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

Abstract

Abstract This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation D q σ c [ k ] ( t ) = w ( t , k ( t ) , c D q ζ [ k ] ( t ) ) ${}^{c}\mathcal{D}_{q}^{\sigma }[k](t) = w (t, k(t), {}^{c} \mathcal{D}_{q}^{\zeta }[k](t) )$ with three-point conditions for t ∈ ( 0 , 1 ) $t \in (0,1)$ on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } $\mathbb{T}_{t_{0}}= \{ t : t =t_{0}q^{n}\}\cup \{0\}$ , where n ∈ N $n\in \mathbb{N}$ , t 0 ∈ R $t_{0} \in \mathbb{R}$ , and 0 < q < 1 $0< q<1$ , 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

• Three-point conditions
• Nonnegative solutions
• Caputo fractional q-derivative
• Numerical results