for j =,k Now choose functions vi(t) inIL) such that Lvi = fi. Then

59 vi-UicT(Tl(T)) and T1(T)(vi-ui) = O. Hence, there exist real num(i) (i) bers cl,..,c such that n Vi(t) = j (i) (t) + ui(t), i = 12o... el=i But Bj(vi) = 0 for j = 1,...,k, so n c Bj() = - Bj(ui), j = l,...,k Q=.1 or n (*) lim c(i) B() = Bj(u), j ek =1o If we let T be the linear transformation from Euclidean n-space into Euclidean k-space induced by the matrix with coefficients Bj(~i), then (*) says that the k-tuple (-Bl(u),...,-Bk(u)) is a limit point of the range of T. But the range of T is certainly a closed linear subspace in Euclidean k-space, and hence, this k-tuple belongs to the range of T. Thus, there exist real numbers c1,..*~cn such that n (**) jc~Bj(() = _ Bj(u), j = 1,...,k. Q=1 Let n v(t) = c>e(t) + u(t). Q=.1 From (**) it follows that vefAL), and also, Lv = T1()V = T(T)u = f. Q.E.D,

60 We have shown that the differential operator L is a closed linear operator in the Hilbert space S = L2(I) and satisfies conditions (Iabc) of Chapter 2. Therefore, the linear operator H = [LI|(L)(nY(L)J-] exists and is a continuous right inverse for L. The basic properties of H are given in Theorem 2.1. We show next that H can be represented as an integral operator. Theorem 3-8 Let B be the kx(n-p) matrix Bi(~p+J) Bs(OP+ )... B(n)| P+2 = B2(p+) B2(P+)... I(Bp+J)...... B ~(n) > and let T be the linear transformation from Euclidean (n-p) - space into Euclidean k-space induced by T: n n T(c.p+,Y — "n) EBi(/i)ci)..., Bk(n)c). -fr i:=p+l Then T is 1-1. Proof. Suppose T(cp+l,...,cn) = 0. Let n f(t) = jci.i(t) for t~I. i=p+l

61 Since the Xi are orthonormal, we must have fe (L)-. Also, we have fcX(iT1(T)), T1(T)f= 0. and n Bj(f) = iB ) = for j = 1,...,k. i=p+j Thus, fe~(L) and Lf = O, or fe2(L)Ok(L). This implies that f = C and ci = O for i = p+l,...,n. Q.E.D. This theorem has three corollaries which are very useful in establishing the integral representation for H. Corollary 1 n-p < k where n is the order of the formal differential operator T, p is the dimension of the null space 2(L), and k is the number of linearly independent boundary conditions defining the differential operator L. Corollary 2 The matrix B has rank n-p. Corollary 3 There exists a (n-p)xk matrix.A A' AP+I k P+1 I P+2 AP+ Ap+. A +2 1 AP+2 2 An1... Ank such that AB = Inp' i.e.,

62 k Z Ai'B2(j) = 6ij for i,j = p+l,...,n. (3.24) R=1 Proof. Since B has rank (n-p), there exists a finite sequence of kxk elementary matrices E,...,Ep which transform B to row-echelon form: li 1 l 1 Ep..El B = n-p Let A be the (n-p)xk matrix obtained from E..p'.E1 by retaining only the first (n-p) rows. Q.E.D. Remark Given the matrix B we can easily compute the matrix A by using elementary row operations as in the proof of Corollary 3. We define functions?i(s), i = 1,...,n on the interval I by b -jW i(t)G(ts)dt i =.1,..,p ~~S~~~~~ (s) Wi~As)b= k. n i - Ean(s)deto(s) =1 j=O p=l The numbers Pi: are just the coefficients of the boundary values B1,..., Bk as given in (3.11). Clearly each function (i(s) is continuous on I. Theorem 5.9 The right inverse operator H for the differential operator L has an integral representation given by

63 n b t Hf(t) I j(t) a/j(s)f(s)ds +/G(t,s)f(s)ds, tcI, aj a (3.26) for all functions f(t) in the range of L. Proof. Take any function f(t) in the range of L and let g = Hf. Let t u(t) = G(t,s)f(s)ds for teI. a Both g and u belong to 4Tj1(T)) and Tl(T)(g-u) = 0. Thus, there exist real numbers c1,... ecn such that n (*) g(t) = >cjcj(t) + u(t). j=l Now ge (L), and hence, (g,Bi) = 0 for i = l,...,p or b ci = - u(t)i (t)dt a b t - (ts)f(s)d i(t)dt for i = 1,... a a This integral can be rewritten using Pubini's Theorem as b b ci J= - i(s) G(t s)dt1 f(s)ds a s or b Ci,= Y~(s)f(s)ds for i = 1...po a Also, we have gc4L), so Bl(g) = 0 for 2 = l,...,k or

64 n )j cyBej) ) - = B(u) for ~ =1,..,k. j=p+l Using (3.24) we get k n k j 3 cjAieB(j) =- AiYBe(u) for i = p+l,...n Q=1 j=p+l e=1 or k Ci =- AijB(u) for i = pl,...,n, Q=1 We shall write out the expression Bj(u) in detail. In the proof of Lemma 3.3 we showed that n t U j)(t) - tj) LanIsdet~s) f(s)ds, j = 0,... n-1. p an (s)det (s)) p=l a and hence, n-1 n-1 e (u) 7U"jU(j)(a) + Be jU(j)(b) j=0 j=o n-1 n a j=0 p=l for i = 1,...,k. Thus, k n-1 n b c = I 3 Aijgpj (b) b an(s)det f(s )ds a=1 j=O p=l a a or ci = f (i(s)f(s)ds for i = p+l,..o,n.

65 Substituting these expressions for the ci into (*), we get the desired result Q.E.D. We define a function K(ts) on the square IxI by n K(t,s) = > j(t) (j(s) + G*(t,s), a < t,s < b, j=l (3.27) where G(t,s) for a < s < t < b G*(t, s) = (3.28) 0 for a < t < s < b. Clearly the function K(t,s) is square-integrable on the square IxI. Corollary The operator H is a completely continuous operator from j(L) into?LL), and has an integral representation given by b Hf(t) = K(t,s)f(s)ds, t-eI, (3~29) a for all functions f(t) in the range of L. Proof~ The integral representation follows from the theorem. For the complete continuity we must show that if (fil is a bounded sequence in i)(L), then the sequence [Hfi] contains a subsequence converging to some limit in L(L). Let (fi] be a bounded sequence in (L). The linear operator H1: S + S defined by b Hif(t) = JK(t,s)f(s)ds for all f~S a

is known to be completely continuous in S. Hence, there exists a subsequence (Hlfij] which converges to a function gcS. But H = H11 i(L), so Hfi. = Hlfi EC(L)' J a and Hfi A g as j - oo. SinceZ(L) is closed in S, we conclude that ge/'t(L)j. This completes the proof. Q.E.D. By means of this theorem and its corollary we can determine two important bounds on the operator H. As in Chapter 2 we let q = dim f(L*) and then choose an integer m > q and functions a1(t),..., om(t) in (L*) such that the xi form an orthonormal set in S and Wl(t),...,cgq(t) form a base for the null space,(L*). Let P be the projection operator given in Definition 2.3, i.e., m Pf = (f',i)wi for all feS. Fix tEI and consider K(t,s) as a function of s on the interval I. Clearly this function is square-integrable on I, and b n b t (ti()d = j ( t)i()d +G(t i()d a j=l a a for i - 1,..o,m while

