Bounded solutions to systems of fractional discrete equations

Abstract

Abstract The article is concerned with systems of fractional discrete equations Δ α x ( n + 1 ) = F n ( n , x ( n ) , x ( n − 1 ) , … , x ( n 0 ) ) , n = n 0 , n 0 + 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 n 0 ∈ Z {n}_{0}\in {\mathbb{Z}} , n n is an independent variable, Δ α {\Delta }^{\alpha } is an α \alpha -order fractional difference, α ∈ R \alpha \in {\mathbb{R}} , F n : { n } × R n − n 0 + 1 → R s {F}_{n}:\left\{n\right\}\times {{\mathbb{R}}}^{n-{n}_{0}+1}\to {{\mathbb{R}}}^{s} , s ⩾ 1 s\geqslant 1 is a fixed integer, and x : { n 0 , n 0 + 1 , … } → R s x:\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 ⩾ n 0 n\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 Δ α x ( n + 1 ) = A ( n ) x ( n ) + δ ( n ) , n = n 0 , n 0 + 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.