Infinitely Many Homoclinic Orbits for 2nth-Order Nonlinear Functional Difference Equations Involving the p-Laplacian

and Applied Analysis 3 F2 F t, xn, . . . , x0 W t, x0 − H t, xn, . . . , x0 , for every t ∈ Z, W,H are continuously differentiable in x0 and xn, . . . , x0, respectively. Moreover, there is a bounded set J ⊂ Z such that H t, xn, . . . , x0 ≥ 0; 2.2 F3 There is a constant μ > p such that 0 < μW t, x0 ≤ W ′ 2 t, x0 x0, ∀ t, x0 ∈ Z × R \ {0} ; 2.3 F4 H t, 0, . . . , 0 ≡ 0, and there is a constant ∈ p, μ such that 0 ∑ i −n H ′ 2 n i t, xn, . . . , x0 x−i ≤ H t, xn, . . . , x0 ; 2.4 F5 There exists a constant b > 0 such that H t, xn, . . . , x0 ≤ bγ , for t ∈ Z, γ > 1, 2.5 where γ ∑n i 0|xi| . F6 F t,−xn, . . . ,−x0 F t, xn, . . . , x0 , for all t, xn, . . . , x0 ∈ Z × R 1. Then 1.1 possesses an unbounded sequence of homoclinic solutions. Theorem 2.2. Assume that r, q, and F satisfy (r), (q), (F1), (F3)–(F5) and the following assumption: F2′ F t, xn, . . . , x0 W t, x0 − H t, xn, . . . , x0 , for every t ∈ Z, W,H are continuously differentiable in x0 and xn, . . . , x0, respectively, and |F t, xn, . . . , x0 | o ( γ ) as γ −→ 0, 2.6 where γ ∑n i 0|xi| 1/p uniformly in t ∈ Z. Then 1.1 possesses an unbounded sequence of homoclinic solutions. Theorem 2.3. Assume that r, q, and F satisfy (r), (q), (F1) and satisfy the following assumptions: F7 For any t ∈ Z, F t, xn, . . . , x0 ≥ F t, x0 ≥ 0; 2.7 4 Abstract and Applied Analysis F8 For any r > 0, there exist a a r , b b r > 0, and ν < p such that ( p 1 a b (∑n i 0|xi| )ν/p ) F t, xn, . . . , x0 ≤ 0 ∑ i −n F ′ 2 n i t, xn, . . . , x0 x−i, ∀t ∈ Z, ( n ∑ i 0 |xi| )1/p ≥ r; 2.8 F9 For any t ∈ Z lim s→ ∞ [ s−pmin |x| 1 F t, sx ] ∞. 2.9 Then there exists an unbounded sequence of homoclinic solutions for 1.1 . 3. Preliminaries 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 S {{u t }t∈Z : u t ∈ R, t ∈ Z}, E { u ∈ S : ∑ t∈Z |r t − 1 Δu t − 1 | q t |u t |p < ∞ } 3.1 and for u ∈ E, let ‖u‖ { ∑ t∈Z |r t − 1 Δu t − 1 | q t |u t |p }1/p . 3.2 Then E is a uniform convex Banach space with this norm. Let I : E → R be defined by I u 1 p ‖u‖ − ∑ t∈Z F t, u t n , . . . , u t . 3.3 If q and F1 hold, then I ∈ C1 E,R and one can easily check that 〈 I ′ u , v 〉 ∑ t∈Z [ r t − 1 |Δu t − 1 |p−2Δnv t − 1 q t |u t |p−2v t −f t , u t n , . . .u t , . . . u t − n v t , ∀u, v ∈ E. 3.4 Abstract and Applied Analysis 5 By usingand Applied Analysis 5 By using Δu t − 1 n ∑ k 0 −1 k ( n k ) u t n − k − 1 , 3.5 we can compute the partial derivative as ∂I u ∂u t Δ ( r t − n φp Δu t − 1 ) q t φp u t − f t, u t n , . . . , u t , . . . , u t − n . 3.6 So, the critical points of I in E are the solutions of 1.1 with u ±∞ 0. Lemma 3.1 see 12 . Let E be a real Banach space and I ∈ C1 E,R satisfy (PS)-condition with I even. 