67 b n b n t [K(t, 2)]1d~ >i (t) j(t) y)i(f)()di + 2 kj(t) G(tq,)~j(i)di a a a i,j=l a j=l a t + [G(t,S)] d. a Thus, these two integrals give continuous functions of t on the interval I. We define a function Km(t,s) on the square IxI by m b Km(t,s) = K(t,s) K(tt)0()i(s), a<t,s<b, (3,30) Clearly this function is square-integrable on the square IxI. If we fix teI and consider Km(t,s) as a function of s on the interval I, then this function is square-integrable on I with b b b m /b a [Km(tj)]d~ =a [K(t,~)]2d~ (- b) a a (3o31) Thus, this last integral defines a continuous function of t on the interval I. Choose constants k/and j' such that bb [Knm(ts)]2dsdt </ 3 a a and a [Km(t,)]2 d < d (333) CI a Theorem 3.10 The linear operator H(I - P) has an integral representation given by

68 b H(I - P)f(t) = Km(t,s)f(s)ds, teI, (3.34) a for all functions f(t) in S. Also, for each f in S: I|H(I - P)f 11 < llf 11 (335) and |H(I - P)f(t)l <'&Ifll| for all teI. (3.36) Proof. Take any function f(t) in S. Then b m b H(I - P)f(t) = K(t,s)f(s) -i f( w ds a i=l a b m b b = K(t,s)f(s)ds - ( (t, sb)i (s)d f ( )Wi() = K(t,s)f(s)ds - j K(t, j)0i( )d wi(s) f(s)ds -J bK(ts) s(td,i('I w i a b - Km(t,s)f(s)ds, tCI, a and by the Schwartz inequality we have b b |H(I - P)f(t)| < [Km(t,s)] 2d 112 [f(s)] d 1 a a <' 11|fI for all teI. Using this same argument we get

69 b H(I - P)fI2 = IH(I P)f(t)12 dt a b b < Ifll'2 i [Km(t 2s)] ds dt a a < 2 illlf QoE.D.

CHAPTER 4 THE NONLINEAR DIFFERENTIAL EQUATION Lx = Nx 4.1 THE EXISTENCE THEOREMS Let I be the finite interval [ab] of the real axis and let S L2(I) be the real Hilbert space consisting of all real-valued functions f(t) which are square-integrable on I with the usual inner product (f,g) and norm I1f~1. Let S' be the linear subspace in S consisting of all functions which are bounded a.e. and let,u be the uniform norm in S', i.e., i(f) = inf(cl Ifi < c a.e.] for fcS'. The number 4(f) is frequently called the essential supremum of IfIl. n Let T =- ai(t) (dt) be a formal differential operator of order i=O n on I and let L be the differential operator in S obtained from T by the imposition of a set of k linearly independent boundary conditions Bi(f) = 0, i = l,...,k. Let N be an operator in S with/rL)nKoAN) A 6 We are going to combine the results of Chapters 2 and 3 to obtain existence theorems for the nonlinear differential equation Lx = Nx. (4.1) We proceed by carrying out the following steps: 1o Determine the adjoint operator L* using Chapter 3, Section 2. 2. Let p = dim@(L) and choose functions dl(t)> oo" n(t) in l1(T)) = H (I) to form an orthonormal base for the null space 70

71 /T1((>T)) in such a way that the functions dl(t),..op(t) are in4<L) and form a base for the null space e(L). 3. Let q = dim (L*) and choose functions wfl(t),...,Lq(t) in /7$L*) to form an orthonormal base for the null space 2(L*). 4. Determine the function G(t,s) using (3.23); determine the matrix A. using Corollary 3 of Theorem 3.8; determine the functions Wl(t),..., 0 (t) using (3.25); determine the function K(t,s) using (3.27); and finally, determine the integral representation for the operator H using Theorem 3.9 and its Corollary. 5. Choose an integer m > q and choose functions wq+l(t),..., a/ wm(t) in 7(L*) such that the functions cil(t),...,mt(t) form an orthonormal set in S. 6. Compute the functions L*wq+l.,L*am and the functions HW q +l l''' Hol' 7. Determine the projection operators P and Q using Definition 2.3. 8, Determine the function Km(ts) using (3.30). 9. Choose constants Wand 6 by means of (3.32) and (3533), respectively. 10. Let So be the linear subspace <l.,y. pHyq+lb.., Ha and check that So is a subset of Y(N). 11. Apply the Gram-Schmidt process to the functions Hwq+l.~ Ha~ to obtain orthornormal functions Ylq+1~... oim with

72 m ri - AjaHa cHe for i = q+l,...,m. j=q+l 12. Let M = p+m-q and determine the linear operators 1': EM + So and F2~ <U1...,JU O> + Em which are given by (2.29) and (2.30), respective ly 13. Let /Jand A be the operators given in Definition 2.6 and Definition 2o7, respectively. Determine * explicitly using (2.34). 14. Choose a function xocSo and let p m xo - Zboii + Coirii i=l. i=q+1 Let to = (bol,...,bop,coq+l,'',com)cEM. (We usually choose xo such that %l(xo) = O, that is, f(5o) = 0). 15. Determine the function y(t) = H(I - P)Nxo(t) and choose constants e and e' by means of (2.13). We have compiled this list in order to facilitate the application of the theory of Chapter 2 to Equation (4.1). Each of the steps listed above can easily be carried out in practice. Most of them are independent of the operator N; only in the tenth step is a restriction placed on N, namely that SocI e4N). This is a very mild restriction which is satisfied in most applications since N is usually defined for at least all the continuous functions in S. This condition guarantees that xo is an element ina4N), and hence, we can determine the element y = H(I - P)Nxo0 We can eliminate this condition completely if we

73 assume that condition (IIc) is satisfied, which automatically guarantees that xo/ 4N). Using Theorem 2.9 and Theorem 2.10, we obtain the following two existence theorems for Equation (4.1). Theorem 4.1 Let m = 1. If there exist constants c, d, r, Ro,', c, and 6 with O<c<d, O<r<Ro, 0>O, c>0, and 6>0 such that (a) The set So = (xeS' I Ix-xoJJ < d, kt(x-xo) < Ro) is a subset of /~(N) and IINx-NyII K< ejx-ylI for all x,yeSo, (b) < < 1, c+e < (1-P)d, r+e' < Ro -'d, (c) The set 2./= { cEEJ IJ-Eojl _ e] is mapped by Fl into the set V = (xcSoj I|x-xo0j _ c, p(x-xo) < r), (d) [-5,bs]Cf() and (id+e)d < 6, then there exists x~ (L)f]~if N) which is a solution to Equation (4.1), and 1x-xoll < d, IIQx-xoI < c (4.2) | p(x-xo) K Ro, i(Qxxo) r. Proof. Clearly conditions (IIabc) are satisfied and relations (2.20) are valid. Condition (IId) has been shown to hold by Cesari [3,p.404]. We have only to apply Theorem 2.9 to complete the proof. Q.E. D.

74 Theorem 4.2 Let p > q and let *r(o) = 0. If there exist constants c, d, r, %,Ro,, and t with 0 < C < c <d r < Ro, e > O, c > 0, and s > 0 such that (a) The set ~o = (x~S' l lx-xoll < d, (x-xo) < Ro] is a subset of rjx)(N) and IINx-Nyll < fllx-yIL for all x,yES0; (b) i < 1, c+e < (l-.b)d, r+e' < Ro - 0d; (c) The set ~W= [E I L4-0oll < c} is mapped by Fl into the set V = [xeSol Ix-xoII < c, Il(x-xo) < r), the first order partial derivatives of' exist and are continuous on 2(the Jacobian matrix for * has rank m at to; (d) The set W = (u2Eml I lul <b] is a subset of V(2, there exists a continuous mapping A: W —-- with'V [A(u)] = u for all ucW, and (-id+e), < 5; then there exists xe4,L)n$)O(N) which is a solution to Equation (4.1), and 1x-xoll < di, IlQx-xoll < C (4.3) jL(X-xo) < Ro, Ix(Qx-xo) < r. Proof. This follows immediately from Theorem 2.10. 4.2 THE SELF-ADJOINT CASE Consider the special case in which the differential operator L is self-adjoint, i.e.,

75 L L*. (4.4) In this case we have //L) = XL*), and'v<(L) = AL*), p = q. Thus, by Lemma 2.2 the Hi.lbert space S = L2(I) is the orthogonal direct sum of 7L(L) and'e(L), and hence, H: (L) -- ( L) When we choose the functions ~i(t), i = 1,...,p and the functions ci (t), i = l,...,q, we can take i(t) -= xi(t) for i = i,...,p = q. (4.6) The functions cp+1(t),...,cm(t) can be chosen in a manner which will simplify our work. Theorem 4.3 If the differential operator L is self-adjoint, then the operator H is a completely continuous self-adjoint linear operator on P/(L). Proof. By the Corollary to Theorem 3.9 the operator H is a completely continuous operator from XS(L) into'(L) = kL). Take ft(L, geI(L) and let u = Hf, v = Hg. Then uc,/(L), vcA(L), and Lu = f, Lv = g. Hence, (Hf,g) = (u,Lv) = (Lu,v) = (f,Hg) or (Hf,g) = (f,Hg) for all f,ge' (L). Therefore, H is self-adjoint. Q.E.D.

