Initial Value Problems of Semilinear Supdiffusion Equations

Abstract

We consider the existence of a mild solution of supdiffusion equations and obtain some results under some growth and noncompactness conditions of nonlinearity without coefficient restriction; and some new results for Ulam-Hyers-Rassias stability are obtained.

1. Introduction

The fractional evolution equations with order $\alpha \in (0,1)$ have attracted increasing attention in recent years. In reference [1], Chen and Li concerned the existence of mild solutions for a class of fractional evolution equations; Chen, Zhang and Li dealt with nonlinear time fractional non-autonomous evolution equations with delay; as a result, the existence of a mild solution was obtained in reference [2]; Wang, Zhou and Fečkan discussed Cauchy problems and boundary value problems of nonlinear impulsive problems for fractional differential equations and Ulam stability in reference [3]. Zhou, Wang and Zhang wrote a book [4] about the basic theory of fractional differential equations.

If a coefficient operator is a $C_0$-semigroup, ones needs to construct a corresponding operator to deal with the fractional semilinear evolution equations with order $\alpha \in (0,1)$, which are usually described by the $C_0$-semigroup and the probability density function. However, at present, this way is very difficult for fractional semilinear evolution equations with order $\alpha \in (1,2)$. Well-posed solutions are obtained for a class of supdiffusion equations of order $\alpha \in (1,2)$ in [5]. To the best of our knowledge, not many results are available for the mild solutions of supdiffusion with order $\alpha \in (1,2)$. In 2001, the solution operator for which there is no semigroup property was firstly introduced by Bajlekova [6] to deal with following fractional evolution equations in Banach space E:

$$\left\{\begin{array}{c}{D}_t^{\alpha }u\left(t\right)=Au\left(t\right),t\ge 0,\alpha >0,\hfill \\ u\left(0\right)=x,u^{\left(k\right)}\left(0\right)=\theta ,k=1,2,\cdots ,m-1,\hfill \end{array}\right.$$

(1)

where $m=\left[\alpha \right]$; $\theta$ is a zero element in Banach space E; $D_t^{\alpha }$ is the regularized Caputo fractional derivative by

$$D_t^{\alpha }u\left(t\right):=D_t^m{J}_t^{m-\alpha }\left(u\left(t\right)-\sum _{i=0}^{m-1}{u}^{\left(i\right)}\left(0\right)g_{i+1}\left(t\right)\right),$$

where $D_t^m:=\frac{d^m}{d{t}^m}$, $J_t^{m-\alpha }$ is Riemann-Liouville integral of order $m-\alpha$ of u.

The notion of solution operator is introduced in [6] as follows:

Definition 1.

A family ${\left\{{\mathcal{T}}_{\alpha }\left(t\right)\right\}}_{t\ge 0}$ of bounded linear operators on E is called a solution operator for problem (1) if the following conditions are satisfied:

$\left(a\right)$${\mathcal{T}}_{\alpha }\left(t\right)$ is strongly continuous for $t\ge 0$ and ${\mathcal{T}}_{\alpha }\left(0\right)=I$ (the identity operator);

$\left(b\right)$${\mathcal{T}}_{\alpha }\left(t\right)D\left(A\right)\subset D\left(A\right)$ and $A{\mathcal{T}}_{\alpha }\left(t\right)x={\mathcal{T}}_{\alpha }\left(t\right)Ax$ for all $x\in D\left(A\right)$, $t\ge 0$;

$\left(c\right)$${\mathcal{T}}_{\alpha }\left(t\right)x$ is a solution of $u\left(t\right)=x+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}Au\left(s\right)ds$ for all $x\in D\left(A\right)$, $t\ge 0$.

In fact, problem (1) is well-posed if and only if it admits a solution operator. Moreover, if problem (1) admits a solution operator ${\left\{{\mathcal{T}}_{\alpha }\left(t\right)\right\}}_{t\ge 0}$, then the following fractional evolution equation

$$\left\{\begin{array}{c}{D}_t^{\alpha }u\left(t\right)=Au\left(t\right),t\ge 0,\hfill \\ u^{\left(k\right)}\left(0\right)=x_k,k=0,1,2,\cdots ,m-1,\hfill \end{array}\right.$$

