We study the semilinear second order ODE u00 + g(t, u) = 0 under the following Sturm- Liouville boundary condition au(0) + bu0(0) = u0 and cu(T) + du0(T) = uT . We obtain solutions by topological methods. Moreover, we show that a solution may be constructed recursively, under appropriate conditions

Amster, Pablo

Cárdenas, Pedro P.

English

Biblioteca Digital de la Universidad del Valle

Vol. XV No 1 Junio (2007)Matemáticas: 311Matemáticas:Enseñanza Universitariac©Escuela Regional de MatemáticasUniversidad del Valle - ColombiaSturm-Liouville boundary conditions for a second order ODEPablo Amster Pedro P. CárdenasReceived Sept. 28, 2006 Accepted Feb. 2, 2007AbstractWe study the semilinear second order ODE u′′ + g(t, u) = 0 under the following Sturm-Liouville boundary condition au(0) + bu′(0) = u0 and cu(T ) + du′(T ) = uT . We obtainsolutions by topological methods. Moreover, we show that a solution may be constructedrecursively, under appropriate conditions.Keywords: Sturm-Liouville boundary conditions - Topological methodsMSC(2000): 34B151 IntroductionWe study the semilinear second order problemu′′ + g(t, u) = 0au(0) + bu′(0) = u0cu(T ) + du′(T ) = uT(1)with g : [0, T ]×R→ R continuous, and ad−bc 6= 0. Problems of this kind havebeen considered since the fifties by, among others, Ehrmann [4] and Struwe [7]using shooting arguments, and by Bahri-Berestycki [1], Rabinowitz [6], usingcritical point theory. In the nineties, Capietto, Henrard, Mawhin and Zanolin[2], [3] applied the Leray-Schauder continuation method for a nonlinearity ofthe type g = g1(u)+ p(t, u, u′), where g1 is superlinear and p satisfies a lineargrowth condition.Throughout the paper, we shall assume that all the eigenvalues {λn}n∈Nof the problem−u′′ = λu, au(0) + bu′(0) = cu(T ) + du′(T ) = 0are non-negative. Writing u = γert + δe−rt as a possible eigenfunction (cor-responding to an eigenvalue λ = −r2), it is easy to verify that the previousnon-negativity assumption is equivalent to the following condition:(a+ br)(c− dr) 6= (a− br)(c+ dr)e2rT for r > 0. (2)If furthermore ad − bc + acT 6= 0, then λ1 > 0, and the problem is callednon-resonant. On the other hand, if ad − bc + acT = 0, then λ1 = 0. This4 P. Amster and P. Cárdenassituation corresponds to the resonant case, for which a simple computationshows that the corresponding (normalized) eigenfunction ϕ1 is given byϕ1(t) =(a2T 33− abT 2 + b2T)−1/2(b− at). (3)We shall prove the existence of solutions of (1) by topological methods.More precisely, for the non-resonant case we obtain in section 2.1 an existenceresult under a linear growth condition on g using Schauder's fixed point theo-rem. On the other hand, we shall prove the existence of at least one solutionwhen g is subquadratic and satisfies the one-sided growth conditiong(t, u)− g(t, v)u− v ≤ γ < λ1. (4)We recall that the first eigenvalue can be computed by the Rayleigh quotient:λ1 = infu∈E−{0}− ∫ T0 u′′(t)u(t)dt∫ T0 u2(t)dt(5)with E = {u ∈ H2(0, T ) : au(0) + bu′(0) = cu(T ) + du′(T ) = 0}.In section 2.2 we shall embed problem (1) in a family (1)σ of problemswith a parameter σ ∈ [0, 1]. Thus, starting at a solution uσ for some σ < 1we shall define recursively a sequence which converges to a solution of (1)σ+εfor some appropriate step ε. In particular, when ε does not depend on uσ,we obtain recursively solutions for 0 = σ0 < σ1 < . . . < σN = 1, which givesa solution of the original problem. Finally, in section 3 we obtain solutionsfor the resonant case under the so-called Landesman-Lazer type conditions.Remark 1.1. For simplicity, we consider only the case g = g(t, u), althoughthe methods presented in this paper can be extended to the non-variationalcase g = g(t, u, u′).2 The non-resonant caseIn this section we study the non-resonant case, in which conditionad− bc+ acT 6= 0 (6)holds. In section 2.1 we establish two existence results by topological methods,and in section 2.2 we define an iterative scheme that converges to a solutionof (1).Sturm-Liouville boundary conditions for a second order ODE 52.1 Solutions by fixed point methodsWe shall define a fixed point operator in order to obtain solutions of (1) bytopological methods, under the assumption ad − bc + acT 6= 0. In this case,for any θ ∈ L2(0, T ) there exists a unique solution of the problemu′′ = θ, au(0) + bu′(0) = cu(T ) + du′(T ) = 0given by the integral formulau(t) =∫ T0G(t, s)θ(s)ds,where G is the following Green function:G(t, s) =(b− at)(c(T − s) + d)ad− bc+ acT +max{t− s, 0}.Thus, the solutions of (1) can be regarded as fixed points of the operator Tgiven byTu(t) = αt+ β −∫ T0G(t, s)g(s, u(s))ds, (7)whereα =auT − cu0ad− bc+ acT , β =(cT + d)u0 − buTad− bc+ acT .Thus we obtain:Theorem 2.1. Let (2) and (6) hold, and assume that |g(t, u)| ≤ k|u| + l,with k < λ1. Then problem (1) admits at least one solution.Proof. From the assumption on g, it follows that T : L2(0, T ) → L2(0, T ) iswell defined. Furthermore, by Arzelá-Ascoli's Theorem we deduce that T iscompact. Moreover, from the Rayleigh quotient (5) we get, for fixed u˜:‖Tu− T u˜‖L2 ≤1λ1‖(Tu− T u˜)′′‖L2 =1λ1‖g(·, u)− g(·, u˜)‖L2 ≤kλ1‖u‖L2 + sfor some constant s ≥ 0. Thus, for R large enough we conclude thatT (BR(0)) ⊂ BR(0), and the proof follows from Schauder's Fixed Point theo-rem.Theorem 2.2. Let (2) and (6) hold. Further, assume that g satisfies (4),and that |g(t, u)| ≤ k|u|r + l for some constants k, l and some r < 2. Thenproblem (1) admits a unique solution.6 P. Amster and P. CárdenasProof. From the assumptions, if u ∈ L2(0, T ) then g(·, u) ∈ Lp(0, T ) for somep > 1, and the operator T : L2(0, T )→ L2(0, T ) given by (7) is well defined.Moreover, if u = σTu for some σ ∈ [0, 1], thenSσu := u′′+σg(t, u) = 0, au(0)+ bu′(0) = σu0, cu(T )+ du′(T ) = σuT .Let u˜ ∈ H2(0, T ) satisfy au˜(0) + bu˜′(0) = σu0, cu˜(T ) + du˜′(T ) = σuT . Then:‖Sσu− Sσu˜‖L2‖u− u˜‖L2 ≥ −∫ T0(Sσu− Sσu˜)(u− u˜)dt≥ λ1‖u− u˜‖2L2 −∫ T0(g(t, u)− g(t, u˜))(u− u˜)dt ≥ (λ1 − γ)‖u− u˜‖2L2 .It follows that‖u− u˜‖L2 ≤1λ1 − γ ‖Sσu− Sσu˜‖L2 =1λ1 − γ ‖Sσu˜‖L2 .Thus, if we fix z ∈ H2(0, T ) such that az(0)+bz′(0) = u0, cz(T )+dz′(T ) = uT ,then setting u˜ = σz we obtain:‖u− σz‖L2 ≤σλ1 − γ ‖z′′ + g(·, σz)‖L2 ≤ Cfor some constant C independent of σ. This implies that all solutions of theproblem u = σTu satisfy ‖u‖L2 ≤M for some constant M , and the existenceof a fixed point of T follows from the Leray-Schauder theorem (see e.g. [5]).Finally, if u and u˜ are solutions of (1), then S1u = S1u˜ = 0. As before,‖u− u˜‖L2 ≤1λ1 − γ ‖S1u− S1u˜‖L2 = 0.2.2 An iterative procedure for problem (1)In what follows of this section we shall embed problem (1) in a family ofproblems(1)σu′′(t) + σg(t, u) = 0au(0) + bu′(0) = u0cu(T ) + du′(T ) = uT .Starting at a solution uσ for σ < 1 we shall define recursively a sequence thatconverges to a solution of (1)σ+ε for some step ε ≤ 1− σ.As a basic assumption, we shall assume that g is C2 with respect to u,and ∂g∂u ≤ γ < λ1. In particular, note that (4) holds.Sturm-Liouville boundary conditions for a second order ODE 7Let uσ be a solution of (1)σ and consider the sequence {un} ⊂ H2(0, T )given recursively by u1 = uσ, and un+1 the unique solution of the linearproblem: u′′n+1 + (σ + ε)(g(t, un) + ∂g∂u(t, un)(un+1 − un))= 0aun+1(0) + bu′n+1(0) = u0cun+1(T ) + du′n+1(T ) = uT .(8)From the Fredholm alternative for linear operators (and also as a particularcase of Theorem 2.2) sequence {un} is well defined. Moreover, if un → u inthe L2-norm, then it is easy to see that u is a solution of (1)σ+ε.Let zn = un+1 − un, then for n ≥ 2 we have:z′′n+(σ+ε)∂g∂u(t, un)zn = −(σ+ε)[g(t, un)−g(t, un−1)−∂g∂u(t, un−1)(un−un−1)]= −12(σ + ε)∂2g∂u2(t, ξ)z2n−1for some mean value ξ(t) between un(t) and un−1(t). Then, for some constantµ (independent of σ):‖zn‖H1 ≤ µ∥∥∥∥z′′n + (σ + ε)∂g∂u(·, un)zn∥∥∥∥L2≤ µ2∥∥∥∥∂2g∂u2 (·, ξ)z2n−1∥∥∥∥L2≤ Cn‖zn−1‖2H1for some constant Cn. In particular, if∂2g∂u2is bounded, we may considerCn = C := µν2 ‖ ∂2g∂u2‖L∞ for every n, where ν is the constant of the imbeddingH1(0, T ) ↪→ L4(0, T ). On the other hand,z′′1 + (σ + ε)∂g∂u(t, u1)z1 = −u′′1 − (σ + ε)g(t, u1) = −εg(t, u1),whence ‖z1‖H1 ≤ µε‖g(·, u1)‖L2 . Thus we obtain:Theorem 2.3. Assume that (2) and (6) hold, and let u1 = uσ be a solutionof (1)σ for some σ ∈ [0, 1). Furthermore, assume that ∂g∂u ≤ γ < λ1 for someconstant γ, and that ∂2g∂u2is bounded. Then the iterative scheme defined by (8)converges to a solution of (1)σ+ε, provided that µεC‖g(·, uσ)‖L2 < 1, with Cand µ as before.Proof. From the previous computations, we deduce that‖zn+1‖H1 ≤ C2n−1‖z1‖2nH1 ≤1C(µεC‖g(·, uσ)‖L2)2n.Then {un} is a Cauchy sequence in H1(0, T ), and the proof follows.8 P. Amster and P. CárdenasCorollary 2.4. Let the assumptions of the previous theorem hold. Further,assume that g is bounded. Then the step ε in the iterative scheme defined