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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 951251, 12 pages
http://dx.doi.org/10.1155/2012/951251
Research Article

Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 20 December 2011; Accepted 22 February 2012

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Chenghua Gao. 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

This paper is concerned with the existence of solutions for the discrete second-order boundary value problem , , , where is an integer, , is bounded and continuous, and is the first eigenvalue of the eigenvalue problem , , .

1. Introduction

Let and be continuous. The nonlinear two-point boundary value problem of ordinary differential equation is very important in applications. Let us mention the problems arising in viscosity, nonlinear oscillations, electric circuits, and so forth. The term may be regarded as a nonlinear damping term in resonance problems and its appears, for example, in Rayleigh’s equation (which is closely connected with a theory of oscillation of violin string), in oscillations of a simple pendulum under the action of viscous damping, in dry (Coulomb) friction (which occurs when the surfaces of two solids are contact and relative motion without lubrication), and in some cases of van der Pol oscillator, see [14] and the references therein.

Since the pioneer work of Landesman and Lazer [5], the problems of the type (where is independent of ) have been extensively studied in the past forty years, see Iannacci and Nkashama [6] and the references therein.

It has been remarked (see [7, 8]) that conditions of the Landesmen-Lazer type are not appropriated to yield the existence of solutions to (1.1). Thus, it is usually much more difficult to deal with (1.1) than to deal with (1.2), see Kannan et al. [7], Cañada and Drábek [8], Habets and Sanchez [9], Drábek et al. [10], and Del Toro and Roca [11].

In [8], Cañada and Drábek used the well-known Lyapunov-Schmidt method and the Schauder fixed point theorem to find a necessary and sufficient condition for the existence of solutions of (1.1). To wit, they proved

Theorem A (See [8, Theorem  3.1]). Let be continuous and let

Let be continuous and bounded with and for , where

Then for any with , there exists a real number such that (1.1) has at least one solution if and only if

It is the purpose of this paper to establish the similar results for the discrete analogue of (1.1) of the form where is an integer, , is bounded and continuous, , is the first eigenvalue of the linear eigenvalue problem Finally, it is worth remarking that the existence of solutions for nonlinear problem which is a discrete analogue of (1.2), has been studied by Rodriguez [12] and Ma [13]. For other recent results on the existence of solutions of discrete problems, see [1421] and the reference therein.

The rest of this paper is arranged as follows. In Section 2, we give some preliminaries and develop the methods of lower and upper solutions for the more generalized problems, that is, the case of the nonlinearity ; in Section 3, we state our main result and provide the proof.

2. Preliminaries

Recall that . Let . Let , be equipped with the norm

respectively. It is easy to see that and are Banach spaces.

Assume that is a continuous function, bounded by a constant :

for and . Consider the following problem:

Definition 2.1. If satisfies then one says is a lower solution of (2.3), (2.4). If satisfies then one says is an upper solution of (2.3), (2.4).

Theorem 2.2. Suppose that , are the lower and upper solutions of (2.3), (2.4), respectively, and , . Then BVP (2.3) and (2.4) have at least one solution satisfies

Proof. Define the function by Set . Consider the auxiliary problems: From (2.8) and the boundness of , we know is bounded. So, by the Schauder fixed point theorem, (2.9) has a solution .
Now, we only prove , the other case is similar.
Set . Suppose that , for , and , , where , .
On the other hand, by the definition of upper solution, for Then Now, by the convexity of on , we get , , that is, , . This contradicts , . Thus, , .

Lemma 2.3. See

Proof. Let . Then and . Since , we have . This together with the fact that implies the assertion holds.

Now, let , , denote the positive eigenfunction corresponding to the first eigenvalue of (1.7). Then by Lemma 2.3, .

Since is located on , by the direct computation, we can obtain the following result.

Lemma 2.4. If is an odd number, then if is an even number, then ,

Define the operator by

where .

Define by

Then (2.3), (2.4) is equivalent to the operator equation .