76 Theorem 4.4 If the differential. operator L is self-adjoint, then there exists an orthonormal sequence of functions wji(t), i = p+l, p+2,... in,(L) and a sequence of non zero real numbers hi, i = p+l,p+2,... such that: (a) Hii = hi)i for i = p+l,p+2,.; (b) Each eigenvalue for H occurs in the sequence [hi); 00 (c) Hf = hi(f,l)i)u)i for all fet)(L). i=p+l Proof. This follows from the fact that H is a completely continuous self-adjoint linear operator in the Hilbert space Wi(L). See [23., p.336]. Q.E.D. Choose functions xi(t), i = p+lp+2... as in the last theorem. Then for any integer m > p the functions wl(t),... wm(t) have all the required properties of the last section. Note that if fcS and (f,ui) = O for i = 1,2,..., then fe?A(L)j =(L) and 00 Hf = hi(fUi)1i =' i=p+l or f = 0 since H is 1-1. Therefore, the functions jSi(t), i = 1.,2,... form a complete orthonormal sequence in the Hilbert space S = L2(I). Let Xi(t) = ci(t) for i = p+l,p2, o...o, and let

77 0 for i = l,...,p ~ = 1 (4.8) for i = p+l,p+2,... hi The functions,i(t), i = 1,2,... are functions in.v(L), they form a complete orthonormal sequence in S, and L~i =?iti for i = 1,2,... (4.9) The projection operators P and Q are given by Px = (Xj)i i=.l and P m Qx = (x,) + (x,'i )H~i i=l i=p+1 p m i=l i=p+l = > (x,i)i, i=l or P - Q. (4.10) In case the differential operator L is self-adjoint and the Xi and wi are chosen as eigenfunctions as above, the general theory given in the last section reduces to the theory of Cesari [3].

CHAPTER 5 THE BASIC CONDITIONS OF THE EXISTENCE THEOREMS In Chapter 2 we have developed a theory for solving the equation Lx = Nx, which has been applied in Chapter 4 to the differential equation Lx = Nx where L is a differential operator of order n in the Hilbert space S = L2(I) and N is some operator in S. Our basic existence theorem is Theorem 2.7. The existence Theorems 2.9, 2.10, 4.1, and 4.2 are all special cases of this theorem. To apply Theorem 2.7 we have to show that conditions (IIabcd) are satisfied, that the inequalities < < 1, c+e < (l- )d, r+e' < R -'d (5.1) are valid, and that there exists x*eV such that P(L~ x* - N Ix*) = O. (5.2) In the event that the equation Lx = Nx is known to have a solution x,. it is reasonable to ask whether the above conditions are satisfied and if x is one of the solutions given by Theorem 2.7? In this chapter we examine this question in detail for the case of differential operators. As in Chapter 4 let I be the finite interval [a,b] of the real axis and let S = L2(I). Let S' be the linear subspace in S consisting of all functions which are bounded avea and let l. be the uniform norm in S'. Let L be a differential operator of order n in S obtained from 78

79 a formal differential operator T by imposing a set of k linearly independent boundary conditions and let N be an operator in S witha(L).rv'N) i ~. To simplify our calculations we assume thato< (L) is a subset of/6(N). This is usually the case in applications. Suppose there exists a function A(t) int/L)D 4N) such that Lxr~ = Nx. IS2 ~ - Nx.~ ~(5.-3) We shall show that under reasonable conditions it is possible to choose the integer m, the projection operators P and Q, the constants-+ and the function x (t), the constants e and e', and the constants c, d, r, Ro, and ~ such that the hypothesis of Theorem 2.7 is satisfied, and that x is one of the solutions guaranteed by Theorem 2.7. We begin by carrying out part of the program of Chapter 4: 1. Determine the adjoint operator L*. 2. Let p = dimi(L) and choose functions 51(t),...,n(t) inJ(Tl(T)) = Hn(I) to form an orthonormal base for the null space T(T1(T)) in such a way that the functions.l(t),...,tp(t) are inAL) and form a base for the null space 5(L). 35. Determine the function G(t,s), the functions 5L(t),... 9(t), and the function K(t,s) so as to obtain the integral representation for the operator H: b n b t Hf(t) = K(t,s)f(s)ds = 7K(t) j (s)f(s)ds + G(t,s)f(s)ds. a j=l a a (

8o Let q = dimr(L*) and choose a sequence of functions ci(t), i = 1,2,... intx L*) such that the xi form a complete orthonormal sequence in S and the functions cl(t),... Cq (t) form a base for the null space a(L*). We can find such a sequence because,L*) is dense in S. For each integer m > q we define projection operators Pm and Qm in S by m PmX = (xci)wx) xeS, (5.5) i=l and P m Qmx = (x,i)i +) (x, L )i) Hoi, xcS. (5.6) i=l i=q+l These are the same projection operators we have been working with in the last three chapters. For each integer m > q let Km(t,s) be the function on IxI defined by mn b Km(t,s) = K(t,s) -J K(t,) )d i(s)d (5~7) and let and' be the constants given by'Ym = (IJb j ~ m(ts ) } S)3 ds dt (5.8) a a and' = (maxI bI (t,9 2 d) (5.9) By Theorem 3. we have that By Theorem 5.10 we have that

81 b H(I - Pm)f(t) = Km(t,s)f(s)ds, teI, (5.10) a for all functions f(t) in S, and ||H(I - Pm)f 11 < imilfl for all fcS. L1(H(I - Pm) <) <'lf ([]] This is true for each integer m > q. Theorem 5.1 (a) lim 0 (b) lim1 = 0 m> m m-~oo Proof. In Chapter 3 we observed that the integrals b b [K(t,~)]2 d~ and K(t,j)wi(~)d~, i = 1,2,... a a define continuous functions of t on the interval I. Let b f(t) = [K(t,i)]a d5 a and b i(t) = lK(t,S)ei(~)dl for i = 1,2,... a These are continuous functions on I and f(t) > 0 on I. Fix tcI and consider K(t,s) as a function of s on I. Since this function is square-integrable on I, by Parseval's equality we have b m [K(t,~)] d{ = lim (tn)X(i(~)d a i=l or

f(t) = lim ) Ici(t) I2 for each teI. i=l Using Dini's Theorem, we conclude that the sequence of continuous functions m f(t) - jlai(t)2, m = 1,2,... i=l converges uniformly to 0 on I. By this last result and Equation (3.31) we have b m 0 = lim (f(t) -7 (t) dt m La i=1l lim f bab bK(te)d b K(ta)w~(~)d:1 )d moo a i= a J -lim r [Km(t,)] d) dt lim (s12 inl L a i=l a b lim max [Km(t, )] d1 = lim i QE.D. and m-*C =TI a

83 Theorem 5.2 Let x be any function in L). If xm = Qmx for m = q,q+l,... then xm + x, Lxm - Lx, and the functions xm(t) converge uniformly on I to the function x(t). The convergence is determined by: IIx-xm|j < 2 ~ml[Lxl], 2 |m 2 IIX-Xmh = (IrlEX - I(xWin) I i=l Proof. By Theorem 2.4 we have Lxm = LQmx = PmLx or m Lxm = I (Lx,ci)'i ~ i=l Since the wi are complete, we get the desired convergence Lxm + Lx. Note that IILxmII < IILxll by Bessel's inequality. Also by Theorem 2.4 we have H(I - Pm)L(x-xm) = (I - Qm)(x-xm) = X-Xm, so IIx-xm1 < I4l1-TLXmI < 2 7~TmlxI and 1(x-m) < 2 IILXll. Q.E.D.