by(8) can be chosen independently of σ. In particular, there exists a sequence0 = σ0 < σ1 < . . . < σN = 1, with uσj solution of (1)σj constructed recursivelyfrom (8), and uσN is a solution of (1).3 Resonant case: Landesman-Lazer type conditionsIn this section we study problem (1) for u0 = uT = 0 under the assumptionof resonance at the first eigenvalue λ1 = 0; namely, we consider the case inwhich the conditionad− bc+ acT = 0 (9)holds. The proof of following lemma is straightforward:Lemma 3.1. Assume that (2) and (9) hold. Let E ⊂ C2([0, T ]) and F ⊂C([0, T ]) the closed subspaces defined byE = {u ∈ C2([0, T ]) : au(0) + bu′(0) = cu(T ) + du′(T ) = 0,∫ T0u(t)ϕ1(t)dt = 0}and F = {θ ∈ C([0, T ]) : ∫ T0 θ(t)ϕ1(t)dt = 0}. Then the continuous linearoperator L : E → F given by Lu = u′′ is bijective, and hence an isomorphism.In particular, there exists a constant γ such that ‖u‖C2 ≤ γ‖u′′‖C for everyu ∈ E.In order to introduce appropriate Landesman-Lazer conditions for ourproblem, we shall assume that the following limits exist:lims→±∞ g(t, sϕ1(t)) := g±(t). (10)Thus, the main result of this section reads:Theorem 3.2. Assume that (2) and (9) hold, and that the limits (10) exist.Then problem (1) for u0 = uT = 0 admits at least one solution, provided thatone of the following conditions holds:∫ T0g+(t)ϕ1(t)dt < 0 <∫ T0g−(t)ϕ1(t)dt, (11)∫ T0g−(t)ϕ1(t)dt < 0 <∫ T0g+(t)ϕ1(t)dt. (12)Sturm-Liouville boundary conditions for a second order ODE 9Proof. Let us first observe that, for σ > 0, problem{u′′ + σg(t, u) = 0au(0) + bu′(0) = cu(T ) + du′(T ) = 0 (13)is equivalent to the fixed point problemu = 〈u− g(·, u), ϕ1〉ϕ1 − σK(g(·, u)− 〈g(·, u), ϕ1〉ϕ1), (14)where K : F → E is the inverse of the mapping L defined in Lemma3.1, and 〈·, ·〉 denotes the usual inner product of L2(0, T ), namely 〈θ, ξ〉 =∫ T0 θ(t)ξ(t)dt. Indeed, if u is a solution of (13) then 〈u′′, ϕ1〉 = 〈u, ϕ′′1〉 = 0,which implies 〈g(·, u), ϕ1〉 = 0, andu− 〈u, ϕ1〉ϕ1 = −σK(g(·, u)).Conversely, if u solves (14) then u′′ = −σ [g(t, u)− 〈g(·, u), ϕ1〉ϕ1] .Moreover,〈u, ϕ1〉 = 〈u−g(·, u), ϕ1〉, and hence 〈g(·, u), ϕ1〉 = 0. Thus, it suffices to provethat (14) is solvable for σ = 1. On the other hand, observe that if σ = 0 then(14) is equivalent to the equalitiesu = kϕ1, 〈g(·, u), ϕ1〉 = 0.Let Tσ : C([0, T ])→ C([0, T ]) be the compact operator defined byTσu = 〈u− g(·, u), ϕ1〉ϕ1 − σK(g(·, u)− 〈g(·, u), ϕ1〉ϕ1),and consider Fσ(u) = u − Tσu. We claim that F1(u) = 0 for some u, whichcorresponds to a solution of the original problem. Indeed, we shall prove that1. Fσ(u) 6= 0 for ‖u‖C large, and σ ∈ [0, 1].2. degLS(F0, BR, 0) = ±1 for R large enough, where BR ⊂ C([0, T ]) is theball of radius R centered at 0 and degLS denotes the Leray-Schauderdegree.We remark that once 1 and 2 are proved, the result follows from the homo-topy invariance of the Leray-Schauder degree. In order to prove 1, assume firstthat Fσnun = 0, with ‖un‖C →∞ and σn ∈ (0, 1]. Then u′′n+σng(t, un) = 