On multi-valued generalized $ \alpha $-nonexpansive mappings and an application to two-point BVPs

Junaid Ahmad; Imen Ali Kallel; Ahmad Aloqaily; Nabil Mlaiki; Junaid Ahmad; Imen Ali Kallel; Ahmad Aloqaily; Nabil Mlaiki

doi:10.3934/math.2025019

AIMS Mathematics

2025, Volume 10, Issue 1: 403-419. doi: 10.3934/math.2025019

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Research article

On multi-valued generalized $ \alpha $-nonexpansive mappings and an application to two-point BVPs

1.
Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad-44000, Pakistan
2.
Department of Mathematics, College of Science, Northern Border University, Arar, Saudi Arabia
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
4.
School of Computer, Data and Mathematical Sciences, Western Sydney University, Sydney 2150, Australia

Received: 19 September 2024 Revised: 23 December 2024 Accepted: 30 December 2024 Published: 08 January 2025
MSC : 47H09, 47H10

In recent years, many researchers have studied the fixed points of generalized $ \alpha $-nonexpansive (GAN) mappings, yet there has been limited research on multi-valued GAN mappings. In this paper, we focused on the class of multi-valued GAN mappings in the context of Banach spaces. We introduced a novel iterative process to find fixed points of these mappings and established weak and strong convergence theorems under mild conditions. To demonstrate the numerical efficiency of our new approach, we presented a supportive example and compare the accuracy of our method with existing iteration processes. To show practical usability of the research, we considered a larger class of boundary value problems (BVPs) and proved a convergence result with a supportive example. Our results are new and unify several results of the current literature.
- iteration,
- approximate solution,
- fixed point,
- condition (I),
- Banach space
Citation: Junaid Ahmad, Imen Ali Kallel, Ahmad Aloqaily, Nabil Mlaiki. On multi-valued generalized $ \alpha $-nonexpansive mappings and an application to two-point BVPs[J]. AIMS Mathematics, 2025, 10(1): 403-419. doi: 10.3934/math.2025019

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Abstract

In recent years, many researchers have studied the fixed points of generalized $ \alpha $-nonexpansive (GAN) mappings, yet there has been limited research on multi-valued GAN mappings. In this paper, we focused on the class of multi-valued GAN mappings in the context of Banach spaces. We introduced a novel iterative process to find fixed points of these mappings and established weak and strong convergence theorems under mild conditions. To demonstrate the numerical efficiency of our new approach, we presented a supportive example and compare the accuracy of our method with existing iteration processes. To show practical usability of the research, we considered a larger class of boundary value problems (BVPs) and proved a convergence result with a supportive example. Our results are new and unify several results of the current literature.

References

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