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 E′ ⊂ E, there is r r E′ > 0 such that I u ≤ 0 for u ∈ E′ \ Br 0 , where Br 0 is an open ball in E of radius r centered at 0. Then I possesses an unbounded sequence of critical values. Lemma 3.2. For u ∈ E, β‖u‖p∞ ≤ β‖u‖plp ≤ ‖u‖, 3.7 where β inft∈Zq t . Lemma 3.3. Assume that (F3) holds. Then for every t, x ∈ Z×R, s−μW t, sx is nondecreasing on 0, ∞ . The proof of Lemmas 3.2 and 3.3 is routine and so we omit it. 4. Proofs of Theorems Proof of Theorem 2.1. It is clear that I 0 0. Our proof is devided into three steps. Step 1 PS Condition . Assume that {uk}k∈N ⊂ E is a sequence such that {I uk }k∈N is bounded and I ′ uk → 0 as k → ∞. Then there exists a constant c > 0 such that |I uk | ≤ c, ∥I ′ uk ∥ E∗ ≤ c for k ∈ N. 4.1 6 Abstract and Applied Analysis From 2.2 , 2.3 , 2.4 , 4.1 , F3 , and F4 , we obtain pc pc‖uk‖ ≥ pI uk − p 〈 I ′ uk , uk 〉 − p ‖uk‖ − p ∑ t∈Z [ W t, uk t − 1 W ′ 2 t, uk t uk t ] p ∑ t∈Z H t, uk t n , . . . , uk t − p ∑ t∈Z 0 ∑ i −n H ′ 2 n i t i, uk t n i , . . . , uk t i uk t − p ‖uk‖ − p ∑ n∈Z [ W t, uk t − 1 W ′ 2 t, uk t uk t ] p ∑ t∈Z H t, uk t n , . . . , uk t − p ∑ t∈Z 0 ∑ i −n H ′ 2 n i t, uk t n , . . . , uk t uk t − i ≥ − p ‖uk‖, k ∈ N. 4.2 By 4.2 , there exists a constant A > 0 such that ‖uk‖ ≤ A for k ∈ N. 4.3 It can be assumed that uk ⇀ u0 in E. For any given number ε > 0, by F1 , we can choose ζ > 0 such that ∣f t, u t n , . . . , u t , . . . , u t − n ∣ ≤ εξp−1 for t ∈ Z\J, u t n , . . . , u t , . . . , u t−n ∈Rn 1, 4.4 where ξ ∑n i −n|u t i |p 1/p ≤ ζ. Since q t → ∞, we can also choose an integer Π > max{|k| : k ∈ J} such that q t ≥ 2n 1 A p ζp , |t| ≥ Π. 4.5 By 4.2 and 4.4 , we have |uk t | 1 q t q t uk t p ≤ ζ p 2n 1 Ap ‖uk‖ ≤ ζ p 2n 1 for |t| ≥ Π, k ∈ N. 4.6 Abstract and Applied Analysis 7 Since uk ⇀ u0 in E, it is easy to verify that uk t converges to u0 t pointwise for all t ∈ Z, that is lim k→∞ uk t u0 t , ∀t ∈ Z. 4.7and Applied Analysis 7 Since uk ⇀ u0 in E, it is easy to verify that uk t converges to u0 t pointwise for all t ∈ Z, that is lim k→∞ uk t u0 t , ∀t ∈ Z. 4.7 Hence, we have by 4.6 and 4.7 |u0 t | ≤ ζ p 2n 1 for |t| ≥ Π. 4.8 It follows from 4.7 and the continuity of f t, u t 1 , . . . , u t , . . . , u t − n on u t 1 , . . . , u t , . . . , u t − n that there exists k0 ∈ N such that Π ∑ t −Π ∣f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n ∣<ε for k ≥ k0. 4.9 On the other hand, it follows from F1 , 2.4 , 4.2 , 4.4 , 4.6 , and 4.8 that ∑ |t|>Π ∣f t, uk t n , . . . , uk t , . . . , uk t − n − f t, u0 t n , . . . , u0 t , . . . , u0 t − n ∣ × |uk t − u0 t | ≤ ∑ |t|>Π (∣∣f t, uk t n , . . . , uk t , . . . , uk t − n ∣ ∣f t, u0 t n , . . . , u0 t , . . . , u0 t − n ∣ ) × |uk t | |u0 t | ≤ ε ∑ |t|>Π [( n ∑ i −n |uk t i |p−1 ) ( n ∑ i −n |u0 t i |p−1 )] |uk t | |u0 t | ≤ 2n 1 ε ∑ t∈Z ( |uk t |p−1 |u0 t |p−1 ) |uk t | |u0 t | ≤ 4n 2 ε ∑ t∈Z |uk t | |u0 t | ) ≤ 4n 2 ε β ( A ‖u0‖ ) . 4.10 Since ε is arbitrary, combining 4.9 with 4.10 , we get ∑ t∈Z ∣f t, uk t n , . . . , uk t , . . . , uk t − n − f t, u0 t n , . . . , u0 t , . . . , u0 t − n ∣ |uk t − u0 t | −→ 0 as k −→ ∞. 