The $ AA $-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales

Aynur Şahin; Zeynep Kalkan; Aynur Şahin; Zeynep Kalkan

doi:10.3934/math.20241192

AIMS Mathematics

2024, Volume 9, Issue 9: 24480-24506. doi: 10.3934/math.20241192

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The $ AA $-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales

Aynur Şahin ^,,
Zeynep Kalkan

Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Türkiye

Received: 18 March 2024 Revised: 08 August 2024 Accepted: 16 August 2024 Published: 21 August 2024
MSC : 45B05, 47H09, 47H10

We explored the $ AA $-iterative algorithm within the hyperbolic spaces (HSs), aiming to unveil a stability outcome for contraction maps and convergence outcomes for generalized $ (\alpha, \beta) $-nonexpansive ($ G\alpha \beta N $) maps in such spaces. Through this algorithm, we derived compelling outcomes for both strong and $ \Delta $-convergence and weak $ w^2 $-stability. Furthermore, we provided an illustrative example of $ G\alpha \beta N $ maps and conducted a comparative analysis of convergence rates against alternative iterative methods. Additionally, we demonstrated the practical relevance of our findings by applying them to solve the linear Fredholm integral equations (FIEs) and nonlinear Fredholm-Hammerstein integral equations (FHIEs) on time scales.
- hyperbolic space,
- fixed point,
- generalized $ (\alpha, \beta) $-nonexpansive map,
- the AA-iterative algorithm,
- time scale,
- integral equations,
- weak $ w^2 $-stability
Citation: Aynur Şahin, Zeynep Kalkan. The $ AA $-iterative algorithm in hyperbolic spaces with applications to integral equations on time scales[J]. AIMS Mathematics, 2024, 9(9): 24480-24506. doi: 10.3934/math.20241192

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Abstract

We explored the $ AA $-iterative algorithm within the hyperbolic spaces (HSs), aiming to unveil a stability outcome for contraction maps and convergence outcomes for generalized $ (\alpha, \beta) $-nonexpansive ($ G\alpha \beta N $) maps in such spaces. Through this algorithm, we derived compelling outcomes for both strong and $ \Delta $-convergence and weak $ w^2 $-stability. Furthermore, we provided an illustrative example of $ G\alpha \beta N $ maps and conducted a comparative analysis of convergence rates against alternative iterative methods. Additionally, we demonstrated the practical relevance of our findings by applying them to solve the linear Fredholm integral equations (FIEs) and nonlinear Fredholm-Hammerstein integral equations (FHIEs) on time scales.

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