Abstract
We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian. Our approach is based on three critical points theorems.
Highlights
In these last years, the study of discrete problems subject to various boundary value conditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degree see, e.g., 1–3 and the reference given therein
For every λ lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions
We require that the primitive F of f is p-sublinear at infinity and satisfies appropriate local growth condition
Summary
The study of discrete problems subject to various boundary value conditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degree see, e.g., 1–3 and the reference given therein. The critical point theory has aroused the attention of many authors in the study of these problems 4–12. −Δ φp Δuk−1 qkφp uk λf k, uk , k ∈ 1, N , Δu0 ΔuN 0, Pλf where N is a fixed positive integer, 1, N is the discrete interval {1, . N}, qk > 0 for all k ∈ 1, N , λ is a positive real parameter, Δuk : uk 1 − uk, k 0, 1, . −Δ φp Δuk−1 qkφp uk λf k, uk , k ∈ 1, N , Δu0 ΔuN 0, Pλf where N is a fixed positive integer, 1, N is the discrete interval {1, . . . , N}, qk > 0 for all k ∈ 1, N , λ is a positive real parameter, Δuk : uk 1 − uk, k 0, 1, . . . , N 1, is the forward difference operator, φp s : |s|p−2s, 1 < p < ∞, and f : 1, N × Ê → Ê is a continuous function
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