Admissible multivalued hybrid $\mathcal{Z}$-contractions with applications

Monairah Alansari; Mohammed Shehu Shagari; Akbar Azam; Nawab Hussain; Monairah Alansari; Mohammed Shehu Shagari; Akbar Azam; Nawab Hussain

doi:10.3934/math.2021026

AIMS Mathematics

2021, Volume 6, Issue 1: 420-441. doi: 10.3934/math.2021026

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Research article

Admissible multivalued hybrid $\mathcal{Z}$-contractions with applications

1.
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Nigeria
3.
Department of Mathematics, COMSATS University, Chak Shahzad, Islamabad, 44000, Pakistan

Received: 08 July 2020 Accepted: 12 October 2020 Published: 16 October 2020
MSC : 46S40, 47H10, 54H25, 34A12, 46J10

In this paper, we introduce new concepts, admissible multivalued hybrid $\mathcal{Z}$-contractions and multivalued hybrid $\mathcal{Z}$-contractions in the framework of $b$-metric spaces and establish sufficient conditions for existence of fixed points for such contractions. A few consequences of our main theorem involving linear and nonlinear contractions are pointed out and discussed by using variants of simulation functions. In the case where our notions are reduced to their single-valued counterparts, the results presented herein complement, unify and generalize a number of significant fixed point theorems due to Branciari, Czerwik, Jachymski, Karapinar and Argawal, Khojasteh, Rhoades, among others. Nontrivial illustrative examples are provided to support the assertions of the obtained results. From application point of view, some fixed point theorems of $b$-metric spaces endowed with partial ordering and graph are deduced and solvability conditions of nonlinear matrix equations are investigated.
- $b$-metric space,
- fixed point,
- hybrid contraction,
- multivalued contraction,
- simulation function,
- $\mathcal{Z}$-contraction,
- matrix equation
Citation: Monairah Alansari, Mohammed Shehu Shagari, Akbar Azam, Nawab Hussain. Admissible multivalued hybrid $\mathcal{Z}$-contractions with applications[J]. AIMS Mathematics, 2021, 6(1): 420-441. doi: 10.3934/math.2021026

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Abstract

In this paper, we introduce new concepts, admissible multivalued hybrid $\mathcal{Z}$-contractions and multivalued hybrid $\mathcal{Z}$-contractions in the framework of $b$-metric spaces and establish sufficient conditions for existence of fixed points for such contractions. A few consequences of our main theorem involving linear and nonlinear contractions are pointed out and discussed by using variants of simulation functions. In the case where our notions are reduced to their single-valued counterparts, the results presented herein complement, unify and generalize a number of significant fixed point theorems due to Branciari, Czerwik, Jachymski, Karapinar and Argawal, Khojasteh, Rhoades, among others. Nontrivial illustrative examples are provided to support the assertions of the obtained results. From application point of view, some fixed point theorems of $b$-metric spaces endowed with partial ordering and graph are deduced and solvability conditions of nonlinear matrix equations are investigated.

References

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