Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

Abstract

A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions.

Functional-differential equations with Riemann-Liouville integrals in the nonlinearities

“…Some existence results for the initial value problems corresponding to these equations are proved in the papers [5] and [7]. In the papers [18]- [20], several existence results for fractional differential equations are proved. For basic definitions of fractional calculus and fundamentals of the theory of fractional differential equations see, e.g., the monograph [15].…”

Nonlocal cauchy problem for delay fractional integrodifferential equations of neutral type

“…The fractional differential equations have received increasing attention during recent years and have been studied extensively (see, e.g., [1][2][3][4][5][6][7][8][9][10] and references therein) since they can be used to describe many phenomena arising in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc.…”

“…Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [2,3,6,7,[10][11][12]).…”