Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

and Applied Analysis 3 new techniques, we prove the existence of uncountably many bounded positive solutions of 1.8 . Five examples are constructed to illuminate our results which extend essentially the corresponding results in 1, 7 . 2. Preliminaries Throughout this paper, we assume thatΔ is the forward difference operator defined byΔyn yn 1 − yn, R −∞, ∞ , R 0, ∞ , Z, N and N0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively, β min { n0 − τ, inf{adn : d ∈ Jr , n ∈ Nn0}, inf { fjn : j ∈ Jk, n ∈ Nn0 }} , Zβ { n : n ∈ Z with n ≥ β, Nn0 {n : n ∈ N0 with n ≥ n0}, Jl {1, . . . , l} for l ∈ {r, k}. 2.1 l∞ β denotes the Banach space of all bounded sequences y {yn}n∈Zβ with the norm ∥y ∥ sup n∈Zβ ∣yn ∣ for y { yn } n∈Zβ ∈ l ∞ β . 2.2 For any M > N > 0, put V N { y { yn } n∈Zβ ∈ l ∞ β : yn ≥ N, ∀n ∈ Zβ } , U M { y ∈ V N : ∥y∥ < M, B M,N { y { yn } n∈Zβ ∈ l ∞ β : ∥y −M∥ < N } , A N,M { y { yn } n∈Zβ ∈ l ∞ β : N ≤ yn ≤ M, ∀n ∈ Zβ } . 2.3 It is easy to see that V N is a closed convex subset of l∞ β , U M is a bounded open subset of V N and B M,N is a bounded open convex subset of l∞ β andA N,M is a bounded closed and convex subset of l∞ β . By a solution of 1.8 , we mean a sequence {yn}n∈Zβ ∈ l∞ β with a positive integer T ≥ n0 τ |β| such that 1.8 is satisfied for all n ≥ T . The following Lemmas play important roles in this paper. Lemma 2.1 Discrete Arzela-Ascoli’s Theorem 3 . A bounded, uniformly Cauchy subset Y of l∞ β is relatively compact. Lemma 2.2 Rothe Fixed Point Theorem 10 . LetD be a bounded convex open subset of a Banach space E and A : D → E be a continuous, condensing mapping, and A ∂D ⊆ D. Then A has a fixed point in D. 4 Abstract and Applied Analysis Lemma 2.3 Leray-Schauder Nonlinear Alternative Theorem 1 . Let U be an open subset of a closed convex set K in a Banach space E with p∗ ∈ U. Let f : U → K be a continuous, condensing mapping with f U bounded. Then either a f has a fixed point inU; or b there exist an x ∈ ∂U and a λ ∈ 0, 1 such that x 1 − λ p∗ λfx. Lemma 2.4 Krasnoselskill Fixed Point Theorem 5 . Let Y be a nonempty bounded closed convex subset of a Banach space X and f, g be mappings from Y into X such that fx gy ∈ Y for every pair x, y ∈ Y . If f is a contraction mapping and g is completely continuous, then the equation fx gx x has at least one solution in Y . 3. Main Results Nowwe use the Rothe fixed point theorem to show the existence andmultiplicity of bounded positive solutions of 1.8 . Theorem 3.1. Assume that there exist two constants N and M with M > N > 0 and two positive sequences {an}n∈Nn0 and {pn}n∈Nn0 satisfying |a n, u1, u2, . . . , ur | ≥ an, ∀ n, ud ∈ Nn0 × N,M , d ∈ Jr ; ∣f n, u1, u2, . . . , uk ∣ ≤ pn, ∀ ( n, uj ) ∈ Nn0 × N,M , j ∈ Jk; 3.1 ∞ ∑ s n0 1 as ∞ ∑ i s max { pi, |ci| } < ∞; 3.2 bn 1 eventually. 3.3 Then 1.8 has uncountably many bounded positive solutions in B M,N . Proof. Let L ∈ M − N,M N . First of all, we show that there exists a mapping SL : B M,N → l∞ β with SL ∂B M,N ⊆ B M,N such that SL has a fixed point y {yn}n∈Zβ ∈ B M,N , which is also a bounded positive solution of 1.8 . It follows from 3.2 and 3.3 that there exists T ≥ max{1, n0 τ |β|} satisfying bn 1, ∀n ≥ T ; 3.4 ∞ ∑