On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

Iman Ben Othmane; Lamine Nisse; Thabet Abdeljawad; Iman Ben Othmane; Lamine Nisse; Thabet Abdeljawad

doi:10.3934/math.2024686

AIMS Mathematics

2024, Volume 9, Issue 6: 14106-14129. doi: 10.3934/math.2024686

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Research article

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

1.
Department of Mathematics, Faculty of Exact Sciences, Operators Theory and PDE Foundations and Applications Laboratory, University of El-Oued, P.O. Box 789, El-Oued 39000, Algeria
2.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea
5.
Department of Mathematics and Applied Mathematics School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 18 January 2024 Revised: 23 March 2024 Accepted: 02 April 2024 Published: 18 April 2024
MSC : 26A33, 34A08, 34K37

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.
Citation: Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad. On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems[J]. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686

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Abstract

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.

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