Variational Methods and Critical Point TheoryView this Special Issue
Infinitely Many Homoclinic Orbits for 2th-Order Nonlinear Functional Difference Equations Involving the -Laplacian
By establishing a new proper variational framework and using the critical point theory, we establish some new existence criteria to guarantee that the 2th-order nonlinear difference equation containing both advance and retardation with -Laplacian , , , has infinitely many homoclinic orbits, where is -Laplacian operator;, , are nonperiodic in . Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.
In this paper, we shall be concerned with the existence of homoclinic orbits of the nonlinear difference equation where is the forward difference operator defined by . is -Laplacian operator . As usual, we say that a solution of (1.1) is homoclinic (to 0) if as . In addition, if , then is called a nontrivial homoclinic solution. It is well known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems; we refer the interested reader to [1–15].
We may think of (1.1) as being a discrete analogue of the 2th-order differential equation
In some recent papers [1, 2, 8, 9, 16–18], the authors studied the existence of periodic and homoclinic solutions of second-order nonlinear difference equation by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions of second-order difference equations. Compared to one-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, e.g., [2, 3, 19–21] and references contained therein). But to the best knowledge of the authors, results on existence of homoclinic solutions of (1.1) have not been found in the literature.
Motivated by the recent papers [2, 5, 14, 22], the aim of this paper is to consider problem (1.1) in a more general sense. More exactly our results represent the extensions to equations with -Laplacian. Throughout the paper, for a function , we let denote the partial derivative of on the variable.
2. Main Results
Theorem 2.1. Assume that and satisfy the following assumptions: (r)For every , ;(q)For every , , and ; (F1)There exists a functional which is continuously differentiable in the variable from to for every and satisfy uniformly in , where is defined in (F2);(F2), for every , are continuously differentiable in and , respectively. Moreover, there is a bounded set such that(F3)There is a constant such that(F4), and there is a constant such that(F5)There exists a constant such that where .(F6), for all .Then (1.1) possesses an unbounded sequence of homoclinic solutions.
Theorem 2.2. Assume that , , and satisfy (r), (q), (F1), (F3)–(F5) and the following assumption: (F2’), for every , are continuously differentiable in and , respectively, and
where uniformly in ,
Then (1.1) possesses an unbounded sequence of homoclinic solutions.
Theorem 2.3. Assume that , , and satisfy (r), (q), (F1) and satisfy the following assumptions: (F7)For any ,(F8)For any , there exist , , and such that(F9)For any Then there exists an unbounded sequence of homoclinic solutions for (1.1).
To apply critical point theory to study the existence of homoclinic solutions of (1.1), we shall state some basic notations and lemmas, which will be used in the proofs of our main results.
Let and for , let Then is a uniform convex Banach space with this norm.
Let be defined by If and hold, then and one can easily check that By using we can compute the partial derivative as So, the critical points of in are the solutions of (1.1) with .
Lemma 3.1 (see ). Let be a real Banach space and satisfy (PS)-condition with even. Suppose that satisfies the following conditions: (i);(ii)There exist constants such that ;(iii)For each finite dimensional subspace , there is such that for , where is an open ball in of radius centered at 0.Then possesses an unbounded sequence of critical values.
Lemma 3.2. For , where .
Lemma 3.3. Assume that (F3) holds. Then for every , is nondecreasing on .
4. Proofs of Theorems
Proof of Theorem 2.1. It is clear that . Our proof is devided into three steps.
Step 1 (PS Condition). Assume that is a sequence such that is bounded and as . Then there exists a constant such that From (2.2), (2.3), (2.4), (4.1), (F3), and (F4), we obtain By (4.2), there exists a constant such that It can be assumed that in . For any given number , by (F1), we can choose such that where .
Since , we can also choose an integer such that By (4.2) and (4.4), we have Since in , it is easy to verify that converges to pointwise for all , that is Hence, we have by (4.6) and (4.7) It follows from (4.7) and the continuity of on that there exists such that On the other hand, it follows from (F1), (2.4), (4.2), (4.4), (4.6), and (4.8) that Since is arbitrary, combining (4.9) with (4.10), we get Using the Hölder’s inequality where are nonnegative numbers and , , it follows from (3.3) and (3.4) that Since as and in , it follows from (4.11) and (4.13) that which yields that as . By the uniform convexity of and the fact that in , it follows from the Kadec-Klee property that in . Hence, satisfies (PS)-condition.
Step 2 (Condition (ii) of Lemma 3.1). By (F1), there exists such that Set and . If , then by Lemma 3.2, for . By (4.16) and Lemma 3.2, we have Set . Hence, from (3.3), (4.15), (4.17), (q), (F1), and (F2), we have equation (4.18)shows that implies that , that is, satisfies assumption (ii) of Lemma 3.1.
Step 3 (Condition (iii) of Lemma 3.1). Let be a finite dimensional subspace of . Assume that and is a base of .
For any , there exist , such that Let It is easy to verify that defined by (4.20) is a norm of . Since all the norms of a finite dimensional normed space are equivalent, so there exists a constant such that Assume that
Since , by Lemma 3.2, we can choose an integer such that where is given in (4.15). Set Hence, for , let such that Then, by (4.19)–(4.25), we have This shows that and there exists such that , which, together with (4.23), implies that .
Set Since for all and , and is continuous in , so . It follows from (4.27) and Lemma 3.3 that For any , it follows from (F5) that where
From (3.3), (3.7), (4.28), and (4.29), we have for and Since , we deduce that there is such that That is This shows that (iii) of Lemma 3.1 holds. By Lemma 3.1, possesses an unbounded sequence of critical values with , where is such that for . If is bounded, then there exists such that By a similar fashion for the proof of (4.15), for the given in (4.15), there exists such that Thus, from (F2), (3.7), (4.34), and (4.35), we have It follows that This contradicts the fact that is unbounded, and so is unbounded.
Proof of Theorem 2.2. In the proof of Theorem 2.1, the condition that for , in (F1) is only used in the proofs of assumption (ii) of Lemma 3.1. Therefore, we only prove that assumption (ii) of Lemma 3.1 still holds using (F2') instead of (F2). By (F2'), it follows that If , then by Lemma 3.2, for . Set . Hence, from (3.3), (4.38), and Lemma 3.2, we have equation (4.39) shows that implies that , that is, assumption (ii) of Lemma 3.1 holds. The proof of Theorem 2.2 is completed.
Proof of Theorem 2.3. We first show that satisfies condition (C). Assume that is a (C) sequence of , that is, is bounded and as . Then it follows from (3.3) and (3.4) that
It follows from (F8) that there exists such that (4.15) holds. By (F7) and (F8), we have
and for , , we have
It follows from (3.2), (3.3), (4.38), (4.40), (4.41), and (4.42) that
Since , it follows from (4.43) that is bounded. Similar to the proof of Theorem 2.1, we can prove that has a convergent subsequence in . Hence, satisfies condition (C).
It is obvious that is even and , and so assumption (i) of Lemma 3.1 holds. The proof of assumption (ii) of Lemma 3.1 is the same as in the proof of Theorem 2.2.
Finally, it remains to show that satisfies assumption (iii) of Lemma 3.1. Let be a finite dimensional subspace of . Since all norms of a finite dimensional normed space are equivalent, so there is a constant such that (4.21) holds. Assume that and are the bases of such that (4.22) holds. Let , , and be the same as in the proof of Theorem 2.1. Then (4.23), (4.25), and (4.26) hold. For the given in the proof of Theorem 2.1. By (F9), there exists such that It follows from (F7), (F9), (3.3), and (4.44) that