On nonlinear mixed fractional integrodifferential equations with nonlocal condition in Banach spaces

Abstract

In the present paper we investigate the existence and uniqueness of solutions of nonlinear mixed fractional integrodi erential equations with nonlocal condition in Banach spaces. The technique used in our analysis is based on xed point theorems and Pachpatte's integral inequality.

On mixed fractional integrodifferential equations with fractional non-separated boundary conditions

“…It is obvious that B r , is a closed, bounded and convex subset of the Banach space C(J, R). Firstly, we prove that F : B r → B r , where F x = F 1 x + F 2 x and F 1 , F 2 are as defined by (10), (11) respectively. For any x ∈ B r , we have…”

“…Proof : Let F x = F 1 x + F 2 x where F 1 and F 2 are defined by (10) and (11). We first prove that F maps bounded sets into bounded sets in C(J, R).…”

“…Recently, S. D. Kendre et.al. [10] investigated the existence and uniqueness of solutions of special form of (1) - (3). The aim of the present paper is to prove the existence, uniqueness and boundedness of solution of nonlinear fractional mixed integrodifferential equations (1) - (3).…”

On nonlinear mixed fractional integro-differential equations with positive constant coefficient

“…Recently, many researchers have studied the Cauchy problem and long time behavior for nonlinear fractional differential and integro-differential equations and obtained many interesting results by using all kinds of fixed point theorems, for example, by Aghajani et al [2], Balachandran and Park [4], Barbagallo et al. [7], Cabrera et al [8], Dong et al [11], Furati and Tatar [12], Jagtap and Kharat [14], Kharat [15], Kharat et al [16], Kendre and Kharat [19], Kendre et al [18,20,21], Liang et al [24], N'Guérékata [26,27], Pierri and O'Regan [29], Ragusa and Scapellato [31], Ren et al [32], Ruggieri et al [33], Tate et al [37], Turmetov [40], Wang and Li [41], Zhou et al [42,43], Zhou and Jiao [44] and the references therein.…”