(2)

is well-posed, and solution $u\left(t\right)$ given by

$$u\left(t\right)=\sum _{k=0}^{m-1}\left(J_t^k{\mathcal{T}}_{\alpha }\right)\left(t\right)x_k=\sum _{k=0}^{m-1}(g_k\ast {\mathcal{T}}_{\alpha })\left(t\right)x_k,$$

where $x_k\in D\left(A\right)$, $k=0,1,2,\cdots ,m-1$; $(g_0\ast {\mathcal{T}}_{\alpha })\left(t\right):={\mathcal{T}}_{\alpha }\left(t\right),$ where

$$g_{\beta }\left(t\right)=\frac{t^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)},t>0;$$

$$g_{\beta }\left(t\right)=0,t\le 0;$$

$$\mathsf{\Gamma }\left(\beta \right)={\int }_0^{\infty }{t}^{\beta -1}{e}^{-t}dt,$$

and * denotes the convolution of functions.

For $\alpha \in (1,2)$, at present, properties of the solution operator have been discussed by several researchers such as Li, Chen and Li [7], who studied the fractional powers of generators of fractional resolvent families. Later, Li, Peng and Jia [8] considered the Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives by developing a resolvent family [6]. In 2015, Li [9] investigated the regularity of mild solutions for a linear, inhomogeneous fractional evolution equation:

$$\left\{\begin{array}{c}{D}_t^{\alpha }u\left(t\right)+Au\left(t\right)=h\left(t\right),t\in (0,a],\hfill \\ u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1,\hfill \end{array}\right.$$

(3)

where $D_t^{\alpha }$ is the regularized Caputo fractional derivative for $\alpha \in (1,2)$; $u_0,u_1\in D\left(A\right)$; h is a Bochner integrable function on $(0,a]$; and when $A\in {\mathcal{G}}^{\alpha }(M,\omega )$ (the notation ${\mathcal{G}}^{\alpha }(M,\omega )$ is introduced in next section), the mild solution u is given by

$$u\left(t\right)={\mathcal{S}}_{\alpha }\left(t\right)u_0+(1\ast {\mathcal{S}}_{\alpha })\left(t\right)u_1+(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast h)\left(t\right).$$

where ${\left\{{\mathcal{S}}_{\alpha }\left(t\right)\right\}}_{t\ge 0}$ is the solution operator of the problem (3). Lizama, Pereira and Ponce, in 2016, considered the compactness of fractional resolvent operator functions; see [10]. In particular, in [5], Li, Sun and Feng explored the existence and uniqueness of a classical solution to the linear inhomogeneous problems for order $1<\alpha <2$ and considered the existence and uniqueness of classical solution for nonlinear problems with a resolvent family that they defined. In [11], Wang and Zhou considered Mittag-Leffler-Ulam stabilities of fractional evolution equations of order $0<\alpha <1$. Motivated by the above consideration, in this paper, we study the existence of mild solutions of the following problems:

$$\left\{\begin{array}{c}{D}_t^{\alpha }u\left(t\right)+Au\left(t\right)=f(t,u\left(t\right)),t\in [0,a],\hfill \\ u\left(0\right)=u_0,u^{\prime }\left(0\right)=u_1,\hfill \end{array}\right.$$

(4)

where $D_t^{\alpha }$ is a regularized Caputo fractional derivative for $1<\alpha <2$; $A:D\left(A\right)\subset E\to E$ is a densely defined and closed linear operator in Banach space E. The nonlinear map $f:\varpi \times E\to E$ is continuous and $\varpi =[0,a]$, $a>0$ is a constant, $u_0$ and $u_1$$\in E$. Finally, we consider the stability for problem (4).

2. Preliminaries

Regarding fractional integrals and derivatives, throughout this paper, they are Riemann-Liouville fractional integrals and regularized Caputo fractional derivatives; we refer to the references Kilbas et al. [12]. Assume $u:[0,\infty )\to X$, where X is a Banach space. The fractional integral of order $\alpha >0$ for the function u is defined as

$$J_t^{\alpha }u=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}u\left(s\right)ds,t>0.$$

The standard Caputo fractional derivative of order $1<\alpha <2$ for the function u is defined by

$${}^C{D}_t^{\alpha }u\left(t\right)=\frac{1}{\mathsf{\Gamma }(2-\alpha )}{\int }_0^t{(t-s)}^{1-\alpha }{u}^{″}\left(s\right)ds,t>0.$$

