On The Solution Structure of Fractional İntegro-Differential Equations via A New (Ω, Φ)-Caputo Derivative With The Modified Generalized Mittag-Leffler Kernel

Abstract

This article extends new left-sided (L.S) fractional derivatives and integrals to higher orders with respect to another (wrta) function φ, which involves a weighted function ω and the modified generalized Mittag-Leffler function (GMLF) in the context of Riemann-Liouville (RL) and Caputo operators. Additionally, several operators of non-singular kernels are deduced as special cases of the study. Also, some important properties of the proposed operators are proved. Building on the new general Q-operator of a function wrta function φ, we introduce its reciprocal operator and present the general fundamental relations between the left and right (L&R) sides to any (ω, φ)-fractional integrals and derivatives. As a consequence, we have derived the right-sided (R.S) definitions of our proposed operators, along with their characteristics. Furthermore, fixed point theorems (FPTs) of the Banach and Schaefer are utilized to examine the qualitative results of solutions for (ω, φ)-Caputo fractional integro-differential equations (IDEs). Finally, two mathematical applications are demonstrated to confirm the effectiveness of our outcomes.