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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 794040, 13 pages
http://dx.doi.org/10.1155/2012/794040
Research Article

Two-Point Boundary Value Problems for a Class of Second-Order Ordinary Differential Equations

1Department of Mathematical Sciences, University of Texas at El Paso, Bell Hall 316, El Paso, TX 79968-0514, USA
2Department of Mathematical Sciences, University of Texas at El Paso, Bell Hall 124, El Paso, TX 79968-0514, USA

Received 14 October 2011; Accepted 27 December 2011

Academic Editor: Martin Bohner

Copyright © 2012 Indranil SenGupta and Maria C. Mariani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the general semilinear second-order ODE under different two-point boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on , we prove that the set of solutions of is homeomorphic to the two-dimensional real space.

1. Introduction

The Dirichlet problem for the semilinear second-order ODE has been studied by many authors from the pioneering work of Picard [1], who proved the existence of a solution by an application of the well-known method of successive approximations under a Lipschitz condition on and a smallness condition on . Sharper results were obtained by Hamel [2] in the special case of a forced pendulum equation (see also [3, 4]). The existence of periodic solutions for this case has been first considered by Duffing [5] in 1918. Variational methods have been also applied when by Lichtenstein [6], who considered the functional with . When depends on ,  the problem is nonvariational, and different techniques are required, for example, the shooting method introduced in 1905 by Severini [7] and the more general topological approach, which makes use of Leray-Schauder Degree theory. For an overview of the problem and further results, we refer the reader to [8]. A different kind of nonlinear boundary value PDE (quasilinear elliptic equations) was studied extensively in [9, 10].

This problem is recently studied in [11]. Also, this problem is generalized in [1214]. Several much more general forms of the problem have been studied in [1517] via lower and upper solution method. We will study the existence of solutions of (1.1) under Dirichlet, periodic, and nonlinear boundary conditions of the type where and are given continuous functions. Note that if , then (1.3) corresponds to a particular case of Sturm-Liouville conditions and Neumann conditions when .

In the second section, we impose a growth condition on in order to obtain unique solvability of the Dirichlet problem. Furthermore, we prove that the trace mapping given by is a homeomorphism, and we apply this result to obtain solutions for other boundary conditions in some specific cases.

In the third section, we construct solutions of the aforementioned problems by an iterative method based on the existence of an ordered couple of a lower and an upper solution. This method has been successfully applied to different boundary value problems when does not depend on . For general , existence results have been obtained assuming that , where is a Bernstein function for some , namely,(i) is nondecreasing in ,(ii)for , if for any , then (see, e.g., [18]).

We will assume instead a Lipschitz condition with respect to and construct in each case a nonincreasing (resp., nondecreasing) sequence of upper (lower) solutions that converges to a solution of the problem.

2. A Growth Condition for

For simplicity, let us assume that is continuous. We may write it as with being continuous. We will assume that satisfies a global Lipschitz condition on , namely,

Remark 2.1. Without loss of generality, we may assume that . Indeed, if not, we may multiply (1.1) for any positive such that in order to get the modified equation: Note that in this case the value in (2.2) must be replaced by , where is the first eigenvalue of for the Dirichlet conditions.
Furthermore, assume that satisfies the following one-sided growth condition on : for , with Under these assumptions, the set of solutions of (1.1) is homeomorphic to . More precisely, one has the following.

Theorem 2.2. Assume that (2.2)–(2.4) hold and let . Then there exists a unique solution of (1.1) satisfying the nonhomogeneous Dirichlet condition: Furthermore, the trace mapping given by is a homeomorphism.

Proof. For any let be the unique solution of the linear problem: It is immediate that the operator given by is compact. Moreover, if with , a simple computation shows that the following a priori bound holds for any with : where Hence, if (i.e., ) for some , setting , we obtain for some fixed constant . Thus, existence follows from Leray-Schauder Theorem. Uniqueness is an immediate consequence of (2.8) for .
Hence, is bijective, and its continuity is clear. On the other hand, if , applying (2.8) to and , it is easy to see that for the -norm. As and satisfy (1.1), we conclude that also for the -norm and the proof is complete.

Remark 2.3. The proof of Theorem 2.2 still holds under more general assumptions for . In fact, if satisfies Caratheodory-type conditions, we may assume only that (for and as before), which is not equivalent to (2.2)–(2.4) when is noncontinuous. Thus, the result may be considered a slight extension of well-known results (see, e.g., [19], Corollary V.2).

As a simple consequence we have the following.

Corollary 2.4. Assume that (2.2) and (2.4) hold. Further, assume that there exists a constant such that where is the constant defined by (2.9) and . Then (1.1) admits at least one -periodic solution, which is unique if in (2.4).

Proof. With the notations of the previous theorem, let us consider the mapping given by From Theorem 2.2, is continuous, and it is clear that is a periodic solution of the problem if and only if for some with .
From (2.8), if , we take and . Observe that . Therefore it follows that where in the last inequality we used (2.6). Hence or, equivalently: Let be the unique solution (in distributional sense) of the following problem: where is given by with if . A simple computation shows that is positive, and multiplying the equation by , we obtain Hence by (2.12) for and existence follows from the continuity of . On the other hand, if and are periodic solutions of the problem, then where Now take as the unique solution of the problem , . Multiplying the previous equality by and applying the boundary conditions for , and , we observe Thus we obtain If , we conclude that .

