EXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS. 1. Introduction

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1 EXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS DANIEL MARONCELLI AND JESÚS RODRÍGUEZ Abstract. In this work we provide conditions for the existence of solutions to nonlinear Sturm-Liouville problems of the form (p(t)x (t)) + q(t)x(t) + λx(t) = f(x(t)) subject to non-local boundary conditions ax() + bx () = η (x) cx() + dx () = η (x). Our approach will be topological utilizing Schaefer s fixed point theorem the Lyapunov-Schmidt procedure.. Introduction In this paper we provide criteria for the solvability of nonlinear Sturm-Liouville problems of the form () (p(t)x (t)) + q(t)x(t) + λx(t) = f(x(t)) t [ ] subject to non-local boundary conditions () ax() + bx () = η (x) cx() + dx () = η (x). There are several stard ways in which one may define a solution to problem ()-() so to maintain completeness we mention that in this paper we will be interested in proving the existence of classical solutions to ()-(). Formally by a solution to ()-() we mean a function x : [ ] R such that px is continuously differentiable ()-() are satisfied. Throughout our analysis we will assume that p q : [ ] R are continuous p(t) > for all t [ ] a + b > c + d > λ is an eigenvalue of the associated linear Sturm-Liouville problem f : R R is continuous for i = η i (x) = [] g i(x)dµ i where g g : R R are continuous µ µ are finite Borel measures on [ ]. The focus of this paper is the analysis of nonlinear Sturm-Liouville problems at resonance subject to non-local boundary conditions where by resonance we mean that the linear homogeneous problem (7)-(8) has nontrivial solutions. Since the pioneering work of Lesman-Lazer [] much has been written about resonant nonlinear Sturm-Liouville boundary value problems with linear boundary conditions. Pertinent references from the point of view of this paper are [ ]. Less has been said in regard to problems with nonlocal boundary conditions even for the case of nonresonance; readers ineterested in results in this direction may consult [ ]. Corresponding Author.

2 D. MARONCELLI AND J. RODRÍGUEZ The novelty of this work is due in large part to the generality of the nonlinear boundary conditions η η. As an important special case we point out that by taking µ µ to be point-supported measures our integral boundary conditions allow for nonlinear multipoint boundary conditions of the form n η (x) = f k (x(t k )) η (x) = k= m h j (x(t j )) where each f k h j is a continuous function each t k t j [ ]. j= Our main result Theorem 3. provides conditions for the existence of solutions to ()-() under a suitable interaction of the eigenspace of the linear Sturm-Liouville problem the nonlinearities in both the differential equation the boundary conditions. We would like to remark that the result we obtain in Theorem 3. constitutes a significant extension of the work found in [5] by allowing for much more generality in the boundary conditions ().. Preliminaries The nonlinear boundary value problem ()-() will be viewed as an operator equation. We let C := C[ ] denote the space of real-valued continuous functions topologized by the supremum norm C. As usual L := L [ ] will denote the space of real-valued square-integrable functions defined on [ ]. The topology on L will be that induced by the stard L -norm L. We use H to denote the Sobelov space of functions with two weak derivatives in L ; that is H = {x L x is absolutely continuous x L }. Unless otherwise stated the topology on H will be the subspace topology inherited from L. However we will on several occasions toplologize H with the Sobelov norm x H = x L + x L + x L. On occasion we may also view H as a subspace of C. We will use to denote the Euclidean norm on R S R will denote the inner products on L H R respectively. Weak convergence in L will be denoted by weak convergence in the Sobelov space H will be denoted by. S We make L R an inner product space with inner product (3) h w g v w v := m ( ) w v h g + w v R where m is a positive constant which will be chosen later we will use L R to denote the norm generated by this inner product. Lastly we give C R the product topology we will use C R to denote the stard product norm on this space. Linear boundary operators B B will be defined as follows: B : H R is given by

