Bounded solutions of discrete equations with several fractional differences

Bounded solutions of discrete equations with several fractional differences

Paper in proceedings (conference paper)

In the paper is considered a fractional discrete equation Sigma(s)(pi=1) Delta(beta pi) z(k + 1) = G(k)(k, z(k),..., z(k(0))), k = k(0), k(0) + 1,... where Delta(beta pi), beta(pi) > 0, pi = 1,..., s, are the beta(pi)-order fractional differences, G(k): {k} x Rk-k0+1 -> R, k(0) is an element of Z, k is an element of Z, k >= k(0) and z: {k(0), k(0) + 1,...} -> R. Sufficient conditions are given for the existence of bounded solutions satisfying inequalities b(k) < z(k) < c(k), for all k >= k(0) where b and c are real functions satisfying b(k) < c(k). An application is considered to an equation with several fractional differences Sigma(s)(pi=1) Delta(beta pi) z(k + 1) = G(k)(k, z(k),..., z(k(0))), k = k(0), k(0) + 1,... where xi is an element of R and sigma: {k(0), k(0) + 1,...}-> R. It is proved that there exists a bounded solution satisfying the inequality vertical bar z(k)vertical bar < L, k = k(0), k(0) + 1,..., for a constant L.

In the paper is considered a fractional discrete equation Sigma(s)(pi=1) Delta(beta pi) z(k + 1) = G(k)(k, z(k),..., z(k(0))), k = k(0), k(0) + 1,... where Delta(beta pi), beta(pi) > 0, pi = 1,..., s, are the beta(pi)-order fractional differences, G(k): {k} x Rk-k0+1 -> R, k(0) is an element of Z, k is an element of Z, k >= k(0) and z: {k(0), k(0) + 1,...} -> R. Sufficient conditions are given for the existence of bounded solutions satisfying inequalities b(k) < z(k) < c(k), for all k >= k(0) where b and c are real functions satisfying b(k) < c(k). An application is considered to an equation with several fractional differences Sigma(s)(pi=1) Delta(beta pi) z(k + 1) = G(k)(k, z(k),..., z(k(0))), k = k(0), k(0) + 1,... where xi is an element of R and sigma: {k(0), k(0) + 1,...}-> R. It is proved that there exists a bounded solution satisfying the inequality vertical bar z(k)vertical bar < L, k = k(0), k(0) + 1,..., for a constant L.

discrete fractional equation; bounded solution; fractional difference

discrete fractional equation; bounded solution; fractional difference

BAŠTINEC, J.; DIBLÍK, J.

2025

07.06.2024

American Institute of Physics

USA

9780735449541

AIP Conference Proceedings, Volume 3094, Issue 1, 7 June 2024, International Conference of Numerical Analysis and Applied Mathematics 2022, ICNAAM 2022

0094-243X

AIP conference proceedings

3094

1

United States of America

500044-1

500044-4

4

https://pubs.aip.org/aip/acp/article-abstract/3094/1/500044/3297296/Bounded-solutions-of-discrete-equations-with?redirectedFrom=fulltext