84 For each pair of real numbers c > 0, 6 > 0 let W(EC.) = (xeS'J llx-;'11 < c, (x-Z) < K. (5.13) We assume that the following condition is satisfied: (IV) There exist constants c > O, 8 > 0, and i > 0 such that the set W(E,5) is a subset of IN) and |INx-NyIl < 2Jjx-yll for all x,ycW(e,5). Note that SeW(e,5), so this set is nonempty. Let xm = Qm x for m = q,q+l,.... (5.14) Each function xm belongs to S' and by the last theorem 1lx-11 — o (x- ) — - o. Choose an integer mo > q such that the following conditions are satisfied: (a) xm f W(e/2,6/2) for all m > mo, (b) k e < 1/3 for all m > mo, (c) jm(\e+ jINXIi) < e/6 for all m > mo (d) m'(e ( + jINx2I) < 5/6 for all m > m (e)'m A < 6/3 for all m > mo. Now we choose all the quantities which are needed in order to apply Theorem 2.7. Fix any integer m > mo and let P and Q be the projection operators P = Pom, Q = Qm We choose the constants f and 7t' by taking

Let So be the linear subspace in S spanned by the functions a1,-.., ~p 0q+z *...*'H"m and let xo = Xm Qm x C So. Corresponding to the function y = H(I - P)Nxo, we choose e =;O~(~ e + IINx11), e' = t'm( + IIN+xIi)' Then lyll = |H(I - P)NxoI[ <:bm( INXo-N'II + IINl)II) < m(Z + JINXJI ) < e, and similarly, t(7) < e'. Let c, d, r, and Ro be the constants given by c = s/6, d = c/2, r = 5/6, Ro = 5/2. Clearly 0 < c < d and 0 < r < Ro. As in Chapter 2 we let V and So be the sets given by V = [xeSoi Ilx-xOl <K c, 4(X-xo)< r] and S = (xeSJ lx- || < d, p(x-xo) < Rol. Also, for each x*cV let S(x*) = (xeS' Qx=x*, Ix-xol0 _ d, k(x-xo) < Ro). Note that if x~So, then xeS',

lx-11 < lIx-xol[ + Ilxo,-AI K d + c/2 = and 4(x-x) _< (x-xO) + 4(xo-X) < Ro + /2 = 6, and hence, So CW(e,6)C/N). Therefore, conditions (IIabcd) are satisfied. Also, we have = < 1/, c + e + d < K/6 + e/6 + 1/3'e/2 = d, and r + e' + e' d < 5/6 + 5/6 + 1/2 < R. Therefore, the relations (2.20) are valid. From Definition 2.5 we obtain the continuous operator.;/: V —-..L) o which assigns to each x*cV the unique element xES(x*) which satisfies the equation x = x* + H(I - P)Nx. Note that xoeV, xcS(xo), and by Theorem 2.4 we have xo + H(I - P)Nx = xo + H(I - P)La Qx + (I -Q) = x, and hence, 7 x, = x and P(Lixo-Nrfxo) = 0. Therefore, the hypothesis of Theorem 2.7 is satisfied, and we see that x is one of the solutions which is guaranteed by this theorem.

CHAPTER 6 AN EXAMPLE In this chapter we use the results of Chapter 4 to study the nonlinear boundary value problem (6.1) x(O) - O where ca and P are real constants. From Theorem 4.1 it will follow that this equation has a solution for all (c,P) with jal < i, ImJ <.001. We shall obtain estimates on the norms of these solutions. Let I be the finite interval [0,2ct] and let S = L2(I) be the real Hilbert space consisting of all real-valued functions f(t) which are square-integrable on [0,27t]. The inner product and norm in S will be denoted by (f,g) and I|f|j, respectively. Let S' be the linear subspace in S consisting of all functions which are bounded a.e. and let p be the uniform norm in S'. Let L be the differential operator of order 2 in S obtained from the formal differential operator Tx = x" + x by imposing the single boundary condition Bl(f) = f(O) = 0. (6.2) We have n = 2 and k =.1 Let N: S' —-S be the operator given by Nx = - cx2(t) + 2t, o < t < 2_c, (6.3) where c~ and B are real constants. Clearly.'iN) = S' and L)C 5'N) 87

88 We proceed to carry out the steps listed in Chapter 4. 1. In Example 2, Chapter 3 we showed that T is formally self-adjoint and that the adjoint operator L* is obtained from T = T* by imposing the three linearly independent boundary conditions B*(f) = f(o) - O B*(f) = f'(2xc) = O (6.4) B3(f) = f(2iT) 0o. 2. Let /1(t)= -sin t, g2(t) cos t Clearly l1(t) and /2(t) are functions inAi(Ti(r)) = H (I) which form an orthonormal base for the null space 2(Tl(T)). If x(t) is any function in the null space (L), then x = bj1j+bd2 and x(O) = 0, i.e., b2 = O and x(t) = bljl(t). Thus, 4(L) = < (t)> p =. (6.6) 3. Note that if x(t) is any function in the null space (L*), then x = bjll + bd2 and x(O) b2 o bl x' (2j) 0, x(2c) b2 0, or bl = b2 = O. Thus9

89 9(L*) =< o>, > (L) = S, q = 0. (6.7) 4. We have sin t cos t ~(t) - cos t -sin t detD(t) = - 1 0 cos t W1(t) = det r- 1 cos t, - sin t 0; W2(t) = det =f 1 sin t 1 cost 1 and hence, G(t,s) = t [)1(t)Wl(s) + /2(t)W2(s)] or G(t,s) = sin t cos s - cos t sin s, 0 < t,s < 2t. (6.8) Now B = [B1(~2)] = 1, so we can take A = [fC ]. Thus,

9o 2r Yt/(s) = 1 sin t (sin t cos s - cos t sin s)dt S = x (-cos s + cos s - 1 sin s) and YLJ2(S) = Q since B1o = 11 = 0, or (s) -= J (- cos s + s cos s 1 sin s) 2Je 2r2 0 < s < 2Tc (s) = (6.9) Therefore, 2K Hf(t) = sin t S (- cos s + — cos s. sin s)f(s)ds 227 0 (6. 1o) t t + sin t' cos s f(s)ds- cos t' sin s f(s)ds, 0 0 0 < t < 2tr, for all functions f(t) in P(L) S. 5o Consider the function c(t) = (t-2gc)sin t. Clearly w(O) = w(2w) = 0 and w' (2r) = [(t-23)cos t + sin t] = 0, 23 and hence, aC (L*). We choose m = 1 and cwl(t) = 1 ci(t). Now 2i 2 2 IIoII = (t-2c) sin t dt 0 6 so

