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Existence of a positive solution to a system of discrete fractional boundary value problems. (English) Zbl 1215.39003

This paper is concerned with a pair of discrete fractional difference equations

$\begin{array}{cc}& -{{\Delta }}^{{\nu }_{1}}{y}_{1}\left(t\right)={\lambda }_{1}{a}_{1}\left(t+{\nu }_{1}-1\right){f}_{1}\left({y}_{1}\left(t+{\nu }_{1}-1\right),{y}_{2}\left(t+{\nu }_{2}-1\right)\right),\hfill \\ & -{{\Delta }}^{{\nu }_{2}}{y}_{2}\left(t\right)={\lambda }_{2}{a}_{2}\left(t+{\nu }_{2}-1\right){f}_{2}\left({y}_{1}\left(t+{\nu }_{1}-1\right),{y}_{2}\left(t+{\nu }_{2}-1\right)\right),\hfill \end{array}$

for $t\in {\left[0,b\right]}_{{ℕ}_{0}},$ subject to boundary conditions

${y}_{1}\left({\nu }_{1}-2\right)={\psi }_{1}\left({y}_{1}\right),\phantom{\rule{4pt}{0ex}}{y}_{2}\left({\nu }_{2}-2\right)={\psi }_{2}\left({y}_{2}\right),\phantom{\rule{4pt}{0ex}}{y}_{1}\left({\nu }_{1}+b\right)={\phi }_{1}\left({y}_{1}\right),\phantom{\rule{4pt}{0ex}}{y}_{2}\left({\nu }_{2}+b\right)={\phi }_{2}\left({y}_{2}\right),$

where ${\lambda }_{1},{\lambda }_{2}>0;$ ${a}_{1},{a}_{2}:ℝ\to \left[0,+\infty \right)$ and ${\nu }_{1},{\nu }_{2}\in \left(1,2\right];$ and ${\psi }_{1},{\psi }_{2},{\phi }_{1},{\phi }_{2}:{ℝ}^{b+3}\to ℝ$ are functionals. A standard transformation of the problem is achieved by means of the Green’s function method, and then using the Krasnoselski fixed point theorem for mappings on Banach spaces with cones, two existence theorems are derived for positive solutions of the above boundary value problem based on further restrictions on ${\lambda }_{1},{\lambda }_{2},{a}_{1},{a}_{2},{f}_{1},{f}_{2}$ and the functionals. Since the system is more general than some existing ones, the existence results are extensions of some, if not all, of the existing results for second order difference boundary value problems.

##### MSC:
 39A12 Discrete version of topics in analysis 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A70 Difference operators 47B39 Difference operators (operator theory)