Infinitely many homoclinic solutions for a class of nonlinear difference equations

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee a class of nonlinear difference equation has infinitely many homoclinic orbits. Our conditions on the nonlinear term are rather relaxed and we generalize some existing results in the literature.


Introduction
Difference equations have attracted the interest of many researchers in the past twenty years since they provided a natural description of several discrete models.Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural network, ecology, cybernetics, biological systems, optimal control, population dynamics, etc.These studies cover many of the branches of difference equation, such as stability, attractiveness, periodicity, oscillation and boundary value problem.Recently, there are some new results on periodic solutions and homoclinic solutions of nonlinear difference equations by using the critical point theory in the literature, see [1-3, 7-15, 20, 21, 30-33].
Consider the nonlinear difference equation of the form where ∆ is the forward difference operator defined by ∆u(n is the ratio of odd positive integers, {p(n)} and {q(n)} are real sequences, {p(n)} = 0. f : Z × R → R. As usual, we say that a solution u(n) of (1.1) is homoclinic (to 0) if u(n) → 0 as n → ±∞.In addition, if u(n) ≡ 0 then u(n) is called a nontrivial homoclinic solution.
In general, equation (1.1) may be regarded as a discrete analogue of the following second order differential equation (p(t)ϕ(x ′ )) ′ + q(t)x(t) + f (t, x) = 0, t ∈ R. (1.2) Equation (1.2) can be regarded as the more general form of the Emden-Fowler equation, appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically * Corresponding author.Emails: pengchen729@sina.com(P.Chen), zhengmeiwang@126.com(Z.Wang) EJQTDE, 2012 No. 47, p. 1 reacting system in terms of various special forms of f (t, x(t)), for example, see [33] and the reference therein.
It is well-known that the existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized from Poincaré, homoclinic orbits play an important role in analyzing the chaos of dynamical system.In the past decade, this problem has been intensively studied using critical point theory and variational methods.
In some recent papers [7, 8, 10, 13-15, 20-21, 30], the authors studied the existence of periodic solutions, subharmonic solutions and homoclinic solutions of some special forms of (1.1) by using the critical point theory.These papers show that the critical point method is an effective approach to the study of periodic solutions for difference equations.
When δ = 1, (1.1) reduces to the following equation: which has been studied in [21].Ma and Guo applied the critical point theory to prove the existence of homoclinic solutions of (1.3) and obtained the following theorems.
When δ = 1, it seems that no similar results were obtained in the literature on the existence of homoclinic solutions.When F (n, x) is an even function on x, however, generalize or improve Theorem A by using the Symmetric Mountain Pass Theorem, there has not been much work done up to now, because it is often very difficult to verify the last condition of the Symmetric Mountain Pass Theorem, different from the Mountain Pass Theorem.
Motivated by the above papers, we will obtain some new criteria for guaranteeing that (1.1) has infinitely many homoclinic orbits without any periodicity and generalize Theorem A. Especially, F (n, x) satisfies a kind of new superquadratic condition which is different from the corresponding condition in the known literature.
In this paper, we always assume that F (n, x) x 0 f 2 (n, s)ds.Our main results are the following theorems.
Theorem 1.3.Assume that p, q and F satisfy (p), (q), (f3), (F4), (F5) and the following assumption: , for every n ∈ Z, F 1 and F 2 are continuously differentiable in x and there is a bounded set J ⊂ Z such that Then Eq.(1.1) possesses an unbounded sequence of homoclinic solutions.
Then E is a uniform convex Banach space with this norm and is a reflexive Banach space, see details in ref. [36] or Lemma 2.4.
As usual, for 1 ≤ p < +∞, let and their norms are defined by respectively.
Let I : E → R be defined by If (p), (q) and (F1) or (F1') or (F1") hold, then I ∈ C 1 (E, R) and one can easily check that Furthermore, the critical points of I in E are classical solutions of (1.1) with u(±∞) = 0.
We will obtain the critical points of I by using the Symmetric Mountain Pass Theorem.We recall it and a minimization theorem as: Lemma 2.1 [17, 25] .Let E be a real Banach space and I ∈ C 1 (E, R) satisfy (PS)-condition.Suppose that I satisfies the following conditions: (i) I(0) = 0; (ii) There exist constants ρ, α > 0 such that I| ∂Bρ(0) ≥ α; (iii) For each finite dimensional subspace is an open ball in E of radius r centered at 0.
Then I possesses an unbounded sequence of critical values.
Remark 2.1.As shown in [6], a deformation lemma can be proved with condition (C) replacing the usual (PS)-condition, and it turns out that Lemma 2.1 hold true under condition (C).We say I satisfies condition (C), i.e., for every sequence where q = inf n∈Z q(n), λ = q − 1 δ+1 .
Lemma 2.3.Assume that (F2) and (F3) hold.Then for every The proof of Lemma 2.3 is routine and so we omit it.
Lemma 2.5 [16] Let E be a uniformly convex Banach space, x n ∈ E, then x n → x if and only if x n ⇀ x and x n → x .

