IJNSNS 2019; aop

In the present work, we consider the nonlinear secondorder difference problem ⎪⎪⎨ ⎪⎪⎩ –B ( a(k)6p(Bu(k – 1)) ) + b(k)6p(u(k)) = f (k, u(k)) for all k ∈ Z u(k) → 0 as |k| → ∞. (Pf ) Here, p > 1 is a real number, 6p(t) = |t|p–2t for all t ∈ R, a, b : Z → (0, +∞), while f : Z × R → R is a continuous function. Moreover, the forward difference operator is defined as Bu(k) = u(k + 1) – u(k). We say any function u = {u(k)} is homoclinic if lim|k|→∞ u(k) = 0. The theory of nonlinear difference equations has been widely used to examine discrete models appearing in many fields such as computing, computer science, economics, neural networks, biology, ecology, cybernetics, physics, and so on. Discrete boundary value problems

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