Existence and uniqueness of solution of Cauchy-type problem for Hilfer fractional differential equations

Abstract

The Cauchy-type problem for a nonlinear differential equation involving Hilfer fractional derivative is considered. We prove existence, uniqueness and continuous dependence of a solution for Cauchy-type problem using successive approximations and generalized Gronwall inequality.

Hilfer–Katugampola fractional derivatives

“…The number of operators of fractional integration and differentiation has been increasing during the last years [3,4,5,8,13,14,15,17,20,21,22]. In order to solve fractional differential equations, we mention the following works [1,2,6,11,18], where the authors propose and prove the equivalence between an initial value problem and the Volterra integral equation.…”

Hilfer-Hadamard-Type Fractional Differential Equation with Cauchy-Type Problem

“…Recently, the theory and applications of fractional derivatives have received considerable attention by researchers and authors. They have studied the existence and uniqueness of solutions of fractional differential equations on the different finite intervals such as the examples in [5][6][7][8][13][14][15][17][18][19][20][21]23]and references therein. Some uses of the Gronwall inequality with its applications to the fractional derivatives and the continuous dependence for a solution on the order of fractional differential equations under the initial conditions are studied in [9,11,12,16,17].…”

On existence and approximation of solution of nonlinear Hilfer fractional differential equation

“…The existence and uniqueness of solution of fractional differential equations comprehensively studied using variety of techniques by imposing and/or generalizing different conditions, for example see [1,2,7,9] and references therein. For fractional differential equation containing Hilfer derivative, the existence and uniqueness results can be found in [2,4]. Using fixed point approach and some suitable conditions on the nonlinear function f, results are obtained in weighted space of continuous functions.…”