4.11 8 Abstract and Applied Analysis Using the Hölder’s inequality ac bd ≤ a b 1/p c d , 4.12 where a, b, c, d are nonnegative numbers and 1/p 1/q 1, p > 1, it follows from 3.3 and 3.4 that 〈 I ′ uk − I ′ u0 , uk − u0 〉 ∑ t∈Z |r t − 1 Δuk t − 1 |p−2 Δuk t − 1 ,Δuk t − 1 −Δu0 t − 1 ∑ t∈Z q t |uk t |p−2 uk t , uk t − u0 t − ∑ t∈Z |r t − 1 Δu0 t − 1 |p−2 Δu0 t − 1 ,Δuk t − 1 −Δu0 t − 1 − ∑ t∈Z q t |u0 t |p−2 u0 n , uk t − u0 t − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk n − u0 n ) ‖uk‖ ‖u0‖ − ∑ n∈Z |r t − 1 Δuk t − 1 |p−2 Δuk t − 1 ,Δu0 t − 1 − ∑ t∈Z q t |uk t |p−2 uk t , u0 t − ∑ t∈Z |r t − 1 Δu0 t − 1 |p−2 Δu0 t − 1 ,Δuk t − 1 − ∑ t∈Z q t |u0 t |p−2 u0 t , uk t − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk n − u0 n ) ≥ ‖uk‖ ‖u0‖ − ( ∑ t∈Z |r t − 1 Δu0 t − 1 | 1/p∑ t∈Z |r t − 1 Δuk t − 1 | )1/q − ( ∑ t∈Z q t |u0 t | 1/p∑ t∈Z q t |uk t | )1/q − ( ∑ t∈Z |r t − 1 Δuk t − 1 | 1/p∑ t∈Z |r t − 1 Δu0 t − 1 | )1/q Abstract and Applied Analysis 9and Applied Analysis 9 − ( ∑ t∈Z q t |uk t | 1/p∑ t∈Z q t |u0 t | )1/q − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk t − u0 t ) ≥ ‖uk‖ ‖u0‖ − ( ∑ t∈Z |r t − 1 Δu0 t − 1 | q t |u0 t | ] )1/p · ( ∑ t∈Z |r t − 1 Δuk t − 1 | q t |uk t | ] )1/q − ( ∑ t∈Z |r t − 1 Δuk n − 1 | q t |uk t | ] )1/p · ( ∑ t∈Z |r t − 1 Δu0 n − 1 | q t |u0 t | ] )1/q − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk t − u0 t ) ‖uk‖ ‖u0‖ − ‖u0‖‖uk‖ − ‖uk‖‖u0‖ − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk t − u0 t ) ( ‖uk‖ − ‖u0‖ ) ‖uk‖ − ‖u0‖ − ∑ t∈Z ( f t, uk t n , . . . , uk t , . . . , uk t − n −f t, u0 t n , . . . , u0 t , . . . , u0 t − n , uk t − u0 t ) . 4.13 Since I ′ uk → 0 as k → ∞ and uk ⇀ u0 in E, it follows from 4.11 and 4.13 that 〈 I ′ uk − I ′ u0 , uk − u0 〉 −→ 0 as k → ∞, 4.14 which yields that ‖uk‖ → ‖u‖ as k → ∞. By the uniform convexity of E and the fact that uk ⇀ u0 in E, it follows from the Kadec-Klee property that uk → u0 in E. Hence, I satisfies PS -condition. 10 Abstract and Applied Analysis Step 2 Condition ii of Lemma 3.1 . By F1 , there exists η ∈ 0, 1 such that |F t, u t n , . . . , u t | ≤ 2 −pβ 4p n 1 n ∑ i 0 |u t i | for t ∈ Z \ J, ( n ∑ i 0 |u t i | )1/p ≤ η. 4.15 Set M sup{W t, u | t ∈ J, u ∈ R, |u| 1}, 4.16 and δ min{ β/ 4pM 1 μ−p , η}. If ‖u‖ β1/pδ : ρ, then by Lemma 3.2, |u t | ≤ δ ≤ η < 1 for t ∈ Z. By 4.16 and Lemma 3.2, we have ∑ t∈J W t, u t ≤ ∑ t∈J,u t / 0 W ( t, u t |u t | ) |u t | ≤ M ∑ t∈J |u t | ≤ Mδμ−p ∑ t∈J |u t | ≤ Mδ μ−p β ∑ t∈J q t |u t | ≤ 1 4p ∑ t∈J q t |u t |. 4.17 Set α βδ/2p. Hence, from 3.3 , 4.15 , 4.17 , q , F1 , and F2 , we have I u 1 p ‖u‖ − ∑ t∈Z F t, u t n , . . . , u t 1 p ‖u‖ − ∑ t∈Z\J F t, u t n , . . . , u t − ∑ t∈J F t, u t n , . . . , u t ≥ 1 p ‖u‖ − 2 −pβ 4p n 1 ∑ t∈Z\J ( n ∑ i 0 |u t i | ) − ∑ t∈J W t, u t ∑ t∈J H t, u t n , . . . , u t ≥ 1 p ‖u‖ − β 4p ∑ t∈Z |u t | − 1 4p ∑ t∈J q t |u t | ≥ 1 p ‖u‖ − 1 4p ∑ t∈Z q t |u t | − 1 4p ∑ t∈J q t |u t | Abstract and Applied Analysis 11 ≥ 1 p ‖u‖ − 1 4p ‖u‖ − 1 4p ‖u‖ 1 2p ‖u‖and Applied Analysis 11 ≥ 1 p ‖u‖ − 1 4p ‖u‖ − 1 4p ‖u‖ 1 2p ‖u‖