A faster fixed point iterative algorithm and its application to optimization problems

Hamza Bashir; Junaid Ahmad; Walid Emam; Zhenhua Ma; Muhammad Arshad; Hamza Bashir; Junaid Ahmad; Walid Emam; Zhenhua Ma; Muhammad Arshad

doi:10.3934/math.20241153

AIMS Mathematics

2024, Volume 9, Issue 9: 23724-23751. doi: 10.3934/math.20241153

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Research article

A faster fixed point iterative algorithm and its application to optimization problems

1.
Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad 44000, Pakistan
2.
Department of Statistics and Operations Research, Faculty of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
3.
Department of Mathematics and Physics, Hebei University of Architecture, Zhangjiakou 075024, China

Received: 10 May 2024 Revised: 17 July 2024 Accepted: 25 July 2024 Published: 08 August 2024
MSC : 47H05, 47H09, 47H10

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $ (\alpha, \beta) $-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $ (\alpha, \beta) $-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.
- iteration,
- fixed point,
- differential equation,
- optimization problem,
- Banach space
Citation: Hamza Bashir, Junaid Ahmad, Walid Emam, Zhenhua Ma, Muhammad Arshad. A faster fixed point iterative algorithm and its application to optimization problems[J]. AIMS Mathematics, 2024, 9(9): 23724-23751. doi: 10.3934/math.20241153

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Abstract

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $ (\alpha, \beta) $-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $ (\alpha, \beta) $-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.

References

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