Abstract
This paper studies a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the q-difference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of q-difference equations. The results corresponding to a second-order q-difference equation with anti-periodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical second-order boundary value problems with non-separated boundary conditions in the limit q → 1, which provides a useful check.2010 Mathematics Subject Classification. 39A05, 39A13.
Highlights
1 Introduction In this paper, we discuss the existence of solutions for the second-order q-difference equation with non-separated boundary conditions
The main objective of this paper is to develop some existence and uniqueness results for the boundary value problem (1.1)
Our results are based on a variety of fixed point theorems such as Banach’s contraction principle, Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem
Summary
For the forthcoming analysis, let C = C(I, R) denotes the Banach space of all continuous functions from I to R endowed with the norm defined by ║x║ = sup{|x(t)|, t Î I}. Theorem 2.1 Let f: I × R ® R be a continuous function satisfying the condition The boundary value problem (1.1) has a unique solution, provided Λ = LΛ 1 < 1, where Λ1 is given by (2.9).
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