Bounded solutions to systems of fractional discrete equations
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 inde...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-07-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2022-0260 |
| _version_ | 1828741863662682112 |
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| author | Diblík Josef |
| author_facet | Diblík Josef |
| author_sort | Diblík Josef |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T01:08:48Z |
| format | Article |
| id | doaj.art-4fefa113ed7743d0a8d3d6702ef74e97 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-13T01:08:48Z |
| publishDate | 2022-07-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-4fefa113ed7743d0a8d3d6702ef74e972022-12-22T03:09:15ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2022-07-011111614163010.1515/anona-2022-0260Bounded solutions to systems of fractional discrete equationsDiblík Josef0Brno University of Technology, CEITEC - Central European Institute of Technology, Brno 612 00, Czech RepublicThe 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.https://doi.org/10.1515/anona-2022-0260fractional discrete differenceasymptotic behaviorsystem of fractional discrete equationsestimates of solutions39a0539a0639a22 |
| spellingShingle | Diblík Josef Bounded solutions to systems of fractional discrete equations Advances in Nonlinear Analysis fractional discrete difference asymptotic behavior system of fractional discrete equations estimates of solutions 39a05 39a06 39a22 |
| title | Bounded solutions to systems of fractional discrete equations |
| title_full | Bounded solutions to systems of fractional discrete equations |
| title_fullStr | Bounded solutions to systems of fractional discrete equations |
| title_full_unstemmed | Bounded solutions to systems of fractional discrete equations |
| title_short | Bounded solutions to systems of fractional discrete equations |
| title_sort | bounded solutions to systems of fractional discrete equations |
| topic | fractional discrete difference asymptotic behavior system of fractional discrete equations estimates of solutions 39a05 39a06 39a22 |
| url | https://doi.org/10.1515/anona-2022-0260 |
| work_keys_str_mv | AT diblikjosef boundedsolutionstosystemsoffractionaldiscreteequations |