Coupled systems of fractional \(\nabla\)-difference boundary value problems (Q2823233)

The authors study the existence of solutions for a coupled system of two-point fractional \(\nabla \)-difference boundary value problems of the form NEWLINE\[NEWLINE\begin{pmatrix} \nabla _{a^{+}}^{\alpha }u(t) \\ \nabla _{a^{+}}^{\alpha }v(t)\end{pmatrix}+\begin{pmatrix} f(t,v(t) \\ f(t,u(t)\end{pmatrix}=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{pmatrix} u(a+1) \\ u(b+1) \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \end{pmatrix} =\begin{pmatrix} v(a+1) \\ v(b+1)\end{pmatrix},NEWLINE\]NEWLINE where \(1<\alpha ,\beta \leq 2\), \(t\in [ a+2,b+1]_{\mathbb{N} }=\{a+2,a+3,\dots,b,b+1\}\), \(a,b\in \mathbb{Z}\) such that \(a\geq 0\), \(b\geq 3\) and the functions \(f,g:[a+2,b+1]_{\mathbb{N}}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous. The authors' analysis relies on the Green functions and the nonlinear alternative of Leray-Schauder and Krasnoselskii-Zabreiko fixed point theorems. At the end they give some numerical examples to illustrate the main results.