In this paper, we consider the existence and uniqueness of mild solutions to the following nonlinear fractional integro-differential diffusion equations with nonlocal initial conditions: where $\alpha \in (0,1)$, $\beta \in (0,\infty )$, ${\partial }_t^{\alpha }$ and $I_t^{\beta }$ are the $\alpha$ order partial Caputo derivative and $\beta$ order partial Riemann-Liouville integral with respect to t. $u_0\left(x\right)\in H^{-1}(\Omega )$, which is given in Section 2, $\Omega \subset {\mathbb{R}}^n$ with smooth boundary $\partial \Omega$, $T>0$, the nonlinear term f and the nonlocal term g are given functions. The coefficient linear operator A is defined by: where $a_{ij}\left(x\right)(1\le i,j\le n)$ are some real valued functions in $\Omega$ satisfying conditions: with some constants $\varsigma >0$, $p\left(x\right)\in \Omega$ is also a real valued function satisfying where $L^{\infty }(\Omega )$ denotes the space composed of measurable functions defined on $\Omega$ and almost bounded everywhere.

Fractional differential equations originated in 1695, see [1,2,3]. As we have known that they can provide such excellent descriptive models to resolve various problems in reality, to the extent that they are applied in various fields, such as control engineering [4,5], viscoelastic materials [6], fluid mechanics [7], electrochemistry [8], the analysis of epidemic [9] and complex networks [10], and statistical mechanics [11], etc. The relevant problems for the diffusion equations have been studied by many scholars, see [12,13,14] and recent relevant references [15,16,17].

The existence results to nonlocal initial problems in Banach space was initiated by Byszewski and Lakshmikantham [18]. The motivation for these studies is that the nonlocal condition is better to describe the diffusion phenomenon than using the usual local Cauchy problem $u\left(0\right)=u_0$. For example, $g\left(u\right)$ can be given by where $c_i$$(i=1,2,\dots ,l)$ are given constants and $0<{\tau }_1<{\tau }_2<\cdots <{\tau }_l<T$. In addition, for some applications of nonlocal conditions, please refer to [19,20,21,22,23].

Fractional equations containing only differential terms have been studied widely. For instance, in [24], Mu et al. considered the initial-boundary value problem of fractional diffusion equations in Caputo sense: where f is weighted H$\ddot{\mathrm{o}}$lder continuous. The existence, uniqueness and regularity of solutions of (3) be established in $C([0,b],H^{-1}(\Omega ))\cap C((0,b],D(-A))\cap C^{\alpha }((0,b],H^{-1}(\Omega ))$ with some assumptions. Nevertheless, a lot of practical phenomena can be depicted via appropriate models, which include differential and integral terms. Hence the appearance of integro-differential equations shows that their excellent applicability in some physical or engineering areas. In [25], Amin et al. obtained the solutions of a integro-differential equation with initial condition where $\xi \in (0,1]$, W is the kernel of integral, $r\left(t\right)$ and $b\left(t\right)$ are known. This integral term has certain limitations when describing nonlocal diffusion phenomena, and the applicability of the initial condition is also relatively weak.

According to our knowledge, the mild solutions of fractional differential equations are usually represented by the Mittag-Leffler function or probability density function, see [26,27,28,29,30,31,32]. In [32], Zhou et al. obtained a mild solution, where ${\psi }_q\left(\theta \right)$ is the probability density function defined on $(0,\infty )$ and ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is a $C_0$-semigroup generated by A. On the other hand, if $f(x,t,u(x,t\left)\right)=g\left(u\right(x,t\left)\right)=0$, then a formal solution of (1) is Where $X_i$ is the eigenfunction related to the eigenvalue ${\lambda }_i$ of the corresponding eigenvalue problem, that is, $-A\left(X_i\right)={\lambda }_i{X}_i$, $i=1,2,\dots$, see further details in [29]. Obviously, using these technique directly to solve the problem (1) is quite difficult.

Based on the above, in this paper we apply the $(\alpha ,\beta )$-resolvent family to discuss the mild solutions. Resolvent families are powerful for studying solutions to fractional diffusion equations. Chen et al. [33] established the existence and controllability estimation of mild solutions for a class of evolution equations with nonlocal conditions through a resolvent family. Ponce [23] obtained properties on the behavior of mild solutions for fraction Cauchy problems by a resolvent family. Later, Chang et al. [19] proved that if the source function of a diffusion equation has vector value periodicity or almost periodicity or almost automorphism, then the diffusion equation has a mild solution through a resolvent family. Although there is also an integral of f in [19], some of the proof techniques in which are not applicable to this article due to the derivative order of u being different.

