The ordered implicit relations and related fixed point problems in the cone $ b $-metric spaces

Anam Arif; Muhammad Nazam; Aftab Hussain; Mujahid Abbas; Anam Arif; Muhammad Nazam; Aftab Hussain; Mujahid Abbas

doi:10.3934/math.2022290

AIMS Mathematics

2022, Volume 7, Issue 4: 5199-5219. doi: 10.3934/math.2022290

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Research article

The ordered implicit relations and related fixed point problems in the cone $ b $-metric spaces

1.
Department of Mathematics, Government College University, Lahore, Pakistan
2.
Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan
3.
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 09 September 2021 Revised: 01 December 2021 Accepted: 02 December 2021 Published: 04 January 2022
MSC : 47H09, 47H10, 54H25

In this paper, we introduce an ordered implicit relation. We present some examples for the illustration of the ordered implicit relation. We investigate conditions for the existence of the fixed points of an implicit contraction. We obtain some fixed point theorems in the cone $ b $-metric spaces and hence answer a fixed-point problem. We present several examples and consequences to explain the obtained theorems. We solve an homotopy problem and show existence of solution to a Urysohn Integral Equation as applications of the obtained fixed point theorem.
- fixed point,
- implicit relation,
- cone $ b $-metric space,
- application
Citation: Anam Arif, Muhammad Nazam, Aftab Hussain, Mujahid Abbas. The ordered implicit relations and related fixed point problems in the cone $ b $-metric spaces[J]. AIMS Mathematics, 2022, 7(4): 5199-5219. doi: 10.3934/math.2022290

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Abstract

In this paper, we introduce an ordered implicit relation. We present some examples for the illustration of the ordered implicit relation. We investigate conditions for the existence of the fixed points of an implicit contraction. We obtain some fixed point theorems in the cone $ b $-metric spaces and hence answer a fixed-point problem. We present several examples and consequences to explain the obtained theorems. We solve an homotopy problem and show existence of solution to a Urysohn Integral Equation as applications of the obtained fixed point theorem.

References

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