3 STURM-LIOUVILLE BVP S 3 B : H R is given by We define L : H L R where A : H L is defined by B x = ax() + bx () B x = cx() + dx (). Lx = Ax B x B x Ax(t) = (p(t)x (t)) + (q(t) + λ)x(t). Similarly we define a nonlinear operator G : H L R by F(x) G(x) = η (x) η (x) where F(x)(t) = f(x(t)) as before for i = η i (x) = g i (x)dµ i. Solving the nonlinear boundary value problem ()-() is now equivalent to solving (4) Lx = G(x). The study of the nonlinear boundary value problem ()-() will be intimately related to the linear nonhomogeneous boundary value problem [] (5) (p(t)x (t)) + q(t)x(t) + λx(t) = h(t) t [ ] (6) ax() + bx () = w cx() + dx () = w where h is an element of L w w are elements of R. Using our notation from above we have that solving (5)-(6) is equivalent to solving h Lx = w. w We begin our study of the nonlinear boundary value problem ()-() by analyzing (5)-(6). To aid in this analysis we first recall some well-known facts regarding the linear homogeneous Sturm-Liouville problem (7) (p(t)x (t)) + q(t)x(t) + λx(t) = (8) ax() + bx () = cx() + dx () =. For those readers interested in a more detailed introduction to linear Sturm-Liouville problems we suggest [9]. It is well known that λ is a simple eigenvalue; that is Ker(L) is one-dimensional. We may therefore choose a vector ψ as a basis for Ker(L). Without loss of generality we will assume ψ L =. Since (7) is a second-order linear homogeneous differential equation we may choose φ satisfying (7) so that {ψ φ} forms a basis for the solution space of this linear homogeneous problem. We will assume ψ φ =.

4 4 D. MARONCELLI AND J. RODRÍGUEZ For u v H let wr(u v) denote the Wrońskian of u v; that is wr(u v) = uv vu. It follows from stard ode theory that if u v are linearly independent solutions to (7) then p wr(u v) is a nonzero constant. We will assume that φ has be chosen so that p wr(ψ φ) = define ω : [ ] [ ] R by { ψ(t)φ(s) if t s (9) ω(t s) = ψ(s)φ(t) if s t. As a reminder to the reader ω is often referred to as a fundamental solution of (7). If we define K : L H by () Kh(t) = ω(t s)h(s)ds then it is easy to verify that K is self-adjoint compact satisfies AKh = h for every h L. Differentiating under the integral symbol one easily establishes that for every h L B Kh = h φ B ψ = B Kh = h ψ B φ. Let v = B φ v = B φ. Since φ satisfies (7) is linearly independent of ψ we must have B φ B φ ; this is a consequence of the uniqueness of solutions to initial value problems that fact the linear Sturm-Liouville boundary conditions can be thought of as an orthogonality condition. With the above ideas in h we are now in a position characterize the range of L. We have the following result. Proposition.. Let h L w w R. Then h h := w Im(L) if w only if h ψ = where ψ ψ := v. That is in L R Im(L) = { ψ}. v h Proof. Lx = if only if Ax = h B x = w B x = w. However w w Ax = h if only if x = c ψ + c φ + Kh for some real numbers c c. Applying the boundary map B recalling B Kh = we get B (c ψ + c φ + Kh) = c v. Similarly using B Kh = h ψ B φ we get B (c ψ +c φ+kh) = (c + h ψ )v. Now if only if c v = w (c + h ψ )v = w c = w v h ψ = w v w v = which happens if only if h ψ =. [ w w ] [ ] v v R With this characterization of the Im(L) in h we make the following definitions which will play a crucial role in our ability to analyze the nonlinear Sturm- Liouville problem ()-() using a projection scheme.

