Abstract
Abstract In this paper, we study the boundary value problem for a class of nonlinear fractional q-difference equation with mixed nonlinear conditions involving the fractional q-derivative of Riemann-Liouuville type. By means of the Guo-Krasnosel’skii fixed point theorem on cones, some results concerning the existence of solutions are obtained. Finally, examples are presented to illustrate our main results. MSC:39A13, 34B18, 34A08.
Highlights
1 Introduction Fractional calculus is a generalization of integer order calculus [, ]
There are a large number of papers dealing with the continuous fractional calculus
Some efforts have been made to develop the theory of discrete fractional calculus in various directions
Summary
Fractional calculus is a generalization of integer order calculus [ , ]. There seems to be a new interest in the study of the boundary value problems for fractional q-difference equations [ – ]. In , Zhou and Liu [ ] studied the existence results for fractional q-difference equations with nonlocal q-integral boundary conditions, Dαq u (t) + f t, u(t) = , t ∈ ( , ), u( ) = , u( ) = μIqβ u(η) = μ η (η – qs)(β– ). To the best of our knowledge, very few authors consider the boundary value problem of fractional q-difference equations with mixed nonlinear boundary conditions. Motivated by [ ], we will consider the boundary value problem of the nonlinear fractional q-difference equations. In Section , we introduce some definitions of q-fractional integral and differential operator together with some basic properties and lemmas to prove our main results.
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