A five-step approximation method for a nonlinear delay integral equation in hyperbolic spaces: A qualitative study

Filali Doaa; Mohammad Dilshad; Mohammad Akram; Filali Doaa; Mohammad Dilshad; Mohammad Akram

doi:10.3934/math.2026730

AIMS Mathematics

2026, Volume 11, Issue 6: 17917-17936. doi: 10.3934/math.2026730

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Research Article

A five-step approximation method for a nonlinear delay integral equation in hyperbolic spaces: A qualitative study

1.
Department of Mathematical Science, College of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia

Received: 10 March 2026 Revised: 30 May 2026 Accepted: 08 June 2026 Published: 18 June 2026
MSC : 47H09, 47H10, 47H22, 47H25, 49J40

In this manuscript, we design and present a Jungck-type iterative method to find the common fixed point of a pair of mappings with weak compatibility in hyperbolic metric spaces. Strong convergence is perfomed to estimate the common fixed point, and stability of the designed iterative scheme is established under suitable assumptions. Further, $ \Delta $-convergence is established, and a theoretical result is outlined to exhibit the coherence and effectiveness of our scheme with some of its counterparts. A numerical case study is set forth to evidence the convergence and effectiveness of the proposed method. Finally, the efficacy and applicability of our proposed method is illustrated by implementing it to explore a nonlinear delay integral equation (NDIE).
- hyperbolic metric spaces,
- coincidence point,
- Jungck-type iterative scheme,
- convergence and stability,
- delayed integral equation
Citation: Filali Doaa, Mohammad Dilshad, Mohammad Akram. A five-step approximation method for a nonlinear delay integral equation in hyperbolic spaces: A qualitative study[J]. AIMS Mathematics, 2026, 11(6): 17917-17936. doi: 10.3934/math.2026730

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Abstract

In this manuscript, we design and present a Jungck-type iterative method to find the common fixed point of a pair of mappings with weak compatibility in hyperbolic metric spaces. Strong convergence is perfomed to estimate the common fixed point, and stability of the designed iterative scheme is established under suitable assumptions. Further, $ \Delta $-convergence is established, and a theoretical result is outlined to exhibit the coherence and effectiveness of our scheme with some of its counterparts. A numerical case study is set forth to evidence the convergence and effectiveness of the proposed method. Finally, the efficacy and applicability of our proposed method is illustrated by implementing it to explore a nonlinear delay integral equation (NDIE).

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