A new iterative approximation scheme for Reich–Suzuki-type nonexpansive operators with an application

• Published in: Journal of inequalities and applications Vol. 2022; no. 1; pp. 1 - 26
• Main Authors: Ofem, Austine Efut, Işık, Hüseyin, Ali, Faeem, Ahmad, Junaid
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 02.03.2022; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Approximation; Banach space; Banach spaces; Contractive-like mapping; Convergence; Fixed points (mathematics); Game theory; Integral equations; Iterative methods; Iterative scheme; Mathematics; Mathematics and Statistics; Nonlinear integral equation; Stability; w 2 $w^{2}$ -stability; Iterative scheme; Nonlinear integral equation; 47H09; Contractive-like mapping; Banach space; 47H05; stability; 39B82
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-022-02762-8

In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating fixed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The w 2 -stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w 2 -stability. Then, we prove weak and several strong convergence results of our new iterative scheme for fixed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the efficiency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature.