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The article is concerned with systems of fractional discrete equations Δαx(n+1)=Fn(n,x(n),x(n−1),…,x(n0)),n=n0,n0+1,…,{\Delta }^{\alpha }x\left(n+1)={F}_{n}\left(n,x\left(n),x\left(n-1),\ldots ,x\left({n}_{0})),\hspace{1em}n={n}_{0},{n}_{0}+1,\ldots , where n0∈Z{n}_{0}\in {\mathbb{Z}}, nn is an independent variable, Δα{\Delta }^{\alpha } is an α\alpha -order fractional difference, α∈R\alpha \in {\mathbb{R}}, Fn:{n}×Rn−n0+1→Rs{F}_{n}:\left\{n\right\}\times {{\mathbb{R}}}^{n-{n}_{0}+1}\to {{\mathbb{R}}}^{s}, s⩾1s\geqslant 1 is a fixed integer, and x:{n0,n0+1,…}→Rsx:\left\{{n}_{0},{n}_{0}+1,\ldots \right\}\to {{\mathbb{R}}}^{s} is a dependent (unknown) variable. A retract principle is used to prove the existence of solutions with graphs remaining in a given domain for every n⩾n0n\geqslant {n}_{0}, which then serves as a basis for further proving the existence of bounded solutions to a linear nonhomogeneous system of discrete equations Δαx(n+1)=A(n)x(n)+δ(n),n=n0,n0+1,…,{\Delta }^{\alpha }x\left(n+1)=A\left(n)x\left(n)+\delta \left(n),\hspace{1em}n={n}_{0},{n}_{0}+1,\ldots , where A(n)A\left(n) is a square matrix and δ(n)\delta \left(n) is a vector function. Illustrative examples accompany the statements derived, possible generalizations are discussed, and open problems for future research are formulated as well. |