This paper is organized as follows. In Section 2, by selecting the appropriate space, we transform (1) into an abstract Cauchy problem and provide some necessary definitions and preliminary results that will be used in the sequel. Afterwards, we define the mild solutions for (9) by Laplace transform. In Section 3, the existence and uniqueness of the mild solutions be established by several fixed point theorems under some assumptions.

In this section, we provide some definitions and lemmas about fractional calculus and the $(\alpha ,\beta )$-resolvent family that will be used in this paper.

If $0<\alpha <1$ and $f:[0,\infty )\to \mathbb{R}$ is continuously differentiable, then If f is an abstract function with values in a Banach space, then the integrals and derivatives appearing in (5) and (7) are understood in Bochner’s sense.

Let $X=H^{-1}(\Omega )$ with the norm $\parallel ·\parallel$ and $J=[0,T]$. Here $H^{-1}(\Omega )$ represents the dual space of $H_0^1(\Omega )$, which is the closure of $C_c^{\infty }(\Omega )$ in $H^1(\Omega )$, $C_c^{\infty }(\Omega )$ consists of the functions in $C^{\infty }(\Omega )$ with compact support, $H^1(\Omega )$ is the Sobolev space (see [36]). $C(J,X)$ denotes the Banach space of all continuous functions from J into X with the norm Similarly, $C(J,{\mathbb{R}}^{+})$ denotes all continuous functions from J into ${\mathbb{R}}^{+}$ with the norm We define $A:D\left(A\right)\subset X\to X$ with $D\left(A\right)=H_0^1(\Omega )$ and $\left(Au\right)\left(t\right)x=Au(x,t)$. Then according to [37], A generates an analytic semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on X. Without loss of generality, we can assume that $0\in \rho \left(A\right)$, ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is uniformly bounded, and there exists a constant $N>0$ such that Denote $\rho \left(A\right)$ and $R(\lambda ,A)={(\lambda I-A)}^{-1}$ are the resolvent set and resolvent operator of A, respectively, where I is the identity operator. By [38], there exist $\delta \in (0,\frac{\pi }{2})$ and $C>0$ such that and

Set $u\left(t\right)\left(x\right)=u(x,t)$, $f(t,u(t\left)\right)\left(x\right)=f(x,t,u(x,t\left)\right)$, where $x\in X$, then (1) can be formulated as an abstract problem with nonlocal initial conditions: where $D_t^{\alpha }$ and $I_t^{\beta }$ denote the $\alpha$ order Caputo derivative and $\beta$ order Riemann-Liouville integral, respectively. $g:C(J,X)\to X$, $f:J\times X\to X$ and $u_0\in X$.

It can be seen from the reference [24,31] that the following results can be obtained.

It is noteworthy that the strongly continuity of ${\left\{R_{\alpha ,\beta }\left(t\right)\right\}}_{t\ge 0}$ can not be obtained immediately by Lemma 8, in which $t>0$, not $t\ge 0$.

Consequently, we give the definition of a mild solution to (12) as follows.

In order to obtain the main results, we should introduce some hypotheses.

(${\mathrm{H}}_1$) $f(t,z)$ is continuous with respect to $z\in X$ for almost all $t\in J$, and strongly measurable with respect to $t\in J$ for any $z\in X$;

(${\mathrm{H}}_1^{\prime }$) $f(t,z)$ is strongly measurable with respect to any $z\in X$ and almost all $t\in J$;

(${\mathrm{H}}_2$) $g:C(J,X)\to X$ is completely continuous, and there exists a constant $L>0$ such that for any $u\in C(J,X)$;

(${\mathrm{H}}_2^{\prime }$) For $g:C(J,X)\to X$, there exists a constant $L>0$ such that for any $u\in C(J,X)$;

(${\mathrm{H}}_3$) There exists a continuous function $\phi :J\to {\mathbb{R}}^{+}$ such that for a.e. $t\in J$ and each $z\in X$;

(${\mathrm{H}}_4$) $T\left(t\right)$ is compact for $t>0$;

(${\mathrm{H}}_5$) For any bounded subset $X_1$ of X and each $t\in J$, there exists $a_1>0$ such that and $\nu \left(f(t,X_1)\right)\le \phi \left(t\right)\nu \left(X_1\right)$, where $\phi \left(t\right)$ is defined as in (${\mathrm{H}}_3$);

(${\mathrm{H}}_6$) There exist $L_g>0$ such that and $\parallel f(t,y_1\left(t\right))-f(t,y_2\left(t\right))\parallel \le \phi \left(t\right)\left(t\right)\parallel y_1-y_2{\parallel }_{\infty }$, where $\phi \left(t\right)$ is defined as in (${\mathrm{H}}_3$), $t\in J$, $y_1,y_2\in B_{k_2}$, and