Further, $D_t^{\alpha }$ represents the regularized Caputo fractional derivative of order $1<\alpha <2$ defined by

$$D_t^{\alpha }u\left(t\right)=\frac{d^2}{d{t}^2}\left[J_t^{2-\alpha }(u\left(t\right)-t{u}^{\prime }\left(0\right)-u\left(0\right))\right],t>0.$$

Let $(X,\parallel ·\parallel )$ be a Banach space. Use $C(\varpi ,E)$ for the Banach space of all continuous X-value functions on interval $\varpi$ with the norm ${\parallel u\parallel }_C={sup}_{t\in \varpi }\parallel u\left(t\right)\parallel$, and denote by $\mathfrak{B}\left(E\right)$ the Banach space of all linear and bounded operators in E endowed with the topology defined by the operator norm. Denote

$${\Sigma }_{\vartheta }\left(\omega \right):=\{\lambda \in \mathbb{C}\backslash \left\{0\right\}\mid |\arg (\lambda -\omega )|<\vartheta |\},\vartheta \in [0,\pi ),\omega \in \mathbb{R},$$

and ${\Sigma }_{\vartheta }:={\Sigma }_{\vartheta }\left(0\right).$

Definition 2. 

([6]) Suppose $\alpha \in (1,2)$. The solution operator ${\mathcal{S}}_{\alpha }\left(t\right)\subseteq \mathfrak{B}\left(E\right)$ of (3) is called exponentially bounded if there are constants $M\ge 1$ and $\omega \ge 0$ such that

$$\parallel {\mathcal{S}}_{\alpha }\left(t\right)\parallel \le M{e}^{\omega t},t\ge 0.$$

(5)

If operator ${\mathcal{S}}_{\alpha }\left(t\right)$ satisfies (5), we will write

$A\in {\mathcal{G}}^{\alpha }(M,\omega )$. Denote ${\mathcal{G}}^{\alpha }\left(\omega \right)$$:=\cup \{{\mathcal{G}}^{\alpha }(M,\omega )\mid M\ge 1\}$, ${\mathcal{G}}^{\alpha }$$:=\cup \{{\mathcal{G}}^{\alpha }\left(\omega \right)\mid \omega \ge 0\}.$

Definition 3. 

([6]) Suppose $\alpha \in (1,2)$. A solution operator ${\mathcal{S}}_{\alpha }\left(t\right)$ of $\left(3\right)$ is called analytic if ${\mathcal{S}}_{\alpha }\left(t\right)$ admits an analytic extension to a sector ${\Sigma }_{{\vartheta }_0}$ for some ${\vartheta }_0\in (0,\frac{\pi }{2}]$. An analytic solution operator is said to be of analytic type $({\theta }_0,{\omega }_0)$ if for each $\vartheta <{\vartheta }_0$ and $\omega >{\omega }_0$ there is $M=M(\vartheta ,\omega )$ such that $\parallel {\mathcal{S}}_{\alpha }\left(t\right)\parallel \le M{e}^{\omega \mathrm{Re}t},t\in {\Sigma }_{\vartheta }$. The set of all operators $A\in {\mathcal{G}}^{\alpha }$, generating analytic solution operators ${\mathcal{S}}_{\alpha }\left(t\right)$ of type $({\vartheta }_0,{\omega }_0)$, is denoted by ${\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. In addition, denote ${\mathcal{A}}^{\alpha }\left({\vartheta }_0\right):=\cup \{{\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)\mid {\omega }_0\in {\mathbb{R}}^{+}\}$, ${\mathcal{A}}^{\alpha }:=\cup \{{\mathcal{A}}^{\alpha }\left({\vartheta }_0\right)\mid {\vartheta }_0\in (0,\frac{\pi }{2}]\}$.

From the proof of Theorem 2.14 and Theorem 2.23 in references [6], for $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$ and $t>0$, the estimate

$$\parallel {\mathcal{S}}_{\alpha }{\left(t\right)\parallel }_{\mathfrak{B}\left(E\right)}\le M{e}^{\omega t};{\parallel (1\ast {\mathcal{S}}_{\alpha })\left(t\right)\parallel }_{\mathfrak{B}\left(E\right)}\le M{e}^{\omega t}t;$$

(6)

$$\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })\left(t\right){\parallel }_{\mathfrak{B}\left(E\right)} \le M{e}^{\omega t}{g}_{\alpha }\left(t\right).$$

(7)

is held.

