On the oscillation of q-fractional difference equations

Bahaaeldin Abdalla

doi:10.1186/s13662-017-1316-x

Abstract

In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form \t\t\t{∇0αqx(t)+f1(t,x)=r(t)+f2(t,x),t>0,limt→0+qI0j−αx(t)=bj(j=1,2,…,m),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left \\{ \\textstyle\\begin{array}{l} {}_{q}\\nabla_{0}^{\\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \\quad t>0 ,\\\\ \\lim_{t \\to0^{+}}{{}_{q}I_{0}^{j-\\alpha}x(t)}=b_{j} \\quad(j=1,2,\\ldots,m), \\end{array}\\displaystyle \\right . $$\\end{document} where m=lceilalpharceil, {}_{q}nabla_{0}^{alpha} is the Riemann-Liouville q-differential operator and {}_{q}I_{0}^{m-alpha} is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result.

Highlights

1 Introduction The oscillation theory for fractional differential equations was initiated in [ ] where oscillation criteria were obtained for a nonlinear fractional differential equation of the form
The oscillation of solutions for fractional difference equations, which is the discrete counterpart of the corresponding fractional differential equations, was first studied in [ ]
Sufficient conditions were given for the oscillation of solutions for fractional difference equations of the form

Summary

Introduction

The oscillation theory for fractional differential equations was initiated in [ ] where oscillation criteria were obtained for a nonlinear fractional differential equation of the form. For some of these applications, we refer to [ – ] Following this trend and motivated by the claim that there are no results available in the literature regarding the oscillation of solutions of q-fractional difference equations, we consider equations of the form q∇ αx(t) + f (t, x) = r(t) + f (t, x), t > , limt→ + qI j–αx(t) = bj Section is devoted to the results obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. If α ∈/ N, the α-order Liouville-Caputo (left) q-fractional derivative of a function f is defined by (see [ ]). If α ∈/ N, the α-order Riemann (left) q-fractional derivative of a function f is defined by q∇aαf (t) ∇qnqIa(n–α)f (t).

Oscillation of Riemann q-fractional difference equations

Conclusion