In Theorem 2.2, we established the methods of lower and upper solutions under well order. Now, we can also develop the methods of lower and upper solutions for (2.3), (2.4) when is not necessary, its proofs are based on the following lemma, that is, the connectivity properties of the solution sets of parameterized families of compact vector fields, they are a direct consequence of Mawhin [22, Lemma  2.3].

Lemma 2.5 (see [22, Lemma  2.3]). Let be a Banach space and a nonempty, bounded, closed convex subset. Suppose that is completely continuous. Then the set contains to be a closed connected subset which connects to .

Theorem 2.6. Assume that , are the lower solution and the upper solution of (2.3), (2.4), respectively. Then (2.3) and (2.4) have at least one solution.

Proof. Define the projections , by Then , , and ,. Now, the operator equation is equivalent to the alternative system where is the inverse of mapping .
Writing in the form , , , (2.3) and (2.4) are equivalent to the system Since is finite dimensional, it is easy to see that is completely continuous, by the Schauder fixed point theorem and the fact is bounded, we get that for any fixed , (2.19) and is bounded. Then there exist positive constants such that for all , and . Let for all . Observe that Lemma 2.5 is applicable. Hence there exists a connected subset of (2.19)}, , which connected and . Since is continuous, is an interval. If , then (2.3) and (2.4) have a solution. If , then every with is an upper solution. Indeed, it is obvious that By construction, with satisfies . Hence, from Theorem 2.2, (2.3) and (2.4) have a solution. A similar argument applies if .

Theorem 2.7. Suppose that satisfies where satisfies Then, there exists a nonempty, connected, and bounded set such that (2.3) and (2.4) have at least one solution if and only if.

Proof. As the proof of Theorem 2.6, (2.3) and (2.4) are equivalent to the system (2.19), (2.20). Since is bounded, applying the Schauder fixed point theorem we obtain that for any fixed , there exits at least one such that (2.19) holds.
Now, (2.20) becomes Hence, for a given , , (2.3), (2.4) with has at least one solution if and only if belongs to the range of the (multivalued, in general) function , where (2.19). But is a connected set. In fact, let and belong to and . Then (2.3), (2.4) with and has solutions and , respectively. If we consider (2.3), (2.4) with , where , then is an upper solution and is a lower solution to this problem. By Theorem 2.6, there exists at least one solution, that is, belongs to . Moreover, since is bounded, the range of is bounded.

3. Main Results

In this section, we deal with (1.6). First, let us make the following assumptions:

(H1) is a bounded and continuous function and satisfies and for any ,

(H2) satisfies

Theorem 3.1. Suppose that (H1),  (H2) hold. Then there exists a real number , , such that (1.6) has at least one solution if and only if

Proof. Note that . Due to the consideration in the the proof of Theorem 2.7. It is sufficient to show that for a given with , we have The (possibly multivalued) function has the following form: where and verify (2.19). From the boundedness of and (2.19), there exists a constant (independent of ) such that for any , furthermore, Now, we divide the proof into two cases.Case 1. is an odd number. By Lemma 2.4, we obtain that as . Due to , we get The assumption , , and (3.5) yields for any .Case 2. is an even number. By Lemma 2.4, we know that Hence, By (3.5) and the assumption , , we know that for any , . Thus, for any ,
It is sufficient to prove that this infimum is achieved. Let us denote Suppose that satisfies and is the corresponding minimizing sequence, that is, , are the solution of (1.1), with the right-hand sides .
We claim that is bounded. In fact, if as , then we can get two contradictions in the following two cases.
Case 1. If is an odd number, then by (2.20), letting in (3.13), we get From (H1), we arrive for any , , which together with (3.14) implies that This contradicts (3.12).Case 2. If is an even number, then by (2.20) and , we get This implies that On the other hand, by (3.12) and (H1), we get that for any fixed , Now, we obtain a contradiction. Thus, is bounded.
Since is finite dimensional and is bounded, we obtain that , (at least for a subsequence), and is a solution of (1.1) with . Hence, the infimum is achieved in .

Acknowledgments

This paper is supported by NSFC (11061030, 11101335, 11126296) and the Fundamental Research Funds for the Gansu Universities.

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