Abstract
This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.
1. Introduction and Preliminaries
In this paper, we consider the existence, uniqueness, and multiplicity of solutions for a second-order discrete boundary value problem where is a parameter. Our technique is based on critical point theory, which is successfully used to deal with the existence of solutions for discrete problems (see [1–9]), especially in [7, 9]. Similarly to [7], we denote by , and the sets of all natural numbers, integers, and real numbers, respectively. For , , when . We assume that is nonzero and real-valued for each , is real-valued for each , and is real-valued for each and continuous in . Let be the real Euclidean space with dimension . For any , and , denote the usual norm and inner product in , respectively.
Consider the functional defined on ,where (⋅)T is the transpose of a vector in ,
It is easy to see that is Fréchet differentiable with Fréchet derivativewhere , and there is a one-to-one correspondence between the critical point of functional and the solution of BVP (1.1). Furthermore, is a critical point of if and only if is a solution of (1.1), where [7].
Recently, Yu and Guo [7] studied the BVP,They obtained some existence results for (1.5). One of the main results is as follows.
Theorem (). Suppose that satisfies the
following assumption:
(f3) there exist constants , , , and such
thatThen BVP (1.5) has at least one
solution.
Equation (1.6) shows that for large enough. Since may be negative, then the conclusion of Theorem 1.1 cannot be drawn, which motivates us to consider (1.1). Note that if we take in (1.1), then . Under the similar condition to (1.6) when , we not only obtain the existence of solutions for (1.1), but also the multiplicity.
Let be a Banach space with a direct sum decomposition The functional has a local linking at if for some , and , The functional is said to satisfy the (PS) condition if any sequence for which is bounded, and possesses a convergent subsequence in .
Theorem 1 A (see[10]). Let be a Banach space. Suppose that satisfies the (PS) condition and has a local linking at Assume that is bounded below and Then has at least two nontrivial critical points.
Theorem 1 B (see[6, 11]). Let be a real Banach space, with even, bounded from below, and satisfying (PS) condition. Suppose . There is a set such that is homeomorphic to a unit sphere in () by an odd map, and Then possesses at least distinct pairs of critical points.
2. Main Results
Following conditions will be useful to prove our main results.
(H1)There exist numbers and such that(H2)
Theorem 2.1. Suppose that is positive definite, , for and , and that . Then (1.1) has only trivial solution for .
Proof. Note thatfor , where is the least eigenvalue of , which means that the Nahari manifold is empty. Thus (1.1) has only trivial solution.
Theorem 2.2. Suppose that (H1) and (H2) hold, and is neither positive definite nor negative definite. Then (1.1) has at least two nontrivial solutions for .
Proof. We
will prove that the functional satisfies all
conditions of Theorem A by two steps.
Step 1. is bounded from
below and satisfies (PS) condition. Let denote all the
eigenvalues of , where and . For any , set to be an
eigenvector of corresponding
to the eigenvalue , , such
that
Let and be subspaces of defined
byrespectively. Then has the direct
sum decomposition . In view of (H1), we haveThe second inequality follows
from the elementary inequality where , and which can be
easily obtained by the fact that the function attains its
minimum at Thus is bounded from
below.
Equation (2.5) shows that is coercive, so
we can obtain that any (PS) sequence must be bounded in and, by a standard
argument, has a convergent subsequence.
Step 2. has a local
linking at . Indeed, by (H2) for given and sufficiently
small such that , there exists small enough
such that for ,holds. Then for such that , we have
On the other hand, for with , we haveThe application of
Theorem A
finishes our proof.
Remark 2.3. By the above proof, we see that replacing (H1) with (f3), and by adding the condition that for , Theorem 2.2 still holds, where (f3) is the same as in Theorem 1.1. Indeed, we havewhich means that is bounded from below. The fact that has local linking at may be verified similarly.
If we further impose some condition on and matrix , then the following result can be derived.
Theorem 2.4. Suppose (H1) and (H2) hold, is odd in , that is, for , and that is neither positive definite nor negative definite and has distinct negative eigenvalues. Then (1.1) has at least distinct pairs of solutions for .
Proof. By the proof of Theorem 2.2, is bounded from below and satisfies (PS) condition. In addition, , is even. Consider the subset of :where is a positive number small enough to be determined later, is defined by Theorem 2.2 similarly. Define the mapping bywhere is a unit sphere in . Then is a homeomorphism between and , and is a subset of the finite dimensional space equipped with the Euclidian norm. We can choose and small enough such that for , and then we haveFor above , we havewhich together with Theorem B concludes the proof.
Remark 2.5. The condition that is bounded from below is crucial to prove both Theorems 2.2 and 2.4. As in Remark 2.3, if we replace (H1) by (f3) and the condition that for , then Theorem 2.4 is also true.
Acknowledgments
The authors would like to thank Professor B. G. Zhang and referee's valuable suggestions and comments. This research was partially supported by the NNSF (10571115) of China, China Postdoctoral Science Foundation (20070421106), the Key Project of Chinese Ministry of Education, and the Foundation of Chinese Nationalities Committee (07QH04).