The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach

Abstract

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation ${}^{c}D_q^{\sigma }\left[k\right]\left(t\right)=w(t,k(t),{}^c{D}_q^{\zeta }[k\left]\right(t\left)\right)$ with three-point conditions for $t\in (0,1)$ on a time scale $T_{t_0}=\{t:t=t_0{q}^n\}\cup \left\{0\right\}$ , where $n\in N$ , $t_0\in R$ , and $0<q<1$ , based on the Leray-Schauder nonlinear alternative and Guo-Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Keywords: Caputo fractional q-derivative; Nonnegative solutions; Numerical results; Three-point conditions.