A new type of three dimensional metric spaces with applications to fractional differential equations

Nizar Souayah; Nabil Mlaiki; Salma Haque; Doaa Rizk; Amani S. Baazeem; Wasfi Shatanawi; Nizar Souayah; Nabil Mlaiki; Salma Haque; Doaa Rizk; Amani S. Baazeem; Wasfi Shatanawi

doi:10.3934/math.2022980

AIMS Mathematics

2022, Volume 7, Issue 10: 17802-17814. doi: 10.3934/math.2022980

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A new type of three dimensional metric spaces with applications to fractional differential equations

1.
Department of Natural Sciences, Community College Al-Riyadh, King Saud University
2.
Ecole Supérieure des Sciences Economiques et Commerciales de Tunis, Université de Tunis, Tunisia
3.
Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia
4.
Department of Mathematics, College of Science and Arts, Qassim University, Al-Asyah 52571, Saudi Arabia
5.
Department of Mathematics and Statistics, College of Science, Imam Mohammed Ibn Saud Islamic University, P.O. Box 90950, Riyadh 11623, Saudi Arabia
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Received: 21 June 2022 Revised: 26 July 2022 Accepted: 27 July 2022 Published: 03 August 2022
MSC : 47H10, 54H25

In this manuscript, we introduce a three dimension metric type spaces so called $ J $-metric spaces. We prove the existence and uniqueness of a fixed point for self mappings in such spaces with different types of contractions. We use our result to prove the existence and uniqueness of a solution of the following fractional differential equations such as

$ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) & = & f(t,x(t)) = Fx(t) \;{\rm{ if }}\; t\in I_0 = (0,T] \\ x(0) & = & x(T) = r \\ \end{array} \right\} . $

Moreover, we present other applications to systems of linear equations and Fredholm type integral equation.
- fixed point,
- $ J $-metric space,
- fractional differential equation,
- system of linear equations
Citation: Nizar Souayah, Nabil Mlaiki, Salma Haque, Doaa Rizk, Amani S. Baazeem, Wasfi Shatanawi. A new type of three dimensional metric spaces with applications to fractional differential equations[J]. AIMS Mathematics, 2022, 7(10): 17802-17814. doi: 10.3934/math.2022980

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Abstract

In this manuscript, we introduce a three dimension metric type spaces so called $ J $-metric spaces. We prove the existence and uniqueness of a fixed point for self mappings in such spaces with different types of contractions. We use our result to prove the existence and uniqueness of a solution of the following fractional differential equations such as

$ \mathcal{(P)}:\left\{ \begin{array}{ccl} D^{\lambda}x(t) & = & f(t,x(t)) = Fx(t) \;{\rm{ if }}\; t\in I_0 = (0,T] \\ x(0) & = & x(T) = r \\ \end{array} \right\} . $

Moreover, we present other applications to systems of linear equations and Fredholm type integral equation.

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