5 STURM-LIOUVILLE BVP S 5 Definition.. Define P : L L by P x = x ψ ψ. It is clear that P is the orthogonal projection onto Ker(L). Now choose m see (3) to be [ ] + v. With this choice of m ψ is a unit vector in L R. v Definition.3. Define Q: L R L R by Q h w = h w ψ ψ. w w From Proposition. we have that Q is the orthogonal projection of L R on Im(L). Thus I Q is a projection onto the Im(L). In our analysis of the nonlinear Sturm-Liouville problem we will use a projection scheme often referred to as the Lyapunov-Schmidt procedure. The use of the Lyapunov-Schmidt reduction will allow us to write the operator equation (4) as an equivalent equation in which a fixed point argument may be applied to prove the existence of solutions. Interested readers may consult [3 6] for a more detailed account of these ideas. Proposition.4. Solving Lx = G(x) is equivalent to solving the system (I P )x M(I Q)G(x) = [ ] [ ] η (x) v ( F(x) ψ + η (x) v R )ψ = where M denotes (L H Ker(L) ). Proof. Lx = G(x) { (I Q)(Lx G(x)) = Q(Lx G(x)) = Lx (I Q)G(x) = QG(x) = MLx M(I Q)G(x) = QG(x) = (I P )x M(I Q)G(x) = F(x) ψ η (x) v ψ = η (x) v

6 6 D. MARONCELLI AND J. RODRÍGUEZ (I P )x M(I Q)G(x) = F(x) ψ η (x) v ψ = η (x) v (I P )x M(I Q)G(x) = [ ] [ ] η (x) v. ( F(x) ψ + η (x) v R )ψ = 3. Main Results We now come to our main result. In what follows we will assume that the nonlinear integral boundary operators η η are induced by bounded continuous functions g g. To simplify the statement of the theorem we introduce the following notation. For i = we let g i+ (+ ) := lim sup g i (x) x g i (+ ) := lim inf g i(x) x g i+ ( ) := lim sup g i (x) x g i ( ) := lim inf g i(x). x We define O := {t ψ(t) = } O + := {t ψ(t) > } O := {t ψ(t) < }. From Stard Sturm-Liouville theory we have that O is a finite set consisting of simple zeros. In what follows this fact will be used several times possibly without explicit mention. Finally for i = we let J i± = g i± (+ )µ i (O + ) + g i± ( )µ i (O ). Theorem 3.. Suppose that the following conditions hold: C. The function f is sublinear ; that is there exists real numbers M M β with β < such that for every x R f(x) M x β + M ; C. There exist positive real numbers ẑ J such that for all z > ẑ f( z) J < < J f(z); C3. For i = µ i (O ) = where again u i is the Borel measure in the definition of the boundary operator η i ; Jsgn( v) v C4. J ψ dt < R where for a real number v J sgn(v) v sgn(v) = + if v > sgn(v) = if v < ; then there exists a solution to ()-().

7 STURM-LIOUVILLE BVP S 7 Proof. We start by defining T : L H by η (x) v T (x) = P x ( F(x) ψ + η (x) v R )ψ + M(I Q)G(x). From Proposition.4 we have that the solutions to ()-() are the fixed points of T. Since M is an integral mapping from L into H it is compact thus so is T. We will show that F P := {x H x = γt (x) for some γ ( )} is a priori bounded in L. A fixed point will then follow from an application of Schaefer s fixed point theorem. To this end suppose that there exist sequences {x n } n N {γ n } n N in H ( ) respectively with x n L x n = γ n T (x n ). Let y n = x n. x n H Since the closed unit ball of Sobelov space H is weakly compact by going to a S subsequence if necessary we may assume that y n y for some y H. Again going to a subsequence if necessary we may assume that γ n converges to some γ [ ]. Now x n y n = x n H T (x n ) = γ n x n H η (x P x n ( F(x n ) ψ + n ) v η (x n ) v R )ψ + M(I Q)G(x n ) = γ n. x n H Since f is sublinear (See C) g g are bounded it follows that () G(x) L R K x β L + K () G(x) C R K x β C + K for some positive real numbers K K every x H. Thus from () η (x ( F(x n ) ψ + n ) v η (x n ) v R )ψ + M(I Q)G(x n ) γ n x n H so that [ ] η (x P x n ( F(x n ) ψ + n ) η (x n ) γ n x n H [ ] v v R )ψ + M(I Q)G(x n ) Since y n S y yn y so that we conclude y = γp y. Applying P gives P y = γp y = γp y γp y.