0,and hence0 = 〈u′′n, ϕ1〉 = −σn∫ T0g(t, un)ϕ1(t)dt.On the other hand, we may write un = vn+〈un, ϕ1〉ϕ1, and from the previouslemma‖vn‖C ≤ γ‖v′′n‖C = γ‖u′′n‖C ≤ γ‖g(·, un)‖C ≤M10 P. Amster and P. Cárdenasfor some constant M . We deduce that cn := 〈un, ϕ1〉 → ∞. Taking a subse-quence, assume for example that cn → +∞, then by dominated convergence0 =∫ T0g(t, un)ϕ1(t)dt =∫ T0g(t, vn + cnϕ1)ϕ1(t)dt→∫ T0g+(t)ϕ(t)dt 6= 0,a contradiction. On the other hand, if F0un = 0, with ‖un‖C → ∞, thenun = cnϕ1 and∫ T0 g(t, cnϕ1(t))ϕ1(t)dt = 0. Applying dominated convergenceas before, the claim follows.Finally, we shall compute the Leray-Schauder degree degLS(F0, BR, 0) forR large. As the range of T0 is contained in S := span{ϕ1}, it suffices tocompute the Brouwer degree degB(F0|S , BR∩S, 0). Furthermore, F0|S can beidentified with the mapping φ : R→ R given by φ(r) = ∫ T0 g(t, rϕ1(t))ϕ1(t)dt.Again, by dominated convergence we have thatlimr→±∞φ(r) =∫ T0g±(t)ϕ1(t)dt.Hence, φ(r).φ(−r) < 0 for r  0, and it follows that degB(F0|S , BR ∩S, 0) =±1 for R large enough.Acknowledgements The authors are grateful to the anonymous referee forthe careful reading of the original manuscript and his/her fruitful correctionsand remarks. This work was partially supported by the project PIP 5477,CONICET.References[1] A. Bahri and H. Berestycki, Existence of forced oscillators for some non-linear differential equations, Comm. Pure Appl. Math. 37 (1984), 403-422.[2] A. Capietto, J. Mawhin and F. Zanolin, Boundary value problems forforced superlinear second order ordinary differential equations, in Non-linear partial differential equations, H. Brezis and J.L. Lions eds., Collègede France Seminar, Longman, New York, 1994, 55-64.[3] A. Capietto, M. Henrard, J. Mawhin and F. Zanolin, A continuation ap-proach to some forced superlinear Sturm-Liouville boundary value prob-lems, Topological Methods in Nonlinear Analysis, 3 (1994), 81-100.[4] H. Ehrmann, Ubër die Existenz der Lösungen von Randwertaufgabenbei gewöhnlichen nichtlinearen Differentialglaichungen zweiter Ordnung,Math. Ann. 134 (1957), 167-194.[5] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of secondorder, Springer- Verlag (1983).Sturm-Liouville boundary conditions for a second order ODE 11[6] P. Rabinowitz, Multiple critical points of perturbed symmetric funtionals,Trans. Amer. Math. Soc. 61 (1965), 157-164.[7] M. Struwe, Multiple solutions of anticoercive boundary value problems fora class of ordinary differential equations of second order, J. DifferentialEquations 38 (1980), 285-295.Dirección de los autores: Pablo Amster, Universidad de Buenos Aires, Con-sejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina,pamster@dm.uba.ar.  Pedro P. Cárdenas, Departamento de Matemáticas - Uni-versidad Tecnológica de Pereira, Colombia, ppablo@utp.edu.co.

Sturm-Liouville boundary conditions for a second order ODE

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