Work

Bielecki-Ulam stability of a hammerstein-type difference system

Year: 2025

Type: article

Abstract: In this study, we investigate the Bielecki-Ulam (B-U) stabilities of two forms of Hammerstein-type difference systems (HT-DS). Specifically, we consider the systems: ( 0.1 ) { x m + 1 − x m = M ¯ m x m + F ¯ ( m , x m , x h m ) [ ∑ [ j = 0 ] [ m ] G ¯ ( m , j ) H ¯ ( j , x j , x h j ) ] x 0 = b 0 ,and ( 0.2 ) { x m + 1 − x m = M ¯ m x m + F ¯ ( m , x m , L ¯ x m , J ¯ x m ) x 0 = b 0 , by establishing conditions under which a unique solution exists. We derive sufficient conditions for the existence and uniqueness of solutions that satisfy B-U stability criteria. To demonstrate the theoretical findings, we provide an illustrative example that confirms the validity of our results. • Purpose: In this study, we examine the Bielecki-Ulam (B-U) stabilities of two forms of Hammerstein-type difference systems (HT-DS) to understand the conditions necessary for solution uniqueness and stability. • Methodology: We analyze two specific systems characterized by distinct recursive nonlinear structures and employ the Banach contraction principle under the Bielecki norm to establish stability results. The theoretical development involves verifying boundedness and Lipschitz continuity of the nonlinear terms and ensuring that the involved operators satisfy contractive conditions. • Findings: We derive sufficient conditions (outlined in Theorems 2 and 3 ) under which the systems possess unique solutions and are shown to be Bielecki-Ulam stable ( Theorems 4 and 5 ). Specifically, these conditions include boundedness of system coefficients, Lipschitz continuity of nonlinear mappings, and the fulfillment of a contraction inequality using the Bielecki norm. Illustrative examples are provided to confirm the applicability of the results.

Source: MethodsX

DOI: 10.1016/j.mex.2025.103422

Authors: Gul Rahmat, Sohail Ahmad, Muhammad Sarwar, Kamaleldin Abodayeh, Saowaluck Chasreechai, Thanin Sitthiwirattham