Picard $ S $-type semi implicit mid-point method for fixed point approximation with applications

Doaa Filali; Mohammad Dilshad; Ibrahim Alraddadi; Mohammad Akram; Doaa Filali; Mohammad Dilshad; Ibrahim Alraddadi; Mohammad Akram

doi:10.3934/math.20251359

AIMS Mathematics

2025, Volume 10, Issue 12: 30968-30989. doi: 10.3934/math.20251359

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Picard $ S $-type semi implicit mid-point method for fixed point approximation with applications

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: 16 September 2025 Revised: 15 December 2025 Accepted: 19 December 2025 Published: 31 December 2025
MSC : 47H05, 47H09, 47H10, 47H22, 47H25, 49J40

In this article, our goal is to propose and design a hybrid Picard $ S $-type semi-implicit mid-point method to approximate the fixed point of a contractive-like mapping. The convergence result and stability of the proposed method are established under suitable assumptions. We implemented the newly constructed method to approximate the common element, which is the fixed point of a contractive-like mapping and simultaneously solves a general variational inequality. Finally, the significance of the proposed scheme is illustrated through the study of a fractional diffusion equation.
- convergence,
- stability,
- semi-implicit midpoint method,
- fixed point,
- general variational inequality,
- diffusion equation
Citation: Doaa Filali, Mohammad Dilshad, Ibrahim Alraddadi, Mohammad Akram. Picard $ S $-type semi implicit mid-point method for fixed point approximation with applications[J]. AIMS Mathematics, 2025, 10(12): 30968-30989. doi: 10.3934/math.20251359

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Abstract

In this article, our goal is to propose and design a hybrid Picard $ S $-type semi-implicit mid-point method to approximate the fixed point of a contractive-like mapping. The convergence result and stability of the proposed method are established under suitable assumptions. We implemented the newly constructed method to approximate the common element, which is the fixed point of a contractive-like mapping and simultaneously solves a general variational inequality. Finally, the significance of the proposed scheme is illustrated through the study of a fractional diffusion equation.

References

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