8 8 D. MARONCELLI AND J. RODRÍGUEZ from which we deduce that γ = or P y =. Since y H = it follows that γ =. Thus P y = y we deduce that y = ± ψ. We will assume that ψ H y = ψ as the other case is similar. ψ H Now by the compact embedding of H S in C we have since y n y that yn y in C. Using the fact that y n ψ we have that ψ H (3) y n ψ ψ ψ = ψ H. ψ H However x n ψ = x n H y n ψ so that x n ψ since x n H does (recall x n L ). Without loss of generality we will assume from now on that x n ψ > for each n. From x n = γ n T (x n ) it follows that for each n (I P )x n = γ n M(I Q)G(x n ) η (x P x n = γ n P x n γ n ( F(x n ) ψ + n ) v η (x n ) v R )ψ. This is equivalent to (4) (I P )x n = γ n M(I Q)G(x n ) η (x (5) ( γ n ) x n ψ + γ n ( F(x n ) ψ + n ) v η (x n ) v R ) =. Let v n denote (I P )x n. From () (4) we have that v n C γ n M(I Q) (K x n β C + K ) D x n β C + D where M(I Q) denotes the operator norm of M(I Q) for i = D i = M(I Q) K i. Applying the compact embedding theorem again we may assume by scaling each D i that v n C D x n β H + D. However from (3) we have that x n ψ x n H time we may assume (6) v n C D x n ψ β + D. For the moment fix t O + O. Since we have using (6) that x n (t) x n ψ ψ(t) v n (t) so that by rescaling one more ψ H x n ψ ψ(t) v n C (7) lim x n(t) = ± whenever t O ±.

9 STURM-LIOUVILLE BVP S 9 Define E n = {t ψ (t) ε n } where ε n = ẑ + v n C x n ψ. If t E n then x n (t) x n ψ ψ(t) v n (t) x n ψ ψ(t) v n C x n ψ (ẑ + vn C x n ψ ) v n C = ẑ. This gives using C that f(x n )ψdt = f(x n )ψdt + f(x n )ψdt E n En c J ψ dt + f(x n )ψdt E n En c J ψ dt f(x n )ψ dt E n En c We claim that f(x n )ψ dt so that by Lebesgue s Dominated Convergence Theorem (8) To see that E c n E c n lim inf It then follows from C that f(x n )ψdt lim inf J ψ dt E n = J ψ dt. f(x n )ψ dt first note that for any t E c n x n (t) x n ψ ε n + v n C x n ψ (ẑ + vn C x n ψ ) + v n C = ẑ + v n C ẑ + (D x n ψ β + D ) (Using (6)). f(x n )(t) M x n (t) β + M M (ẑ + (D x n ψ β + D )) β + M which gives that f(x n )ψ dt (M (ẑ + (D x n ψ β + D )) β + M )ε n µ L (En) c E c n where µ L denotes Lebesgue measure on [ ].

10 D. MARONCELLI AND J. RODRÍGUEZ v n Since C we have that En c O. Further since O consists of x n ψ finitely many simple zeros it follows from the Mean Value Theorem that there exists a positive constant say L with µ L (E c n) Lε n. We then have that f(x n )ψ dt (M (ẑ + (D x n ψ β + D )) β + M )Lε n E c n + = (M (ẑ + (D x n ψ β + D )) β vn ) + M )L(ẑ C x n ψ (ẑ (M (ẑ + (D x n ψ β + + D )) β D x n ψ β + M )L + D ) x n ψ so that f(x n )ψ dt R x n ψ β E x n c n ψ for some positive constant R. Letting n using the fact that β < we conclude that f(x n )ψ dt. E c n We now look to analyze lim inf. From (7) if t O + then g i (x n )dµ i lim sup g i (+ ) lim inf g i(x n )(t) lim sup g i (x n )(t) g i+ (+ ). Similarly for each t O each i i = g i ( ) lim inf g i(x n )(t) lim sup g i (x n )(t) g i+ ( ). g i (x n )dµ i for i = Since g g are bounded we have by Fatou s lemma that for each i (9) J i = g i (+ )µ i (O + ) + g i ( )µ i (O ) = g i (+ )dµ i + g i ( )dµ i O + O lim inf g i(x n )dµ i O + O = lim inf g i(x n )dµ i (Using C3) [] lim inf g i (x n )dµ i [] lim sup g i (x n )dµ i [] [] lim sup g i (x n )dµ i lim sup O + O g i (x n )dµ i

