On the Nonlinear $\Psi$-Hilfer Fractional Differential Equations

Abstract

We consider the nonlinear Cauchy problem for $\Psi$- Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $\Psi$- Hilfer fractional derivative and an estimation have been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formula for the solution of linear Cauchy problem for $\Psi$-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas-Saigo) Mittag-Leffler function.

arXiv comment27 Pages

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Cited by in corpus (6)

• On the Nonlinear $\Psi$-Hilfer Hybrid Fractional Differential Equations
• On tempered fractional calculus with respect to functions and the associated fractional differential equations
• Nonlocal Boundary Value Problem for Generalized Hilfer Implicit Fractional Differential Equations
• On the Nonlinear Impulsive $\Psi$--Hilfer Fractional Differential Equations
• Analysis of Impulsive $\phi$--Hilfer Fractional Differential Equations
• Parametric General Fractional Calculus: Nonlocal Operators Acting on Function with Respect to Another Function