Remark 2. 

If A is a positive definite operator then the condition $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$ is easily satisfied.

Further, from references [5,6,9] combined with Definitions 2 and 3, let $f:\varpi \times E\to E$ be continuous and there are ${\vartheta }_0\in (0,\frac{\pi }{2}]$, ${\omega }_0\in \mathbb{R}$, such that $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. We consider problem (4) with the expression of mild solutions of (3). A function $u\in C(\varpi ,E)$ is said to be a mild solution of (4) if it satisfies the following integral equation.

$$u\left(t\right)={\mathcal{S}}_{\alpha }\left(t\right)u_0+(1\ast {\mathcal{S}}_{\alpha })\left(t\right)u_1+(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,u\left(t\right)).$$

(8)

Definition 2. 

([13,14]) The Kuratowski measure of noncompactness $\kappa (·)$ defined on bounded set U of Banach space E is

$$\kappa \left(U\right):=inf\{\delta >0:U={\cup }_{i=1}^m{U}_{i}anddiam\left(U_i\right)\le \delta \mathrm{for}i=1,2,\cdots ,m\}.$$

Lemma 1. 

([13,14]) Let E be a Banach space and let $U_1$, $U_2\subset E$ be bounded. Then the following properties are satisfied:

(1) 

$\kappa \left(U_1\right)=0$ if and only if $\overline{U_1}$ is compact, where $\overline{U_1}$ means the closure hull of $U_1$;

(2) 

$\kappa \left(U_1\right)=\kappa \left(\overline{U_1}\right)=\kappa \left(\mathrm{conv}{U}_1\right)$, where conv $U_1$ means the convex hull of $U_1$;

(3) 

$\kappa \left(\lambda U_1\right)=\left|\lambda \right|\kappa \left(U_1\right)$ for any $\lambda \in \mathbb{R}$;

(4) 

$U_1\subset U_2$ implies $\kappa \left(U_1\right)\le \kappa \left(U_2\right)$;

(5) 

$\kappa (U_1\cup U_2)=max\{\kappa \left(U_1\right),\kappa \left(U_2\right)\}$;

(6) 

$\kappa (U_1+U_2)\le \kappa \left(U_1\right)+\kappa \left(U_2\right)$, where $U_1+U_2=\{x\mid x=y+z,y\in U_1,z\in U_2\}$;

(7) 

If the map $Q:\mathcal{D}\left(Q\right)\subset E\to X$ is Lipschitz continuous with constant k, then $\kappa \left(Q\right(V\left)\right)\le k\kappa \left(V\right)$ for any bounded subset $V\subset \mathcal{D}\left(Q\right)$, where X is another Banach space.

Lemma 2. 

([15]) (1) Let $B_0=\left\{x_n\right\}\subset L^1([c,d],E)$. If there exists a $g\in L^1([c,d],{\mathbb{R}}^{+})$ such that $\parallel x_n\left(t\right)\parallel \le g\left(t\right)$ for $x_n\in B_0$ and almost all $t\in [c,d]$, then $\kappa \left(B_0\left(t\right)\right)\in L^1([c,d],{\mathbb{R}}^{+})$ and

$$\kappa \left(\left\{{\int }_c^t{x}_n\left(s\right)ds\mid n\in \mathbb{N}\right\}\right)\le 2{\int }_c^t\kappa \left(B_0\left(s\right)\right)ds,t\in [c,d].$$

(2) If $B_1\subset C([c,d],E)$ is bounded and equicontinuous, then $\kappa \left(B_1\left(t\right)\right)\in L^1([c,d],$${\mathbb{R}}^{+})$ and

$$\kappa \left({\int }_c^t{B}_1\left(s\right)ds\right)\le {\int }_c^t\kappa \left(B_1\left(s\right)\right)ds,t\in [c,d].$$

Lemma 3. 

([16]) (Mönch fixed point theorem) Let E be a Banach space, $\mathsf{\Omega }$ a bounded open subset in E and ${\vartheta }^{\ast }\in \mathsf{\Omega }$. If operator $\mathcal{F}:\overline{\mathsf{\Omega }}\to E$ is continuous and satisfies the following conditions:

(1) $x\ne \nu \mathcal{F}x$, $\nu \in (0,1),x\in \partial \mathsf{\Omega };$

(2) D is relatively compact if $D\subset \overline{\mathrm{co}}(\left\{\theta \right\}\cup \mathcal{F}\left(D\right))$ for any countable set $D\subset \overline{\mathsf{\Omega }}$.

