- Open Access
Solutions of the boundary value problem for a 2nth-order nonlinear difference equation containing both advance and retardation
© Wang and Zhou; licensee Springer. 2013
- Received: 22 August 2013
- Accepted: 18 October 2013
- Published: 18 November 2013
In this paper, we consider the boundary value problem for a 2n th-order nonlinear difference equation containing both advance and retardation. By using the critical point theory, some sufficient conditions of the existence of solutions of the boundary value problem are obtained. The proof is based on the linking theorem. An example is given to illustrate our results.
- boundary value problem
- difference equation
- linking theorem
- 2n th-order
where T and n are given positive integers with , is continuous for , are nonzero real numbers and Δ is the forward difference operator defined by , .
In the last decade, by using various techniques such as critical point theory, fixed point theory, topological degree theory and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations (see [1–7] and references therein). Among these approaches, the critical point theory seems to be a powerful tool to solving this problem (see [5, 7–9]). However, compared to the boundary value problems of lower order difference equations [6, 8, 10–13], the study of boundary value problems of higher order difference equations is relatively less (see [9, 14, 15]), especially the works done by using the critical point theory. Therefore, there is still spacious room to explore the boundary value problems of higher-order difference equations. For the background on difference equations, we refer to .
with the boundary value conditions (1.2). He distinguished two cases that is superlinear and sublinear in the second variable z, respectively. Equation (1.3) represents a class of no-delay difference equations.
Equation (1.4) has been studied extensively by many scholars. For example, Smets and Willem  obtained the existence of solitary waves of (1.4).
as an example. It represents the amplitude of the motion of every particle in the string.
Before we apply the critical point theory, we shall establish the corresponding variational framework for (1.1) with (1.2).
Let ℕ, ℤ and ℝ denote the sets of all natural numbers, integers and real numbers, respectively. For , define , when .
Then E is a T-dimensional Hilbert space.
where means the transpose of ⋅ .
for . Therefore, is a critical point of J if and only if x is a solution of (1.1) with (1.2).
Definition 2.1 Let E be a real Banach space, the functional is said to satisfy the Palais-Smale (P.S. for short) condition if any sequence in E, such that is bounded and as , contains a convergent subsequence.
Let denote the open ball in E with radius ρ and center 0, and let denote its boundary.
In order to obtain the existence of critical points of J on E, we cite the basic lemma, which is crucial in the proof of our main results.
Lemma 2.1 (Linking theorem )
Let E be a real Hilbert space and , where is a finite-dimensional subspace of E. Assume that satisfies the P.S. condition and the following two conditions.
(J1) There exist constants and such that ;
(J2) There exist an and a constant such that , where .
where and is the identity operator on ∂Q.
Now we state our main results.
Then (1.1) with (1.2) possesses at least two nontrivial solutions.
Corollary 2.1 Assume that (A1) holds, and satisfies (A3) and
Then (1.1) with (1.2) possesses at least two nontrivial solutions.
Remark 2.1 If for , then (2.12) clearly holds.
and (A2) is satisfied. So Corollary 2.1 holds.
Lemma 3.1 Assume that (A2) holds, then the functional J satisfies the P.S. condition.
Let be a sequence in E such that is bounded and as . Then there exists a positive constant C such that .
Since E is a finite-dimensional space, (3.3) implies has a convergent subsequence. Thus P.S. condition is verified. □
Now we give the proof of Theorem 2.1.
Clearly, m is a critical point of J on E. We prove that m is nontrivial.
Therefore, , which implies that . That is, m is a nontrivial critical point of J.
For convenience, we note , . Obviously, .
Note . We will assert that .
By way of contradiction, assume that . Then . Since and , must be an interior point of both sets Q and . However, and for any . This implies that . It is a contradiction, so the conclusion of Theorem 2.1 holds.
Now the proof is complete. □
As an application of Theorem 2.1, we give an example to illustrate our result.
for , we see that (2.13) and (2.14) hold. By Theorem 2.1, we know that (4.1) with (4.2) has at least two nontrivial solutions.
The authors would like to thank the anonymous referee for his/her valuable suggestions. This work is supported by Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1226), the Specialized Fund for the Doctoral Program of Higher Education of China (No. 20114410110002), and SRF of Guangzhou Education Bureau (No. 10A012).
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