91 m = 1, (3)1(t) t 1/2(t-2ir)sin t, 0 < t < 2ito (6.11) 6. To compute L*wcl we have w(t) -= (t-2n)sin t, WI (t) = (t-29)cos t + sin t, o"(t) = -(t-23t) sin t + 2 cos t, and hence, L*Wtl(t) = 2 r. 11 cos t, 0 < t < 2_r. L8 Tc ~- 37C (6.12) To compute Ha1 we first compute the functions H(sin t) and H(t sin t). By (6.10) we have 2g 2 H(sin t) = sin t cos s sin s)sin s ds 23- 2nc t t + sin te ~c.os s sin s ds - cos t sin s ds 0 0 1 sin t t cos t 4 2 and H(t sin t) sin t (-cos s +- cos s - sin s)s sin s ds 2x 2Tc 2 + sin t s sin s cos s ds - cos t s sin s ds 0 0 it sin -sint t cos t 2 4 4 Therefore,

92 Hwl(t) = 3 6 ] [/Lt sin t + ~t cos t cos t 8 - 3 4 4 0 <t < 2rt. (6.13) 7. The projection operators P and Q are given by 2~ Px(t) = 6 ] (t-2ic)sin t./ x(~) (S-2Tc)singd~, 0 < t < 2.t, (6. 14) and 2T Qx(t) = i sin t- x(,)sin~d, + 2 K83 6 ] [t sin t _t 9-3 3 4 0r2 l (6t.15) + cs t cos t - t cos tx()Qos~ d~, o < t < 2x, 0 for all functions x(t) in S. 8. We have K(t,s) = sin t (-cos s + cos s - -sin s) + G*(t,s), O < t,s < 2g 2Tc 2T - - (6.16) where G(t,s) for O < s < t < 2Tc G*(t,s) = 0 for 0 < t < s < 2c Also, 2. *K(t, )wl( )d5 =Hl(t), 0 and hence, Kl(t,s) = K(t,s) - Hcl(t)col(s), O < t,s < 2Xo (6.17)

93 9. To determine the constants and -' we must compute several integrals. We have 2ic L2 i2 22 [7(s)1ds = t(- cos s +-cos s - sin s) ds 2Tc 2ic O O 2 3 8 t t G(t,s) *(s)ds r [sin t cos s - cos t sin s [-cos s + - cos s 0 0 o 1 sin s]ds. t sin t - t sin t + 3 t cos t - sin t, 2 8., _, and tb t [G(ts)] ds = [sin t cos s - cos t sin s] ds 0 0 t 1 ---- sin t cos t. 2 2 Therefore, 27c 2K [K(t,s)]2ds = [~l(t)Y(s) + G*(t,s)]2ds 0 0 ~~2 2t =[1(t ) ] I [((s)] ds + 2 1(t) G(t,sA (s)ds 0 t 0 2 + [G(t,s)] ds 0 or 2 2 [K(t,s)] ds = — + )sin t - sin t cos t - t si.n2t 2 5 8 2 0

94 2 2 + L t sin t cos t + t sin t. (6.18) 4T 4i By (3.31) we have 2 2 22 [Kl(t,s)] ds = [K(t,s)] ds - [HDl(t)], (6.19) 0 0 and hence, 2 2 2xt 2ic 2 2 2 [Kl(t,s)] dsdt = j [K(t,s) ] dsdt - lIHllI 0 0 0 0 where 2 2 [K(t,s)] dsdt - -2 3 8 O0 and 2 II~I = [6 cos 2 [ 2H 6=.6.( t- sin t + X t cos t cos t) dt, 8(3- 4 4 0 2 6 8r 1 r 5 3 1 l ]- L" 6 — j 3(6.20) Therefore, [Kl(t.,s)] 2dsdt = - 5 _ 6 ( 5 03 8 8r - 3x.15 6 3 = 1.998188.!/2 Since [1 998188] = 1.413573, we choose = 1.414. (6.21) If we write out (6.19), we get

95 2Tc [K(tjE)]d~ =.5 t + sin t [1.007409 -t +.078005t ] 0 + sin2t [-.25 +.119366t -.019748t +.001572t ] 2 2 3 + cos t[-.248161t +.039496t -.001572t4]o Using a computer to carry out the calculations, we get the estimates 22.691546 < max [Kj(t,t)] d <.692546. O<t<2nc 0 1/2 Since [.6925461 =.832194, we choose?-.833. (6.22) 10. Let SO be the linear subspace < 1,Hn1>. We note that sO c Y(L) C (N). 11. Using (6.20) we have rl(t) = allJiHl(t), 0 < t < 2g, (6.23) where all =.... L s6>855 t3 3 "-1/2 3 =.502738. Thus,' f$- 75 ]2j. sin t + 5 t cos t _ cos t (6 24) for 0 < t < 2~.

96 12. We have M = p+m-q = 2. (6.25) Let E be a copy of Euclidean 2-space where we denote the points by = (bl,cl) and let E be a copy of Euclidean l-space where we denote the 2 1 points by u = (ul). Then Fl: E - So and r2: <ot~1 >E are given by rl(blcl) = blji + c1 T1 (6.26) and r2(ujj) = 1. (6.27) 13. The function * can be computed using (2.34). We have /(bl,cl) = all1c - N[bll + c1 rl1]l1(t)dt 02T O2 2 = a1le S - W(t)[- c (b3m1(t) + c, rPl(t)) + 3t]dt 0 2 22 = allcl + mbl' O,l(t)[Wl(t)] dt + 2ablcl, l(t)~l(t) ~ll(t) 0 0 2 2 + acl ol(t)[ l(t)[ (t)] dt - t (t)dt. 0 0 These four integrals are given by 02\ 1/2 2T co1(t)Et(t)] dt = i.(6 ) (t-2gc)sin t dt

97 2T 2 W ((t)ol(t).~l(t)dt = all_ 6' (t-2~)sin t ( ~ sin t + g t cos t cos t) dt 4 2 all 6 ) 3;t 98g3 ~ t 2Tc 2r l(t)[ (t) dt =all 2( 3/2 (t-2c)sin t ( tsin t 0 0 3r 4 t2 2 + t cos t - cos t) dt 4 all =a i2 ( 6 >3/2 (54s3 140n) 8(3 - 3TC 27 81 and 2 2 t W(t)dt 6 (t-2t)sin t dt 0 0 0. Therefore, r(bl,cl) = allcl- - 6 )l2 + a1-( 36 C lb c 3 a2 6 38/2( - 40 ) 8 - 37 or 4(b,cl) =.502738cz -.211427ab12 +.093851ab lc (6.28) +.033884ac 12 14o Let xo(t) O, i.e., we take xo = bo?(l + col rT where bol = Col = O. Clearly xocSo,

98 (0,0) a E2, ~=0 and *(~o) = o. 15, We must determine the function y(t) = H(I - P)Nxo(t). We have NXo(t) = ft, - 0, (I - P)NXo(t) t, and H(I - P)Nxo(t) = 1H(t) = f sin t (- cos s cos s +- 1 sin s)s ds 00 Ht 2t 28 sin t + te Thus, y(t) = 28 sin t + 3t, 0 < t < 2~. (6.29) Now 2T 2 2 wl = tIj ~ (2 sin t + t) dt 0 2 3 1 1 ( 8r - 4) or HI1 = I1| ( 8 -I c 4,P)'/ 8.o375592 I!I. 7-,

99 We choose e = 8o3741I1. (6.30) Let f(t) = 2 sin t + t, 0 < t < 2:rc. Then y(t) = Bf(t), f(t) > 0 on the interval [0,2ic], and Iy(t)l = |PjIf(t). Thus, max Iy(t)l = | ~ max f(t). o < t < 2n 0 < t < 2C To maximize f(t) on [0,2t] we have fi(t) = 2 cos t + 1 O, cos t = - 1/2, or =2 4i 3 3 Thus, the maximum for f(t) occurs at one of the four points 0O -2- 4; __ or 2g: 3 3 f(o) = 2Tr 3 3 f(24i) = 3 i4 3 3 f(2g) = 2a e We see that the maximum occurs at 2g, and hence, max Ir(t) = 2~Jf. O < t < 2 We choose