Then $\mathcal{F}$ has a fixed point in $\mathsf{\Omega }$.

Denote $C(\varpi ,{\mathbb{R}}^{+}):=\{u\in C(\varpi ,\mathbb{R})\mid u\left(t\right)\ge 0\}$. Denote $\epsilon >0$ and $\psi \in C(\varpi ,{\mathbb{R}}^{+})$. Consider inequalities

$$\parallel D_t^{\alpha }x\left(t\right)+Ax\left(t\right)-f(t,x\left(t\right))\parallel \le \epsilon \psi \left(t\right),t\in \varpi .$$

(9)

Here, we only introduce the concept of Ulam-Hyers-Rassias stability for problem (4); however, the others notion of Ulam-Hyers stability, generalized Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for problem (4) is similar to the reference [17].

Definition 5.

Let $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$, $u_0,u_1\in D\left(A\right)$. The problem (4) is Ulam-Hyers-Rassias stable with respect to ψ if there exists $c_{f,\psi }>0$ such that for each $\epsilon >0$ and for each solution $x\in C^1(\varpi ,E)\cap C((0,a],D\left(A\right))\cap C^2((0,a],E)$ of inequality (9) there exists a mild solution $u\in C(\varpi ,E)$ of (4) with

$$\parallel x\left(t\right)-u\left(t\right)\parallel \le \epsilon c_{f,\psi }\psi \left(t\right),t\in \varpi .$$

Remark 2.

A function $x\in C^1(\varpi ,E)\cap C((0,a],D\left(A\right))\cap C^2((0,a],E)$ is a solution of the inequality (9) iff there is $H\in C^2(\varpi ,E)\cap C(\varpi ,D\left(A\right))$ such that

(a) $\parallel H\left(t\right)\parallel \le \epsilon \psi \left(t\right);$

(b) $D_t^{\alpha }x\left(t\right)+Ax\left(t\right)-f(t,x\left(t\right))=H\left(t\right),t\in \varpi .$

From the inequality (9), Definition 5 and Remark 2, according to the reference [17], we get the following results.

Remark 3.

A function $x\in C^1(\varpi ,E)\cap C((0,a],D\left(A\right))\cap C^2((0,a],E)$ is a solution of (9), so x is a solution of inequality

$$\begin{array}{c}\parallel x\left(t\right)-{\mathcal{S}}_{\alpha }\left(t\right)x\left(0\right)-(1\ast {\mathcal{S}}_{\alpha })\left(t\right)x^{\prime }\left(0\right)-{\int }_0^t(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)f(s,x\left(s\right))ds\left|\right|\hfill \\ \le \epsilon M{\int }_0^t{e}^{\omega (t-s)}{g}_{\alpha }(t-s)\psi \left(s\right)ds,t\in \varpi .\hfill \end{array}$$

(10)

Make the following assumptions:

$\left(F1\right)$

The function $f:\varpi \times E\to E$ is continuous and for $\xi \in C(\varpi ,{\mathbb{R}}^{+})$ such that

$$\parallel f(t,x)\parallel \le \xi \left(t\right)\parallel x\parallel ,t\in \varpi ,x\in E.$$

$\left(F2\right)$

For any bounded set $D\subset C(\varpi ,E)$, there is a nondecreasing function $\eta \in C(\varpi ,{\mathbb{R}}^{+})$ such that

$$\kappa \left(f\right(t,D\left(t\right)\left)\right)\le \eta \left(t\right)\kappa \left(D\right(t\left)\right),t\in \varpi .$$

$\left(F3\right)$

The function $f:\varpi \times E\to E$ is continuous, $f(t,\theta )=\theta$ for $t\in \varpi$ and there is a constant $L>0$ such that

$$\parallel f(t,x)-f(t,y)\parallel \le L\parallel x-y\parallel ,t\in \varpi ,x,y\in E.$$

$\left(F4\right)$

For $L_f\in C(\varpi ,{\mathbb{R}}^{+})$,

$$\parallel f(t,x)-f(t,y)\parallel \le L_f\left(t\right)\parallel x-y\parallel ,t\in \varpi ,x,y\in E$$

is held.

