Convex interpolative t-simulation contractions in rectangular b-metric spaces with applications

Sarah Aljohani; Nof T. Alharbi; Tawseef Rashid; Sarah Aljohani; Nof T. Alharbi; Tawseef Rashid

doi:10.3934/math.20261392

AIMS Mathematics

2026, Volume 11, Issue 11: 35902-35928. doi: 10.3934/math.20261392

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Convex interpolative t-simulation contractions in rectangular b-metric spaces with applications

1.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
2.
Mathematics Department, University College of Al-Darb, Jazan University, Jazan 82817, Saudi Arabia
3.
Department of Mathematics, University of Kashmir, Hazratbal-190006, Srinagar, India

Received: 21 August 2026 Revised: 13 October 2026 Accepted: 28 October 2026 Published: 02 November 2026
MSC : 47H10, 54H25, 45G10

This work develops the concept of convex interpolative $ t $-simulation contractions within the framework of rectangular $ b $-metric spaces. The proposed condition employs an $ \mathcal{L} $-simulation function and a convex parameter to combine the ordinary distance between two points with an interpolative expression determined by their displacements under the mapping. Suitable hypotheses are formulated under which the Picard sequence converges to the sole fixed point of the underlying operator. A carefully constructed example illustrates the effectiveness of the new condition and shows that it applies even when certain familiar contractive requirements are not satisfied. Its relationships with several existing contraction principles are also examined through appropriate parameter choices. As an application, the obtained theorem is used to guarantee a uniquely determined continuous solution of a nonlinear Fredholm integral equation.
- convex interpolative contraction,
- $ t $-simulation contraction,
- $ L $-simulation function,
- rectangular $ b $-metric space
Citation: Sarah Aljohani, Nof T. Alharbi, Tawseef Rashid. Convex interpolative t-simulation contractions in rectangular b-metric spaces with applications[J]. AIMS Mathematics, 2026, 11(11): 35902-35928. doi: 10.3934/math.20261392

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Abstract

This work develops the concept of convex interpolative $ t $-simulation contractions within the framework of rectangular $ b $-metric spaces. The proposed condition employs an $ \mathcal{L} $-simulation function and a convex parameter to combine the ordinary distance between two points with an interpolative expression determined by their displacements under the mapping. Suitable hypotheses are formulated under which the Picard sequence converges to the sole fixed point of the underlying operator. A carefully constructed example illustrates the effectiveness of the new condition and shows that it applies even when certain familiar contractive requirements are not satisfied. Its relationships with several existing contraction principles are also examined through appropriate parameter choices. As an application, the obtained theorem is used to guarantee a uniquely determined continuous solution of a nonlinear Fredholm integral equation.

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