A unified framework for fixed point results of three-point contractions in multiple perturbed metric spaces with applications to fractional differential equations

Min Wang; Muhammad Din; Mohammad Akram; Mi Zhou; Min Wang; Muhammad Din; Mohammad Akram; Mi Zhou

doi:10.3934/math.2026686

AIMS Mathematics

2026, Volume 11, Issue 6: 16712-16734. doi: 10.3934/math.2026686

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A unified framework for fixed point results of three-point contractions in multiple perturbed metric spaces with applications to fractional differential equations

1.
School of Science, Hainan Tropical Ocean University, Sanya, Hainan 572000, China
2.
Abdus Salam School of Mathematical Sciences, Governement College University, Lahore 54600, Pakistan
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
4.
Center for Mathematical Research, University of Sanya, Hainan, Sanya 572022, China

Received: 16 April 2026 Revised: 25 May 2026 Accepted: 01 June 2026 Published: 11 June 2026
MSC : 47H10, 54H25

In many practical situations, distance measurements are affected by unavoidable inaccuracies due to instrumental limitations and external factors. Although such errors are often small, their accumulation may significantly impact the validity of mathematical models. This motivates the use of perturbed metric spaces as a natural framework to incorporate such imperfections into fixed point theory. In the current study, many theorems are established for three-point mappings contracting the perimeters of triangles under the structure of triple perturbed metric spaces. The findings presented here extend and consolidate numerous known results in fixed point theory by combining three-point contraction techniques with different perturbed metric structures. The use of three distinct perturbed metrics provides a more flexible and generalized contractive framework. Some examples are given showing that these satisfy the proposed conditions, while existing results cannot be applied to these examples, highlighting the wider applicability of the present results. Finally, we derived rigorous existence and uniqueness conditions that guarantee solutions for fractional differential equations and illustrated their relevance in modeling population dynamics, including factors such as memory effects and mortality in rabbit growth. A complementary numerical example further validates the rigorous results and demonstrates the practical applications of the iterative approximation scheme.
- fixed points,
- three-point contractions,
- perturbed metric spaces,
- mappings contracting perimeters of triangles
Citation: Min Wang, Muhammad Din, Mohammad Akram, Mi Zhou. A unified framework for fixed point results of three-point contractions in multiple perturbed metric spaces with applications to fractional differential equations[J]. AIMS Mathematics, 2026, 11(6): 16712-16734. doi: 10.3934/math.2026686

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Abstract

In many practical situations, distance measurements are affected by unavoidable inaccuracies due to instrumental limitations and external factors. Although such errors are often small, their accumulation may significantly impact the validity of mathematical models. This motivates the use of perturbed metric spaces as a natural framework to incorporate such imperfections into fixed point theory. In the current study, many theorems are established for three-point mappings contracting the perimeters of triangles under the structure of triple perturbed metric spaces. The findings presented here extend and consolidate numerous known results in fixed point theory by combining three-point contraction techniques with different perturbed metric structures. The use of three distinct perturbed metrics provides a more flexible and generalized contractive framework. Some examples are given showing that these satisfy the proposed conditions, while existing results cannot be applied to these examples, highlighting the wider applicability of the present results. Finally, we derived rigorous existence and uniqueness conditions that guarantee solutions for fractional differential equations and illustrated their relevance in modeling population dynamics, including factors such as memory effects and mortality in rabbit growth. A complementary numerical example further validates the rigorous results and demonstrates the practical applications of the iterative approximation scheme.

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