$\left(F5\right)$

For a constant ${\zeta }_{\psi }>0$ and nondecreasing function $\psi \in C(\varpi ,{\mathbb{R}}^{+})$,

$${\int }_0^t\psi \left(s\right)ds\le {\zeta }_{\psi }\psi \left(t\right),t\in \varpi$$

is held.

3. Existence and Uniqueness

Theorem 1.

Let E be a Banach space and $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$, $u_0,u_1\in D\left(A\right)$. Suppose ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology, if the functions $f:\varpi \times E\to E$ satisfies $\left(F1\right)$ and $\left(F2\right)$. Then problem (4) has at least one mild solution.

Proof. 

Define the operator $\mathcal{F}:C(\varpi ,E)\to C(\varpi ,E)$ by

$$\left(\mathcal{F}u\right)\left(t\right)=\begin{array}{c}{\mathcal{S}}_{\alpha }\left(t\right)u_0+(1\ast {\mathcal{S}}_{\alpha })\left(t\right)u_1+(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,u\left(t\right)),t\in \varpi .\hfill \end{array}$$

(11)

From (8), the mild solution of (4) is equivalent to the fixed point of $\mathcal{F}$ by (11). Firstly, we prove that $\mathcal{F}$ is continuous in $C(\varpi ,E)$. By the continuity of f with respect to the second variable, for all $s\in \varpi$, we know that

$$\underset{n\to +\infty }{lim}f(s,u_n\left(s\right))=f(s,u\left(s\right)),$$

(12)

where $u_n\in C(\varpi ,E)$ and $\underset{n\to +\infty }{lim}{u}_n=u$ in $C(\varpi ,E)$. By (7) and assumption $\left(F1\right)$, we obtain that

$$\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)(f(s,u_n\left(s\right))-f(s,u\left(s\right)))\parallel \le \frac{M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\xi \left(s\right)(\parallel u_n\parallel +\parallel u\parallel ).$$

(13)

Obviously, the above right part of inequality is Lebesgue integrable for $s\in \varpi$. From (7), (12) and (13) for $t\in \varpi$ and the Lebesgue dominated convergence theorem, we know that

$$\begin{array}{ccc}\hfill \parallel \left(\mathcal{F}{u}_n\right)\left(t\right)-\left(\mathcal{F}u\right)\left(t\right)\parallel & \le & {\int }_0^t\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)(f(s,u_n\left(s\right))-f(s,u\left(s\right)))\parallel ds\hfill \\ & \to & 0\mathrm{as}n\to +\infty .\hfill \end{array}$$

Then

$$\parallel \mathcal{F}{u}_n{-\mathcal{F}u\parallel }_C\to 0\mathrm{as}n\to +\infty ,$$

which means that $\mathcal{F}$ : $C(\varpi ,E)\to C(\varpi ,E)$ is continuous.

Next, we show that the set

$$\mathsf{\Omega }=\{u\in C(\varpi ,E)\mid u=\nu \mathcal{F}u,\nu \in (0,1\left)\right\}$$

is bounded. Conversely, if $u\in \mathsf{\Omega }$, there is ${\nu }_0\in (0,1)$ such that $u={\nu }_0\mathcal{F}u$. From (6) and (7) and assumption $\left(F1\right)$, we have that for $t\in \varpi ,$

$$\begin{array}{ccc}\hfill \parallel u\left(t\right)\parallel & \le & {\nu }_0\parallel \mathcal{F}u\left(t\right)\parallel \hfill \\ & \le & \parallel {\mathcal{S}}_{\alpha }\left(t\right)\parallel \parallel u_0\parallel +\parallel (1\ast {\mathcal{S}}_{\alpha })\left(t\right)\parallel \parallel u_1\parallel +\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,u\left(t\right))\parallel \hfill \\ & \le & M{e}^{\omega a}\parallel u_0\parallel +aM{e}^{\omega a}\parallel u_1\parallel +\frac{M{e}^{\omega a}{a}^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}\parallel u\parallel \parallel \xi \parallel \hfill \\ & :=& c^{\ast }=c^{\ast }({\nu }_0,u).\hfill \end{array}$$

By the above inequality, we get that $\parallel u\left(t\right)\parallel \le c^{\ast }$ for $t\in \varpi$, namely, $\mathsf{\Omega }$ is a bounded set.

