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Bounded solutions to systems of fractional discrete equations
Advances in Nonlinear Analysis ( IF 3.2 ) Pub Date : 2022-07-20 , DOI: 10.1515/anona-2022-0260
Josef Diblík 1
Affiliation  

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.

中文翻译:

分数离散方程组的有界解

这篇文章是关于分数离散方程组的 Δ α X ( 不是 + 1 ) = F 不是 ( 不是 , X ( 不是 ) , X ( 不是 - 1 ) , , X ( 不是 0 ) ) , 不是 = 不是 0 , 不是 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 , 在哪里 不是 0 Z {n}_{0}\in {\mathbb{Z}} , 不是 不是 是一个自变量, Δ α {\Delta }^{\alpha } 是一个 α -阶分数差, α R \alpha \in {\mathbb{R}} , F 不是 { 不是 } × R 不是 - 不是 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 是一个固定整数,并且 X { 不是 0 , 不是 0 + 1 , } R s x:\left\{{n}_{0},{n}_{0}+1,\ldots \right\}\to {{\mathbb{R}}}^{s} 是一个因(未知)变量。缩回原则用于证明解决方案的存在,图保留在给定域中 不是 不是 0 n\geqslant {n}_{0} , 然后作为进一步证明离散方程的线性非齐次系统的有界解存在的基础 Δ α X ( 不是 + 1 ) = ( 不是 ) X ( 不是 ) + δ ( 不是 ) , 不是 = 不是 0 , 不是 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 , 在哪里 ( 不是 ) A\左(n) 是一个方阵并且 δ ( 不是 ) \delta\left(n) 是一个向量函数。说明性示例伴随得出的陈述,讨论了可能的概括,并制定了未来研究的开放问题。
更新日期:2022-07-20
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