11 STURM-LIOUVILLE BVP S g i+ (+ )dµ i + O + g i ( )dµ i O = g i+ (+ )µ i (O + ) + g i+ ( )µ i (O ) = J i+. Suppose for the moment that v > v > let s r be positive real numbers. Using the definitions of limit inferior limit superior see (8) (9) there exists an n s an n r such that if n n s then () J if n n r then () J i r < ψ dt s < f(x n ) ψ = F(x n ) ψ [] g i (x n )dµ i < J i+ + r. Since v > v > it follows that () [ J r v J r v R [] g ] [ ] [ (x n )dµ v [] g J+ + r (x n )dµ v R J + + r However (3) [ J J ] [ ] v v R = [ Jsgn( v) J sgn(v) ] [ ] v v R > J ] ψ dt. [ ] v v R. Thus it follows from ()() () (3) that we may choose r s small enough so that η (x (4) F(x n ) ψ + n ) v η (x n ) v R > for large enough n. The other cases for the sign of v v are similar. In each case the conclusion in (4) holds. Recalling that x n ψ + we have that for large enough n η (x ( γ n ) x n ψ + γ n ( F(x n ) ψ + n ) v η (x n ) v R ) >. However this contradicts the fact that by (5) η (x ( γ n ) x n ψ + γ n ( F(x n ) ψ + n ) v η (x n ) v R ) =. Thus F P := {x H x = γt (x) for some γ ( )} must be a priori bounded the proof is complete. Remark 3.. If η = η = then by choosing for each i i = g i = µ i to be Lebesgue measure on [ ] we have that J i± =. Thus condition C4 of Theorem 3. is trivially satisfied. This shows that Theorem 3. is a generalization of the result found in [5] where they analyze linear homogeneous boundary conditions. The following corollary isolates the special case in which the boundary operators η η are generated by bounded continuous function g g for which we assume that for i = g i (± ) := lim x ± g i (x) exists.

12 D. MARONCELLI AND J. RODRÍGUEZ Corollary 3.3. Suppose that the following conditions hold: C*. The function f is sublinear ; that is there exists real numbers M M β with β < such that for every x R f(x) M x β + M ; C*. There exist positive real numbers ẑ J such that for all z > ẑ f( z) J < < J f(z); C3*. For i = µ i (O ) = where again u i is the Borel measure in the definition of the boundary operator η i ; C4*. For i = g i (± ) := lim x ± g i (x) exists; J+ v C5*. J ψ dt < J + v R ; then there exists a solution to ()-(). Proof. If for i = g i (± ) := lim x ± g i (x) exist then for each of these i J i = J i+. 4. Example In this section we give a concrete example of the application of our main result Theorem 3.. We will use an interval of [ π] to simplify calculations. Consider (5) x + m x = f(x(t)) subject to (6) x() = [π] g (x)du x(π) = [π] g (x)du where f g g are real-valued continuous functions with g g bounded. It is well-known that the L -normalized eigenfunctions corresponding to the Dirichlet problem x + m x = subject to boundary conditions x() = x(π) = are ± π sin(mt). We choose to take ψ(t) = sin(mt). This gives that φ see (9) is π π cos(mt). Thus v = φ() = π v = φ(π) = π. We also have that { j i= O + = ( iπ m (i+)π m ) if m = j + j i= ( iπ ) if m = j O = { j m (i+)π m i= ( (i+)π m j i= ( (i+)π m (i+)π (i+)π m ) if m = j + m ) if m = j.

