Convergence of Fibonacci–Ishikawa iteration procedure for monotone asymptotically nonexpansive mappings

• Published in: Journal of inequalities and applications Vol. 2024; no. 1; pp. 81 - 14
• Main Authors: Alam, Khairul Habib, Rohen, Yumnam, Saleem, Naeem, Aphane, Maggie, Rzzaque, Asima
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.06.2024; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Applied mathematics; Asymptotic properties; Banach spaces; Bounded away sequence; Convergence; Differential equations; Fibonacci numbers; Fibonacci–Ishikawa iteration; Fractional calculus; Mathematics; Mathematics and Statistics; Minimizing sequence; Monotone asymptotically nonexpansive mapping; Weak Opial condition; 47H10; Monotone asymptotically nonexpansive mapping; Bounded away sequence; 47H40; 47H09; 46B20; Fibonacci–Ishikawa iteration; Minimizing sequence; Weak Opial condition
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-024-03156-8

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.