100 e' = 6.2841 I. (6.31) Having chosen xo(t) -0, we must choose the constants c, d, r, Ro, 2, e, and 5 such that the hypothesis of Theorem 4.1 is satisfied. For any number c > 0 and any function x(t) in So with Ilxll < c, we have 2 2 2 x(t) = bl,(t) + c-l1(t) with b1 + cl < c, so Ix(t)l < cl~=(t)l + cll(t)l. But Ilw(t)i = sin tj < and I rl(t)j = alaiHl(t)I =al j ) 2 sin t + t cost tr cost < all('1 (~ + 2 + ) and hence, ix(t)i < c L + all 6 ) / (2 + Let 1/2 r= cr + all(3 6 2 (3 + 2- = 3.049797c (6.32) and ~ = c. (6.33) Then the set

101 V = xGcSo I lx-xoil < c, ~(x-xo) < r} reduces to the set v = (XsO I1Xl < c)3 and the set 2= E 1 11 11 < c! is mapped by Fr into the set V. Also, if d and Ro are numbers with 0 < c < d and 0 < r < Ro, and if so = (xcS' | ||xll < d, Ii(x) < Ro3, then SO is a subset of N) and for xeSo, yeSo we have Nx(t) - Ny(t) = -.x 2(t) + ay2(t) = C[y(t) + x(t)] [y(t) - x(t)], so wNx(t) - Ny(t)l < 2Rol ICx(t) - y(t)I and 2 2 2 2 IINx Nyi < 4Ro< 2 2 lix yll. or INx Nyll < 2Ro lO lix - yll. let = - 2ROIla. (6e34) We assume that l K< Lo The principal inequality that we must satisfy is/e < 1, i.e.,

102 (i 414)(2Ro|al) < I This inequality is satisfied if (1.414)(2Ro) < 1 or Ro <.353607. (6.35) We shall replace the inequality < 1 by the Inequality (6.35) and the assumption JIa < 1 in the work that follows. Now we want to choose c, d, and Ro such that r < Ro, i.e., 3.049797c < Ro. Thus, we must have c <.353607.115944. 3.049797 If c satisfies this inequality, then *(O,c) =.502738c +.033884ac > c(.502738 -.o33884c) and 2 4(O,-c) =-.502738c +.033884acc < - c(.502738 -.033884c), and hence, the interval [-c(.502738 -.o33884c), c(.50278 -.o35884c)] is a subset of xv(2I3. Let 6 =.502738c -.033884c2. (6.36) We must determine a bound on the parameter f and choose constants c, d, and Ro such that

103 O < c < d 35.049797c < Ro Ro <.353607 c + 8.374 | I + (.4.14)(2Ro)d < d 3.049797c + 6.284 iBp + (833)(2Ro)d < Ro 2 [(lo.4l4)(2R)d + 80374 I1I1 (2RO) < o502738c - o033884c or O < c <d 3.049797c < Ro Ro <.353607 (637) c + 8.374!P3 + 2.828 Rod < d 3.049797c + 6.284 IB| + 1.6666ROd < Ro 5.656Ro2d + 16.748 |i|Ro <K.502738c -.033884c2. We assume that the parameters 0 and P satisfy the conditions Jl1 < 1, 11 <.0or0, (6-38) and choose the constants c, d, and Ro as follows: c = 0.1 d =.03 (6.39) Ro =.l Then r = 5o30498 Q.: ~210~ ~(6.40) C.01 =).005024, and the inequalities (6.37) are satisfied. From Theorem 4.1 we conclude that for each pair of real numbers ((cx) with Icl < 1, |I1 <.001 there exists a realiralued function x(t) which is twice continuously differentiable on the interval [0,2it] with

io4 x"(t) + x(t) + sx (t) = t, o0 < t < 2 Lx(o) = O and i1xil <.03, Ix(t)l <.1

APPENDIX I THE THEORY OF CESARI The existence theory for the equation Lx = Nx described in Chapter 2 is closely related to the existence theory of Cesari [3], where the same equation is considered. Cesari gives a system of axioms concerning the existence of linear operators H and P with special properties. These two operators play crucial roles in the development of his theory. The original problem for this thesis was to determine conditions on the linear operator L which would be sufficient to guarantee the existence of the corresponding operators H and P. It was hoped that the resulting class of linear operators L -would be large enough to include all the differential operators on a finite interval [a,b]. However, this was not the case. As long as L was restricted to being a self-adjoint differential operator, the desired operators could be shown to exist; when L was allowed to be non-self-adjoint, then examples arose where it was impossible to construct H and P with the desired properties. If we desire an existence theory which is applicable to all differential operators, then we need a new theory. It is just such a theory which is presented in Chapter 2. In case L is a selfadjoint differential operator, our theory reduces to the Cesari theory. We are going to examine the existence theory in [3] and illustrate the difficulties which can arise when one tries to construct the op. 105

106 erators H and P. Let S be a real Hilbert space with inner product (x,y) and norm i|xlJ, let L be a linear operator in S with domaineL), and let N be an operator in S with domain N) such thatz 4(L)n,(N) # y. Cesari discusses the existence of solutions NeS to the equation Lx = Nx. (1) He assumes that linear operators H and P exist with the following properties: (a) The operator P is a projection operator from S into S with finite-dimensional range So and null space S1; So is a subset of 4(l), and SO is spanned by orthonormal elements l'..s-m; the range of the operator I - P is S1. m (b) Px = 7 (x,$i)di for all xeS. i=l (c) The operator H maps SI into SI, and H(I - P)IL = (I - P)x for all xCeL). (d) PH(I - P)x = 0 for all xeS. Choosing an element xoeSo and a number c > 0, and letting V be the subset of So given by v = (xeSO IIx-xoI < c), Cesari introduces suitable restrictions to show that it is possible to determine a unique solution xeeC(N) to the equation x = x* + H(I - P)Nx (2) corresponding to each element x*eV. He also makes the following assumption:

107 (e) For each x*eV the corresponding element xe'(N) is an element of//(L), and LH(I P)Nx (I - P)Nx LPx = PLx. Under these restrictions the unique element xe/YN) corresponding to x*EV is a solution to Equation (1) provided x also satisfies the equation P(Lx-Nx) = 0. (3) Thus, Cesari proceeds to determine x*eV such that the corresponding x satisfies Equation (3), and hence, yields a solution to Equation (1). We shall make two remarks to illustrate the difficulties which arise when one tries to construct the operators H and P with properties (a) - (e). Remark 1 It is desirable to construct H and P using only the properties of the linear operator L and independent of the operator N. The resulting theory should be general enough to be applicable to a large class of nonlinear operators N. This class of operators should include all the operators Ny: S.>S of the form Nyx = y for all xES where y is an element in S. If the Cesari theory is to be applicable to all the operators Ny, then from (e) we must have H(I - P)y E &L) and LH(I - P)y = (I - P)y for all yeS.

108 The last equation says that LHx = x for all xeS1, which implies that SI is a subset of the range of L and H: SI — SlV(LL). Thus, H must be a partial right inverse for L. Let 7(L) and /L) denote the range and null space for the operator L, respectively. Since S1 is the range of the operator I - P, it follow: that P: S L and I - P: S —- (L) Take any element x belonging to the null space 5(L*) of the adjoint operator L*. For each yeS the element y-Py is an element in,(L), and hence, there exists ze4'iL) such that Lz = y-Py. Thus, (x,y-Py) = (x,Lz) - (L*x,z) -0 or (x,y-Py) = 0 for all yeS. But P is self-adjoint, so (x-Px,y) = 0 for all yeS or x = Px e So.

109 Thus, the null space /(L*) must be a subset ofrAfL). This condition is not satisfied for all differential operators. Example Suppose Lx = x" + x is the differential operator on [0,21r] whose domain 4jL) is determined by the boundary conditions x(0) = 0, x(2iw) = 0, and x'(2t) = 0. It can be shown that L*x = x" +:x with domain (L*) determined by the single boundary condition x(0) = 0. It follows that %(L*) is the 1-dimensional subspace generated by the function sin t. But sin t does not belong to&/L), i.e., 9L*) is not a subset of XL). We examine these particular differential operators in Chapter 6. Remark 2 If the theory of Cesari is applied to the special case where N is given by Nx = 0 for all xcS, then Equation (2) reduces to finding x such that x = x* where x*cV. From (e) we must have Lx* = PLx* c SO for each x*EV, or L maps V into So. For each element yeSo we can choose c > 0 such that lacy[ll < c, and setting z = xO + cay, we have zEV, LzeSo, xoeV, LxoeSo, and aLy = Lz - Lx e So, or LyeSo. Thus, So is a finite-dimensional invariant subspace for the

110 linear operator L. In general, not all differential operators have such invariant subspaces. Example Consider the differential operator Lx = -x" on [O,] whose domain o/<L) is determined by the boundary conditions x(O) + 2x(t) = 0, x'(O) - 2x' (X) = O. Suppose L has a finite-dimensional invariant subspace So. Let ~l(t),...,$m(t) be functions which form a base for So. Then we can write m "- i = 7a ji. for i = l,...,m j=l where the aji are real numbers. Choose a complex number \ and complex numbers c,...,cm not all zero such that m T a jici - X cj for j = 1,...,m, i=l and let m /(t) = 7 cibi(t) for 0 < t _< x. i=l The complex-valued function >(t) has the properties that /(O) + 2>(g) = o, /'(o) - 2 ='(X) = 0, and -'l = — Ci.i i=l

ill But the associated differential operator over the complex numbers is known to have no eigenvalues [6,p.300]. Thus, L has no finite-dimensional invariant subspaces. The above remarks illustrate the difficulties which arise when one tries to use the Cesari theory for an arbitrary differential operator L.

APPENDIX IT THE THEORIES OF BA.RTLE, CRONIN, AND NIRENBERG Bartle [2], Cronin [7], and Nirenberg [19] have each developed existence theories for the equation Lx + F(x,y) = 0 (1) in Banach spaces S and Y, where L is a bounded linear operator defined on all of S with values in S and F(x,y) is a function which maps a neighborhood of the origin in SxY into S with F(O,O) = 0. The purpose of this Appendix is to examine each of these three theories, determining any relationships which might exist between them and determining any relationship which they might have with the existence theory developed in this thesis. Before getting into the details of these three theories, let us make several comments. Note that if we fix an element ycY and let Nx = -F(x,y), then Equation (1) has the same form as the equation we studied in Chapter 2. Thus, when the theories of Bartle, Cronin, and Nirenberg are applied in a Hilbert space, we would expect them to be related to the special case of our theory when we assume L is everywhere defined and bounded. We will find that such a relationship exists when we examine the Nirenberg theory. Nirenberg treats Equation (1) is a very general setting, presenting three methods for establishing the existence of a solution. The 112

113 material for his three methods is based on lectures by K. 0. Friedrichs [10,11]o. Bartle considers Equation (1) under more specialized circumstances, and his theory is essentially the same as Nirenberg's third method. Cronin discusses a very special form of Equation (1); her work is an abstract version of part of the work of Schmidt [20] on nonlinear integral equations, and it is also intimately related to Nirenberg's third method. Now let us begin our discussion of these three theories. We shall examine the general pattern of development which each one uses: the construction of an inverse operator for L, the determination of an auxiliary equation, and the determination of a bifurcation equation. We shall start with the Nirenberg theory because it is the most general. In [19] Nirenberg assumes the following hypothesis for the linear operator L: the null space ~(L) admits a projection operator Q and the range t(L) is a closed subspace in S which also admits a projection operator p. Let P denote the projection operator I - p, so I - P is a projection of S onto i(L). Under the above hypothesis Nirenberg shows that there exists a bounded linear operator H: /L)- S which is a right inverse for L and has the properties: LHx = x for all x~ (L), and HLx = x - Qx for all x~S. The operator H(I - P) gives an extension of H to all of S.

114 Method 1 If xcS is a solution to Equation (1), then applying the operator H(I - P) it follows that x = x* - H(I - P)F(x,y) (2) where x* = QxC 92L). Equation (2) is the auxiliary equation used in Method 1. Under suitable restrictions it can be shown that Equation (2) has a unique solution x = x(y,x*) for x*e (L) and ycY with sufficiently small norms. For such a solution Lx + F(x,y) = PF(x,y), i.e., x = x(y,x*) is a solution to Equation (1) iff PF(x(y,x*),y) = 0. (3) This last equation is the bifurcation equation for Method 1. At this point the element y is usually fixed sufficiently small. To obtain a solution to Equation (1) one only needs to determine a solution x* to the bifurcation Equation (3), which is usually a system of q-equations in p-unknowns where p = dim/(L), q = codim (L). Note that if we let Nx = - F(x,y), then Equations (1), (2), and (3) can be written as x = Nx, (1') x = x* + H(I - P)Nx, (2') and P[Lx(y,x*) - Nx(y,x*)] = 0. (3') When S is a real Hilbert space, codim 4i(L) = q < co, and Q and p are the orthogonal projections onto /2(L) and e(L), respectively, then

115 Method 1 is essentially the same as the special case of our theory when X L) = S and m = q. Both theories use the same operators H, P, and Q., and both theories have the same auxiliary and bifurcation equationso Method 2 The second method that Nirenberg describes is equivalent to the first method. Since it adds nothing pertinent to our discussion, we shall not discuss it here. Method 3 (due to P. Ungar) The third method differs from the first two methods in that it uses a different type of inverse operator for L. It assumes the additional hypothesis that dim Z(L) = codim )(L) = p < o Let B: 2I(L) — P(S) be a 1-1 linear operator from J(L) onto P(S), let W = BQ, and let Ll,: SA —— S be the bounded linear operator given by L L L+ W. L1 is a.11 mapping of S onto itself, and hence, L. has a bounded inverse. Let T 1 = (L + W)-'. The operators L and T are related by TL = I Q. For any solution xeS to Equation (1) it follows that x + TF(x,y) = x* (4)

where x* = Qxc jL). Equation (4) is the auxiliary equation which is used in this method; it is very similar to the auxiliary equation used in Method 1. If x = x(y,x*) is a solution to Equation (4), then (L+W)x + F(x,y) = (L+W)x*, or Lx + F(x,y) = W(x*-x), i.e., x = x(y,x*) is a solution to Equation (1) iff W(x* - x(y,x*)) = 0. (5) Equation (5) is the bifurcation equation for this method. For fixed y it reduces to a system of p-equations in p-unknowns. The one new feature which distinguishes Method 3 from Method 1 is the use of the operator T instead of the operator H. However, T is closely related to H as the following lemma shows. Lemma The linear operator H is the restriction of the linear operator T to the subspace,e(L), and LT = I- P. Proof. First, take any element xc W(L). Then there exists yeS such that Ly = x. Letting z = y-Qy, we have Qz = 0, Wz = 0, and Lz = Ly = x. Thus, Hx - HLz = z-Qz z= = T(L+W)z

117 TLz = Txo Next, take any xeS. Since PxcP(S), there exists zc j (L) such that Bz - Px. We have z = T(L+W)z = TBQz = TPx, so LTx = LT(x-Px) + LTPx = LH(x-Px) + Lz = x-Px. Q.E.D. Let us reexamine Equation (1) under two new sets of hypotheses for the operator L. We shall divide the discussion into two parts. In the first part we shall use the hypothesis of Bartle, establishing the relationship between Bartle and Nirenberg; in the second part we shall use the hypothesis of Cronin, establishing the relationship between Cronin and Nirenberg. I. Consider Equation (1) when the linear operator L satisfies the following hypothesis: the range j(L) is a closed subspace in S, the null space /(L) is finite-dimensional, the null space -(L*) for the adjoint operator L* is finite-dimensional in the dual space S*, and daim (L) = daimt(L*) = p < oo We assert that the linear operator L also satisfies the hypothesis of

