On a coupled system of fractional $ (p, q) $-differential equation with Lipschitzian matrix in generalized metric space

Abdellatif Boutiara; Jehad Alzabut; Mehran Ghaderi; Shahram Rezapour; Abdellatif Boutiara; Jehad Alzabut; Mehran Ghaderi; Shahram Rezapour

doi:10.3934/math.2023079

AIMS Mathematics

2023, Volume 8, Issue 1: 1566-1591. doi: 10.3934/math.2023079

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On a coupled system of fractional $ (p, q) $-differential equation with Lipschitzian matrix in generalized metric space

1.
Laboratory of Mathematics and Applied Sciences, University of Ghardaia 47000, Algeria
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Türkiye
4.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 24 June 2022 Revised: 22 September 2022 Accepted: 26 September 2022 Published: 21 October 2022
MSC : 34A08, 34A12, 34C10

This work is concerned with the study of the existing solution for the fractional $ (p, q) $-difference equation under first order $ (p, q) $-difference boundary conditions in generalized metric space. To achieve the solution, we combine some contraction techniques in fixed point theory with the numerical techniques of the Lipschitz matrix and vector norms. To do this, we first associate a matrix to a desired boundary value problem. Then we present sufficient conditions for the convergence of this matrix to zero. Also, we design some algorithms to use the computer for calculate the eigenvalues of such matrices and different values of $ (p, q) $-Gamma function. Finally, by presenting two numerical examples, we examine the performance and correctness of the proposed method. Some tables and figures are provided to better understand the issues.
- $ (p, q) $-difference,
- boundary value problem,
- fixed point,
- Lipschitzian matrix,
- quantum calculus
Citation: Abdellatif Boutiara, Jehad Alzabut, Mehran Ghaderi, Shahram Rezapour. On a coupled system of fractional $ (p, q) $-differential equation with Lipschitzian matrix in generalized metric space[J]. AIMS Mathematics, 2023, 8(1): 1566-1591. doi: 10.3934/math.2023079

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Abstract

This work is concerned with the study of the existing solution for the fractional $ (p, q) $-difference equation under first order $ (p, q) $-difference boundary conditions in generalized metric space. To achieve the solution, we combine some contraction techniques in fixed point theory with the numerical techniques of the Lipschitz matrix and vector norms. To do this, we first associate a matrix to a desired boundary value problem. Then we present sufficient conditions for the convergence of this matrix to zero. Also, we design some algorithms to use the computer for calculate the eigenvalues of such matrices and different values of $ (p, q) $-Gamma function. Finally, by presenting two numerical examples, we examine the performance and correctness of the proposed method. Some tables and figures are provided to better understand the issues.

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