We are concerned with the nonlocal nonlinear fractional problem

where $\frac{{d}^{\alpha}}{d{t}^{\alpha}}$, $0<\alpha \le 1$ is the Riemann-Liouville fractional derivative, $0\le {t}_{1}<\cdots <{t}_{p}\le a$, ${c}_{1},\dots ,{c}_{p}$ are real numbers, *B* and *A* are linear closed operators with domains contained in a Banach space *X* and ranges contained in a Banach space *Y*, $W(t)=({B}_{1}(t)u(t),\dots ,{B}_{r}(t)u(t))$, $\{{B}_{i}(t):i=1,\dots ,r,t\in I=[0,a]\}$ is a family of linear closed operators defined on dense sets ${S}_{1},\dots ,{S}_{r}\supset D(A)\supset D(B)$ respectively in *X* into *X*, $f:I\times {X}^{r}\to Y$ and $g:\mathrm{\Delta}\times {X}^{r}\to Y$ are given abstract functions. Here $\mathrm{\Delta}=\{(s,t):0\le s\le t\le a\}$.

Fractional differential equations have attracted many authors [1, 6–8, 21, 24, 25, 30]. This is mostly because it efficiently describes many phenomena arising in engineering, physics, economics and science. In fact, we can find several applications in viscoelasticity, electrochemistry, electromagnetic, *etc.* For example, Machado [22] gave a novel method for the design of fractional order digital controllers.

Following Gelfand and Shilov [

20], we define the fractional integral of order

$\alpha >0$ as

${I}_{a}^{\alpha}f(t)=\frac{1}{\mathrm{\Gamma}(\alpha )}{\int}_{a}^{t}{(t-s)}^{\alpha -1}f(s)\phantom{\rule{0.2em}{0ex}}ds,$

also, the (Riemann-Liouville) fractional derivative of the function

*f* of order

$0<\alpha <1$ as

${}_{a}D_{t}^{\alpha}f(t)=\frac{1}{\mathrm{\Gamma}(1-\alpha )}\frac{d}{dt}{\int}_{a}^{t}{(t-s)}^{-\alpha}f(s)\phantom{\rule{0.2em}{0ex}}ds,$

where *f* is an abstract continuous function on the interval $[a,b]$ and $\mathrm{\Gamma}(\alpha )$ is the Gamma function, see also [14, 24].

The existence results to evolution equations with nonlocal conditions in a Banach space was studied first by Byszewski [9, 10]; subsequently, many authors were pointed to the same field, see for instance [2–4, 11–13, 19, 28].

Deng [

15] indicated that using the nonlocal condition

$u(0)+h(u)={u}_{0}$ to describe, for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give a better result than using the usual local Cauchy problem

$u(0)={u}_{0}$. Let us observe also that since Deng’s papers, the function

*h* is considered

$h(u)=\sum _{k=1}^{p}{c}_{k}u({t}_{k}),$

(1.3)

where ${c}_{k}$$k=1,2,\dots ,p$ are given constants and $0\le {t}_{1}<\cdots <{t}_{p}\le a$.

However, among the previous research on nonlocal Cauchy problems, few authors have been concerned with mild solutions of fractional semilinear differential equations [23].

Recently, many authors have extended this work to various kinds of nonlinear evolution equations [2, 3, 5, 11, 12, 18]. Balachandran and Uchiyama [3] proved the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition.

In this paper, motivated by [3, 13, 17, 19], we use Schauder fixed point theorem and the semigroup theory to investigate the existence and uniqueness of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, the solutions were obtained by using Gelfand-Shilov approach in fractional calculus and are given in terms of some probability density functions such that their Laplace transforms are indicated [17].

Our paper is organized as follows. Section 2 is devoted to the review of some essential results. In Section 3, we state and prove our main results; the last section deals with giving an example to illustrate the abstract results.