Multi-strip fractional q-integral boundary value problems for nonlinear fractional q-difference equations

Nittaya Pongarm,Jessada Tariboon,Sotiris K Ntouyas,Suphawat Asawasamrit

doi:10.1186/1687-1847-2014-193

Abstract

Abstract In this article, we study the existence and uniqueness of solutions for multi-strip fractional q-integral boundary value problems of nonlinear fractional q-difference equations. By using the Banach contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory some interesting results are obtained. Some examples are presented to illustrate the results. MSC:34A08, 34B18, 39A13.

Highlights

1 Introduction In this article, we investigate the following nonlinear fractional q-difference equation for multi-strip fractional q-integral boundary condition:
Applying the Banach contraction mapping principle, we will show that the operator A has a fixed point which is a unique solution of problem ( . )
By nonlinear alternative of Leray-Schauder type (Lemma . ), we deduce that A has a fixed point in U, which is a solution of the boundary value problem ( . )

Summary

Introduction

M are given constants, Dαq is the fractional q-derivative of Riemann-Liouville type of order α, Iqβii is the fractional qi-integral of order βi and f : [ , T] × R → R is a continuous function. 4 Main results Let C = C([ , T], R) denote the Banach space of all continuous functions from [ , T] to R endowed with the supremum norm defined by u = supt∈[ ,T] |u(t)|. Let f : [ , T] × R → R be a continuous function satisfying the assumption (H ) there exists a constant L > such that |f (t, u) – f (t, v)| ≤ L|u – v| for each t ∈ [ , T] and u, v ∈ R.

Results

Conclusion