Nirenberg's Method 3. In order to see this let ul,...,up be elements in S which form a base for /I(L), and let fl,...,fp be functionals in S* which form a base for ~(L*). Then there exist functionals g,..., gp in S* and elements zl,...,Zp in S such that gi(uj) = bij, fi(zj) = bij. Define projection operators P and Q by p Px = fi(x)zi, xES, (6) i =.1 and p Qx _ gi(x)ui, xES, (7) i=.l and let p be the projection operator I - P. Clearly 21(L) admits Q as a projection operator. Also, we can show that pxcC(L) for all xeS, and that S is the direct sum of the subspace j(L) and the subspace <zl..Z,zp>. It follows that,(L) admits p as a projection operator, and dim (L) = codim )(L) p < oo. Hence, Nirenberg's Method 3 is applicable when the linear operator L satisfies the above hypothesis. Suppose we determine the particular form that Method 3 takes when P and Q are given by (6) and (7), respectively, and the linear operator B:,t(L) — P(S) is defined by B(ui) = Ezi for i = l,...,p. (8)

119 The operator W = BQ is just p Wx = -1 gi(x)zi for all xeS, (9) i=l and the operator L1 = L + W is p Lix = Lx -7gi(x)zi for all xeS (10) i-=l Again we obtain the inverse operator T = (L+W)-', the auxiliary Equation (4), and the bifurcation Equation (5). Now the bifurcation equation can be rewritten in a different form. Let x = x(y,x*) be a solution to Equation (4), i.e., x + TF(xy) = x*eCZ(L). Applying the operator L and using the lemma, we get Lx + F(xy) = PF(x,y). On the otherhand we observed in Method 3 that Lx + F(xy) =W(x*-x) and hence, PF(x(yx*),y) = W(x*x(y,x)). (L1) Thus, the bifurcation Equation (5) can also be written as PF(x(y,x*),y) = 0. (5') The existence theory which we have developed in part I using Nirenberg's Method 3 is identical to the existence theory which Bartle developes in [2]~ The hypothesis for the linear operator L is the same as his hypothesis for L; the operators P, Q, L1, and T are the

120 same as the operators Z, U, L', and R, respectively, which he constructs in his theory; and Equations (4) and (5') are the auxiliary and bifurcation equations which he uses. II. Consider Equation (1) when the linear operator L and the function F(x,y) satisfy the following hypothesis: the operator L is of the form L = I+C where C is a completely continuous linear operator in S, and the function F(x,y) is of the form F(x,y) = R(x) - y where R is a function which maps a neighborhood of the origin in S into S with R(0) = 0 and ycY = S. Equation (1) takes on the special form (I + C)x + R(x) = y. (l") For such a linear operator L it is well known that the range ~(L) is a closed subspace in S, the null spaces ~(L) and,(L*) are finitedimensional, and dim 2(L) = dim (L*) = p < Thus, we can proceed to apply Method 3 exactly as we did in part I. As before we choose the elements ul,.. Up zl,...,zp in S and the functionals f.o-.,fp,~ gL-o-,~gp in S*, and define the projection operators P and Q by means of Equations (6) and (7), respectively. Instead of defining the operator B by means of Equation (8), we define B: kl(L) —-P(S) by B(ui) = zi for i = l,...,p. ($8) For this choice of B the operators W = BQ and L1 = L + W are given by

121 p Wx = ) gi(x)zi for all xcS, (9') i1= and p Ljx = Lx +X gi(x)zi for all xeS, (lo') 1=1 and again the inverse operator T = (L+W) exists. Up to this point the theory described in part II is identical to the existence theory of Cronin [7]. She studies Equation (1") under the same hypothesis that we used in part II, and she constructs the same operators P, Q, L1, and T. However, at this point she proceeds to derive a different pair of auxiliary and bifurcation equations. Note that if xCS is a solution to Equation (1"), then applying T we have T(I + C)x + TR(x) = Ty, or x-Qx + TR(x) = Ty. Letting x* = Qxc (L) and z = x-x*, this last equation can be written as z + TR(x*+z) = Ty; applying the operator I-Q we obtain the equation z + (I - Q)TR(x*+z) = (I - Q)Ty. (12) Equation (12) serves as the auxiliary equation in Cronin's theory; it is to be solved for z = z(y,x*) with x*E-(L), yES. Observe that

122 if z = z(y,x*) is a solution to Equation (12), then Qz = O, (I Q)z z, and T[L(x*+z) + R(x*+z) - y] = (x*+z) - Q(x*+z) + TR(x*+z) - Ty = z + TR(x*+z) - Ty (I - Q) [z + TR(x*+z) - Ty] + Q[z + TR(x*+z) - Ty] QTR(x*+z) - QTy. Since T is 1-1, it follows that x = x*+z is a solution to Equation (1") iff QTR(x*+z) - QTy = O, or QTR(x* + z(y,x*)) - QTy - 0. (13) Equation (13) is the bifurcation equation used by Cronin. For fixed y one only needs to determine a solution x*cE(L) to Equation (13) in order to obtain a solution x = x* + z(y,x*) to Equation (l").

BIBLI OGRAPHY 1. N. I. Akhiezer and L. M. Glazman, Theory of Linear Operators in Hilbert Space, I and II, F. Ungar, New York, 1961. 2. R. G. Bartle, Singular points of functional equations, Trans. Amer. Math. Soc. 75 (1953), 336-384. 3. L. Cesari, Functional analysis and Galerkin's method, Mich. Math. J. 11 (1964), 385-414. 4. L. Cesari, A nonlinear problem in potential theory, Report No. 2, N.S.Fo Research Project GP-57. 5. L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1, J. Wiley, New York, 1963, 149-187. 6. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. 7. J. Cronin, Fixed Points and TopologicalDegree in Nonlinear Analysis, Math. Surveys, No. 11, Amer. Math. Soc., Providence, R.I., 1964. 8. J. Cronin, Branch points of solutions of equations in Banach space, Trans. Amer. Math. Soc. 69 (1950), 208-231. 9. N. Dunford and J. T. Schwartz, Linear Operators, I and II, Interscience, New York, 1958. 10. K. 0. Friedrichs, Special Topics in Analysis, New York University lecture notes, 1953-1954. 11. Ko 0o Friedrichs, Lectures on Advanced Ordinary Differential Equations, New York University, 1948-1949. 12. I. M. Glazman, On the theory of singular differential operators, Amer. Math. Soc. Translations (1) 4 (1962), 331-372. 13. M. Golomb, Zur Theorie der nichtlinearen Integralgleichungssysteme und allgemeinen Funktionalgleichungen, Math. Z. 39 (1935), 45-75. 14. M. Golomb, Uber Systeme von nichtlinearen Integralgleichungen, Publications Mathematiques de L'Universite de Belgrade 5 (1936), 52-83. 123

124 BIBLIOGRAPHY (Concluded) 15. I. Halperin, Closures and adjoints of linear differential operators, Ann. of Math. (2) 38 (1937), 880-919. 16. E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soco Colloq. Pub., Vol. 31, Revised Edition, Providence, R.I., 1957. 17. M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964. 18. M. A. Neumark, Lineare Differentialoperatoren, Akademie-Verlag, Berlin, 1963. 19. L. Nirenberg, Functional Analysis, New York University Lectures, 1960- 1961. 20. E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann. 65 (1907-1908), 370-399. 21. J. Schwartz, Perturbations of spectral operators, and applications I, Pacific J. Math., 4 (1954), 415-458. 22. M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, Amer. Math. Soc. Colloq. Pub., Vol. 15, New York, 1932. 23. A. E. Taylor, Introduction to Functional Analysis, J. Wiley, New York, 1961. 24. A. C. Zaanen, An Introduction to the Theory of Integration, NorthHolland, Amsterdam, 1958.