Abstract
This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.
Highlights
Introduction and preliminariesIn this paper, we consider the existence, uniqueness, and multiplicity of solutions for a secondorder discrete boundary value problem p n 1 u n 1 cnunpnun − 1 λf n, u n, n ∈ Z 1, k, 1.1 u 0 αu 1 A, u k 1 βu k B, where λ ∈ R is a parameter
Our technique is based on critical point theory, which is successfully used to deal with the existence of solutions for discrete problems see 1–9, especially in 7, 9
We assume that p n is nonzero and real-valued for each n ∈ Z 1, k, c n is real-valued for each n ∈ Z 1, k, and f n, u is real-valued for each n, u ∈ Z 1, k × R and continuous in u
Summary
We consider the existence, uniqueness, and multiplicity of solutions for a secondorder discrete boundary value problem p n 1 u n 1 cnunpnun − 1 λf n, u n , n ∈ Z 1, k , 1.1 u 0 αu 1 A, u k 1 βu k B, where λ ∈ R is a parameter. Our technique is based on critical point theory, which is successfully used to deal with the existence of solutions for discrete problems see 1–9 , especially in 7, 9. To 7 , we denote by N, Z, and R the sets of all natural numbers, integers, and real numbers, respectively. V ∈ Rk, u and u, v , denote the usual norm and inner product in Rk, respectively
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