Remark 2.5. In the previous proof, note that the sign condition on is only used for . Thus, (2.12) may be replaced by the weaker condition where for some , with

Remark 2.6. As a particular case of Corollary 2.4, we deduce the existence of -periodic solutions under the following Landesman-Lazer type conditions (see, e.g., [20]): As in the standard Duffing equation , the asymptotic condition (2.6) can be dropped if the sign in (2.12) is reversed. More precisely, we have the following.

Corollary 2.7. Assume that (2.2) and (2.4) hold. Further, assume that there exists a constant such that Then (1.1) is solvable under periodic or Sturm-Liouville conditions: Furthermore, if in (2.4), then the respective solutions are unique.

Proof. For the periodic problem, define as in the previous corollary. For , if for some , we may assume that is maximum, and hence a contradiction. Thus, , which implies that . In the same way, we deduce that for . Uniqueness follows as in Corollary 2.4.
For (2.27) conditions, let us first note that if , the linear problem is uniquely solvable, and setting problem (1.1)–(2.27) is equivalent to where satisfies the hypothesis. Hence, it suffices to consider only the homogeneous case . In the same way as before, define by For , we obtain that , and for , it holds that . By the generalized intermediate value theorem, we deduce that has a zero in . Uniqueness can be proved as in the periodic case.

3. Iterative Sequences of Upper and Lower Solutions

In this section, we construct solutions of (1.1) under the mentioned two-point boundary conditions by an iterative method. As before, consider where and is globally Lipschitz on with constant .

We will need the following auxiliary lemmas.

Lemma 3.1. Assume that (2.2) holds and let be large enough. Then for any , is uniquely solvable under Dirichlet, periodic, or (2.27) conditions. Furthermore, the application given by is compact.

Proof. Taking , existence and uniqueness follow as a particular case of Theorem 2.2 and Corollary 2.7 for Let , and set , . Then where . Hence, it suffices to prove that the following a priori bound holds for any satisfying periodic or homogeneous Dirichlet or (2.27) conditions: where the constant depends only on . For Dirichlet and (2.27) conditions, apply Cauchy-Schwartz inequality to the integral where and , and observe that for some and depending only on (note that under homogeneous (2.27) conditions, it holds that ). For periodic conditions, take such that with constant and and the proof follows.

Lemma 3.2. Let and assume that a.e. for . Then , provided that satisfies one of the boundary conditions:(i),(ii),(iii).

Proof. For , the result is the well-known maximum principle for Dirichlet conditions. If (ii) holds and , as cannot achieve a positive maximum on , we have that and over a maximal interval . Taking , we deduce that is nondecreasing on , a contradiction. If (iii) holds and, for example, , restricting up to its first zero if necessary, it suffices to consider only the case . As before, we get a contradiction from the fact that is nondecreasing. The proof is similar if we assume that .

In order to prove the main result of this section, we recall that is an ordered couple of a lower and an upper solution for (1.1) if and under the following boundary conditions.

For the Dirichlet problem,For the periodic problem,For the problem (1.3),

We make the following extra assumption.

There exists a constant such that for any such that , and for (1.3): there exists a constant such that

Then we have the following.

Theorem 3.3. Assume that there exists an ordered couple of a lower and an upper solution for Dirichlet, periodic, or (1.3) conditions. Further, assume that (2.2) and (3.9) hold (and also (3.10), for the (1.3) case). Then the respective boundary value problem admits at least one solution with .

Remark 3.4. Observe that a Lipschitz condition is a particular case of a Nagumo condition. This result can also be obtained as a Corollary of Theorem  3.2 in [21].

Proof. For large enough and define to be the unique solution of the following problem: satisfying, respectively, Dirichlet, periodic, or the Sturm-Liouville condition: Compactness of follows easily from Lemma 3.1. Moreover, if , then Hence, setting we deduce that For Dirichlet and periodic cases, it follows that . For (1.3), note that Hence, and from Lemma 3.2, we also obtain that . In the same way, if , we obtain that and the proof follows from Schauder Fixed Point Theorem.

Remark 3.5. Existence conditions in Corollary 2.7 are easily improved by applying Theorem 3.3. Indeed, under condition (2.26), it is immediate that is an ordered couple of a lower and an upper solution.

Remark 3.6. In the context of the previous theorem, from Lemma 3.1, we deduce the existence of a constant such that if for , then

Example 3.7. It is easy to see that the problem has no solution, although is a lower solution. From the previous theorem, we deduce that no upper solution exists. This can be proved directly from the following conditions: Indeed, as no negative minimum exists, if , we may take maximum such that over . Hence, , a contradiction. On the other hand, if , then on and , a contradiction since . The case can be easily reduced to the previous one.

In order to construct solutions by iteration, we need a stronger assumption on .

There exists a constant such that for such that , and .

Corollary 3.8. Assume that there exists an ordered couple of a lower and an upper solution for Dirichlet, periodic, or (1.3) conditions. Further, assume that (2.2) and (3.21) hold (and also (3.10), for the (1.3) case). Set large enough, and define the sequences and given by and , are the (unique) solutions of the following problems: under the respective boundary conditions. Then is an ordered couple of a lower and an upper solution. Furthermore, (resp., ) is nonincreasing (nondecreasing) and converges to a solution of the problem.

Remark 3.9. Observe that this is also a classical result that can be found in the works of Adje [22] or Cabada [23], for example.

Proof. From the previous theorem, we know that . Moreover, Hence, is an upper solution of the problem. Inductively, it follows that is an upper solution for every , with . Hence converges pointwise to a function .
From pointwise. Moreover, by Lemma 3.1, we know that is bounded in ; hence in , it follows easily that Thus, is a solution of the problem. The proof for is analogous. Moreover, if we assume that , it is immediate that .

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