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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 $$\align & -\Delta ^{\nu _{1}}y_{1}(t)=\lambda _{1}a_{1}(t+\nu _{1}-1)f_{1}\left( y_{1}(t+\nu _{1}-1),y_{2}(t+\nu _{2}-1)\right) , \\ & -\Delta ^{\nu _{2}}y_{2}(t)=\lambda _{2}a_{2}(t+\nu _{2}-1)f_{2}\left( y_{1}(t+\nu _{1}-1),y_{2}(t+\nu _{2}-1)\right) , \endalign $$ for $t\in \lbrack 0,b]_{\mathbb{N}_{0}},$ subject to boundary conditions $$y_{1}(\nu _{1}-2)=\psi _{1}(y_{1}),\ y_{2}(\nu _{2}-2)=\psi _{2}(y_{2}),\ y_{1}(\nu _{1}+b)=\phi _{1}(y_{1}),\ y_{2}(\nu _{2}+b)=\phi _{2}(y_{2}),$$ where $\lambda _{1},\lambda _{2}>0;$ $a_{1},a_{2}:\Bbb R\rightarrow \lbrack 0,+\infty )$ and $\nu _{1},\nu _{2}\in (1,2];$ and $\psi _{1},\psi _{2},\phi _{1},\phi _{2}:\Bbb R^{b+3}\rightarrow\Bbb R$ 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.

39A12Discrete version of topics in analysis
39A22Growth, boundedness, comparison of solutions (difference equations)
39A70Difference operators
47B39Difference operators (operator theory)
Full Text: DOI
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