Bounded solutions of fractional discrete equations of positive non-integer orders
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 18 publications
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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2025 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
