Browsing by Author "Gul Rahmat"

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    Bielecki-Ulam stability of a hammerstein-type difference system
    (Elsevier B.V., 2025) Gul Rahmat; Sohail Ahmad; Muhammad Sarwar; Kamaleldin Abodayeh; Saowaluck Chasreechai; Thanin Sitthiwirattham; M. Sarwar; Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan; email: sarwar@uom.edu.pk
    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){xm+1−xm=M¯mxm+F¯(m,xm,xhm)[∑[j=0][m]G¯(m,j)H¯(j,xj,xhj)]x0=b0,and (0.2){xm+1−xm=M¯mxm+F¯(m,xm,L¯xm,J¯xm)x0=b0,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. © 2025 The Author(s)
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    Hyers-Ulam Stability, Exponential Stability, and Relative Controllability of Non-Singular Delay Difference Equations
    (Hindawi Limited, 2022) Sawitree Moonsuwan; Gul Rahmat; Atta Ullah; Muhammad Yasin Khan; ÊKamran Ê; Kamal Shah; K. Shah; Department of Mathematics and Sciences, Prince Sultan University, Riyadh, P. O. Box 66833, 11586, Saudi Arabia; email: kamalshah408@gmail.com
    In this paper, we study the uniqueness and existence of the solutions of four types of non-singular delay difference equations by using the Banach contraction principles, fixed point theory, and Gronwall's inequality. Furthermore, we discussed the Hyers-Ulam stability of all the given systems over bounded and unbounded discrete intervals. The exponential stability and controllability of some of the given systems are also characterized in terms of spectrum of a matrix concerning the system. The spectrum of a matrix can be easily obtained and can help us to characterize different types of stabilities of the given systems. At the end, few examples are provided to illustrate the theoretical results. © 2022 Sawitree Moonsuwan et al.