13 STURM-LIOUVILLE BVP S 3 Suppose for the moment that conditions C-C3 hold since these can be trivially satisfied by any number of choices for f µ µ. Condition C4 of Theorem 3. in this specific problem becomes 4 [ ] π J < J+ J + [ ] π R π which is equivalent to (J + + J + ) < J. It is clear that there are several bounded continuous functions g g Borel measures µ µ which make the above inequality valid. As a concrete example let E m = {t sin(mt) = } fix t E m. Take µ := µ = µ to be the measure point-supported at t ; that is for a subset A of [ ] { if t A µ(a) = if t / A. Since t E m we have that t is in O + or O. If for each i i = g i (± ) := lim x ± g i (x) exists then when t O + J i+ = g i (+ ). Similarly when t O then J i+ = g i ( ). Thus if t O ± then provided g (± ) + g (± ) < J we have from Corollary 3.3 that the nonlinear boundary value problem (5)-(6) has a solution. It is interesting to note that if t / m E m both g (+ )+g (+ ) < J g ( ) + g ( ) < J then (5)-(6) has a solution for all eigenvalues m. References [] A. Boucherif. Second-order boundary value problems with integral boundary conditions. Nonlinear Anal [] X. Chen Z. Du. Existence of Positive Periodic Solutions for a neutral delay predator-prey model with Hassell-Varley type functional response impulse. Qual. Theory Dyn. Syst. 7 DOI:.7/s [3] S. Chow J.K. Hale Methods of Bifurcation Theory Springer Berlin 98. [4] P. Drábek. On the resonance problem with nonlinearity which has arbitrary linear growth. J. Math. Anal. Appl. 7: [5] P. Drábek. Lesman-Lazer type condition nonlinearities with linear growth. Czechoslovak Math. J. 4: [6] P. Drábek S. B. Robinson. Resonance problems for the p-laplacian. J. Funct. Anal. 69: [7] Z.i Du J. Yin. A Second Order Differential Equation with Generalized Sturm- Liouville Integral Boundary Conditions at Resonance. Filomat 8: [8] R. Iannacci M.N. Nkashama. Nonlinear two point boundary value problems without Lesman-Lazer condition. Proc. Amer. Math. Soc. 6: [9] W. G. Kelley A.C. Peterson The theory of differential Equations Springer. [] R.A. Khan. The generalized method of quasilinearization nonlinear boundary value problems with integral boundary conditions. Electronic. J. Qual. Theory Differ. Equ. -5. [] E. M. Lesman A. C. Lazer. Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 9 9: [] A. C. Lazer D. F. Leach. Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pure Appl. 8: [3] X. Lin Q. Zhang. Existence of solution for a p-laplacian multi-point boundary value problem at resonance. Qual. Theory Dyn. Syst. 7 DOI:.7/s [4] R. Ma. Nonlinear discrete Sturm-Liouville problems at resonance. Nonlinear Anal. 67:

14 4 D. MARONCELLI AND J. RODRÍGUEZ [5] D. Maroncelli J. Rodríguez. Existence theory for nonlinear Sturm-Liouville problems with unbounded nonlinearities. Differ. Equ. Appl. 6(4): [6] D. Maroncelli J. Rodríguez. On the solvability of nonlinear impulsive boundary value problems. Topol. Methods Nonlinear Anal. 44(): [7] H. Pang Y. Tong. Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 3(5) [8] Jesús F. Rodríguez. Existence theory for nonlinear eigenvalue problems. Appl. Anal. 87: no. 3. [9] J. Rodríguez. Nonlinear discrete Sturm-Liouville problems. J. Math. Anal. Appl. 38 Issue : [] J. Rodríguez A. Suarez. Existence of solutions to nonlinear boundary value problems. Differ. Equ. Appl. 9():- 7. [] J. Rodríguez A. Suarez. On nonlinear perturbations of Sturm-Liouville problems in discrete continuous settings. Differ. Equ. Appl. 8(3): [] J. Rodríguez Z. Abernathy. On the solvability of nonlinear Sturm-Liouville problems. J. Math. Anal. Appl. 387:3 39. [3] X.M. Zhang W.G. Ge. Positive solutions for a class of boundary-value problems with single integral boundary conditions. Computers Mathematics with Applications. 58() (D. Maroncelli) Department of Mathematics College of Charleston Charleston SC 944- U.S.A (J. Rodríguez) Department of Mathematics Box 85 North Carolina State University Raleigh NC U.S.A address D. Maroncelli: maroncellidm@cofc.edu address J. Rodríguez: rodrigu@ncsu.edu

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