Proofs of theorems
By (F1), there exists η ∈ (0, 1) such that Since F (n, 0) = 0, it follows that By (F2), we have and It follows from (F2), ( ) and (3.5) that Since ν < δ + 1, it follows that there exists a constant A > 0 such that So passing to a subsequence if necessary, it can be assumed that u k ⇀ u 0 in E (Since E is a reflexive Banach space).For any given number ε > 0, by (F1), we can choose ξ > 0 such that Since q(n) → ∞, we can also choose an integer Π > 0 such that  It follows from (3.11) and the continuity of f (n, x) on x that there exists k 0 ∈ N such that On the other hand, it follows from (3.2), (3.9), (3.10), (3.11) and (3.12) that Using Hölder's inequality where a, b, c, d are nonnegative numbers and 1/p + 1/q = 1, p > 1, it follows from (2.2) that Since I ′ (u k ) → 0 as k → +∞ and u k ⇀ u 0 in E, it follows from (3.16) that which, together with (3.15) and (3.16), yields that u k → u as k → +∞.By the uniform convexity of E and the fact that u k ⇀ u 0 in E, it follows from the Kadec-Klee property [16] or Lemma 2.5 that u k → u 0 in E. Hence, I satisfies (C)-condition.
Finally, it remains to show that I satisfies assumption (iii) of Lemma 2.1.Let E ′ be a finite dimensional subspace of E. Since all the norms of a finite dimensional normed space are equivalent, so there exists a EJQTDE, 2012 No. 47, p. 9 Assume that dim E ′ = m and u 1 , u 2 , . . ., u m is a base of E ′ such that For any u ∈ E ′ , there exist λ i ∈ R, i = 1, 2, . . ., m such that It is easy to verify that • * defined by (3.21) is a norm of E ′ .Hence, there exists d ′ > 0 such that where η is given in This shows that and there exists i 0 ∈ {1, 2, . . ., m} such that |u i0 (n 0 )| ≥ d ′ .By (F3), there exists It follows from (F3), (2.1) and (3.27) that We deduce that there is σ 0 = σ 0 (d, Π 1 ) = σ 0 (E ′ ) > 1 such that I(σu) < 0 for u ∈ Θ and σ ≥ σ 0 .
That is This shows that condition (iii) of Lemma 2.1 holds.By Lemma 2.1, I possesses an unbounded sequence By a similar fashion for the proof of (3.3), for the given η in (3.3), there exists Π 2 > 0 such that Thus, from (2.1), (2.3) and (3.3), we have It follows that This contradicts to the fact that {d k } ∞ k=1 is unbounded, and so { u k } is unbounded.The proof is complete.
Proof of Theorem 1.2.
It is clear that I(0) = 0. We first show that I satisfies the (PS)-condition.
Assume that {u k } k∈N ⊂ E is a sequence such that {I(u k )} k∈N is bounded and Then there exists a constant c > 0 such that From (2.1), (2.2), (3.32), (F4) and (F5), we obtain EJQTDE, 2012 No. 47, p. 11 It follows that there exists a constant A > 0 such that Similar to the proof of Theorem 1.1, we can prove that {u k } has a convergent subsequence in E. Hence, I satisfies condition (PS)-condition.By a similar fashion for the proof in Theorem, we can verify that I satisfies assumption (ii) of Lemma 2.1.
Finally, it remains to show that I satisfies assumption (iii) of Lemma 2.1.Let E ′ be a finite dimensional subspace of E. Since all norms of a finite dimensional normed space are equivalent, so there is a constant d ′ > 0 such that (3.22) holds.Let η, Π 1 and Θ be the same as in the proof of Theorem 1.1, then ( Set where d ′ is given in (3.22).
Since F 1 (n, x) > 0 for all n ∈ Z and x ∈ R \ {0}, and F 1 (n, x) is continuous in x, so τ > 0. It follows from (3.26), (3.34) and Lemma 2.3 (i) that For any u ∈ E, it follows from (2.3) and Lemma 2.3 (ii) that where It follows that This contradicts to the fact that {d k } k∈N is unbounded, and so { u k } k∈N is unbounded.
Proof of Theorem 1.3.