In the following and in the sequel, we denote by , and the sets of all natural numbers, integers and real numbers, respectively. For all , , we define when . Also, the symbol * will denote the transpose of a vector.
Consider the nonlinear second-order difference system (1)
with boundary value conditions (2)
where is the forward difference operator, , is the ratio of odd positive integers, and are real valued for each , is a given positive integer, and are constants, . The boundary value problem eq. (1) with eq. (2) contains the following special Dirichlet boundary value conditions, mixed boundary value conditions and Neumann boundary value conditions:
We may regard eq. (1) as being a discrete analog of the following second-order differential equation (3)
Equation (3) includes the following equation (4)
which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems and adiabatic reactor . Equations similar in structure to eq. (4) arise in the study of periodic solutions and homoclinic orbits [10, 11, 12] of differential equations [21, 27, 28, 31].
Difference equations [1, 2, 3, 4, 5, 8, 9, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 29, 30, 32, 33, 34] have attracted the interest of many researchers in the past 20 years since they provide a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as finance insurance, biological populations, disease control, genetic study, physical field and computer application technology. Therefore, it is worthwhile to explore this topic.
When , Thandapani and Ravi  obtained results on the asymptotic behavior of solutions of (5)
including sufficient conditions for all solutions to be bounded or unbounded. Some results on the existence and behavior of nonincreasing solutions of eq. (5) are also obtained.
Cai and Yu  in 2006 concerned with the existence of solutions of boundary value problems for nonlinear second-order difference equations of the type
They applied the linking theorem and the mountain pass lemma in the critical point theory and give some new results for the existence of solutions.
Recently, Shi, Liu and Zhang  considered the following forward and backward difference equation (6)
and obtained the existence of a nontrivial homoclinic orbit for eq. (6).
Motivated by the recent papers [6, 7, 18], the intention of this paper is to investigate a question of the existence of solutions to boundary value problems for a class of difference systems eq. (1) with eq. (2). The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.
Let be a real Banach space, , i.e. is a continuously Fréchet-differentiable functional defined on . is said to satisfy the Palais–Smale condition (P.S. condition for short) if any sequence for which is bounded and possesses a convergent subsequence in .
Let denote the open ball in about 0 of radius and let denote its boundary.
The rest of this paper is organized as follows. First of all, Section 2 presents variational structure. Next, in Section 3, we shall recall some related fundamental results and present some lemmas. Then, Section 4 is dedicated to the proof of main results. Finally, in Section 5, we shall give two examples to illustrate the applicability of the main results.
2 Variational structure
Our main tool is the critical point theory. We shall establish the corresponding variational framework for the boundary value problem (1) with eq. (2). We start by some basic notations for the reader’s convenience.
On one hand, let be the real Euclidean space with dimension . can be equipped with the inner product and norm as follows, (7)
On the other hand, we define the norm on as follows:
for all and .
Since and are equivalent, there exist constants such that , and (9)
For all , define the functional on as follows:
It is easy to see that and for any , by using and the summation by parts
we can compute the partial derivative as
Thus, is a critical point of on if and only if
We reduce the existence of the boundary value problem (1) with eq. (2) to the existence of critical points of on . That is, the functional is just the variational framework of the boundary value problem (1) with eq. (2).
be a matrix.
For convenience, we identify with
3 Main results
In this section, we shall state and prove our main results by using the variational methods.
Suppose that the following assumptions are satisfied:
for any , ;
for any , ;
there exist constants and such that
For any , combining with , it is easy to see that
as . By the continuity of , we have from the above inequality that there exist upper bounds of values of functional . Classical calculus shows that attains its maximal value at some point which is just the critical point of and the result follows. This completes the proof of Theorem 3.1. □
Assume that , , and the following assumptions: and ;
for all ,
there exist constants such that
there exists such that .
Then, possesses a critical value given by (10)
Assume that , , , and are satisfied. Then satisfies the P.S. condition.
By , we have that
It is obvious that is positive definite. Denote (12) (13)
Let be such that is bounded and as Then, there exists a positive constant such that . It comes from , eq. (12) that
Since , there exists a positive constant such that
Therefore, is bounded on . As a consequence, possesses a convergent subsequence in . Therefore, the P.S. condition is satisfied. □
Proof of Theorem 3.2
It follows from that for any , there exists , such that
For any and , we have . Then,
Take Therefore, At the same time, we have also proved that there exist constants and such that . That is to say, satisfies the condition of the mountain pass lemma.
The rest of the proof is similar to that of [16, Theorem 1.2], but for the sake of completeness, we give the details.
For our setting, clearly . In order to exploit the mountain pass lemma in critical point theory, we need to verify other conditions of the mountain pass lemma. By Lemma 3.2, satisfies the P.S. condition. So it suffices to verify the condition .
From the proof of the P.S. condition, we know Since , we can choose large enough to ensure that . By the mountain pass lemma, possesses a critical value , where and
Let be a critical point associated to the critical value of , i.e., . Similar to the proof of the P.S. condition, we know that there exists such that
Clearly, . If , then the conclusion of Theorem 1.2 holds. Otherwise, . Then, . That is, Therefore,
By the continuity of with respect to , and imply that there exists such that Choose such that is empty, then there exists such that Thus, we get two different critical points of on denoted by The above argument implies that the boundary value problem (1) with eq. (2) possesses at least two nontrivial solutions. □
Assume that , , and the following assumptions are satisfied:
for any , ;
As an application of Theorems 3.2 and 3.3, we give two examples to illustrate our main results.
For , consider the boundary value problem: (16)
with boundary value conditions (2) and , , where is the ratio of odd positive integers, . We have
For , consider the boundary value problem: (17)
with boundary value conditions (2) and , , where is the ratio of odd positive integers.
Difference equations occur widely in numerous settings and forms both in mathematics itself and in its applications. The boundary value problem discussed in this paper has important analog in the continuous case of the second-order differential equation. Such problem is of special significance for the study of a result that describes dynamically changing phenomena, evolution and variation. The problem discussed in this paper can be extended to boundary value problem for 2-order -Laplacian  difference equations.
The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.
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About the article
Published Online: 2018-06-26
Published in Print: 2018-07-26
This project is supported by the National Natural Science Foundation of China (No. 11501194). This work was carried out while visiting Central South University.