Choose $R>c^{\ast }$ and denote ${\mathsf{\Omega }}_R{=\{u\in C(\varpi ,E)\mid \parallel u\parallel }_C<R\}$; then ${\mathsf{\Omega }}_R$ is a bounded open set and $\theta \in {\mathsf{\Omega }}_R.$ Since $R>c^{\ast }$, we have that $u\ne \nu \mathcal{F}u,u\in \partial {\mathsf{\Omega }}_R,\nu \in (0,1).$ Let $D\subset \overline{{\mathsf{\Omega }}_R}$ be a countable set and $D\subset \overline{\mathrm{co}}(\left\{\theta \right\}\cup \mathcal{F}\left(D\right))$. Then

$$D\left(t\right)\subset \overline{\mathrm{co}}(\left\{\theta \right\}\cup \mathcal{F}\left(D\left(t\right)\right)).$$

(14)

For $0\le t^{\prime }<t^{″}\le a$ and $u\in D$, we have

$$\begin{array}{ccc}& & \parallel \left(\mathcal{F}u\right)\left(t^{″}\right)-\left(\mathcal{F}u\right)\left(t^{\prime }\right)\parallel \hfill \\ & \le & \parallel {\mathcal{S}}_{\alpha }\left(t^{″}\right)-{\mathcal{S}}_{\alpha }\left(t^{\prime }\right)u_0\parallel +\parallel (1\ast {\mathcal{S}}_{\alpha })\left(t^{″}\right)u_1-(1\ast {\mathcal{S}}_{\alpha })\left(t^{\prime }\right)u_1\parallel \hfill \\ & & +{\int }_{t^{\prime }}^{t^{″}}\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t^{″}-s)\parallel \parallel f(s,u\left(s\right))\parallel ds\hfill \\ & & +{\int }_0^{t^{\prime }}\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t^{″}-s)-(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t^{\prime }-s)\parallel \parallel f(s,u\left(s\right))\parallel ds.\hfill \end{array}$$

Given the continuity of solution operator ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$, the boundedness of $(1\ast {\mathcal{S}}_{\alpha })\left(t\right)(t\ge 0)$, the differentiability of the $(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })\left(t\right)(t\ge 0)$ and $\left(F1\right)$, we easily know that if $t^{″}-t^{\prime }\to 0$, then

$$\parallel \left(\mathcal{F}u\right)\left(t^{″}\right)-\left(\mathcal{F}u\right)\left(t^{\prime }\right)\parallel \to 0$$

for $u\in D$, which means that the $\mathcal{F}\left(D\right)$ in interval $\varpi$ is equicontinuous.

Finally, we show that D is relatively compact. From (7), (11), (14), $\left(F2\right)$ and measures of noncompactness properties and Lemma 2, for $t\in \varpi$, we get that

$$\begin{array}{ccc}\hfill \kappa \left(D\right(t\left)\right)& \le & \kappa \left(\mathcal{F}\right(D\left)\right(t\left)\right)\hfill \\ & =& \kappa \left(\right\{{\mathcal{S}}_{\alpha }\left(t\right)u_0+(1\ast {\mathcal{S}}_{\alpha })\left(t\right)u_1\hfill \\ & & +{\int }_0^t(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)f(s,u\left(s\right))ds\mid u\in D\left\}\right)\hfill \\ & \le & \kappa \left(\left\{{\int }_0^t(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)f(s,u\left(s\right))ds\mid u\in D\right\}\right)\hfill \\ & \le & 2{\int }_0^t(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)\kappa \left(\{f(s,u\left(s\right))\mid u\in D\}\right)ds\hfill \\ & \le & \frac{2M}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{e}^{\omega (t-s)}{(t-s)}^{\alpha -1}\eta \left(s\right)\kappa \left(D\left(s\right)\right)ds\hfill \\ & \le & \frac{2M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t\eta \left(s\right)\kappa \left(D\left(s\right)\right)ds.\hfill \end{array}$$

The Bellman inequality implies that $\kappa \left(D\right(t\left)\right)\equiv 0$ on $\varpi$, which is the $D\left(t\right)$ relatively compact on $\overline{\mathrm{co}}(\left\{\theta \right\}\cup \mathcal{F}\left(D\left(t\right)\right))$. Namely, D is a relative compact on set $\overline{{\mathsf{\Omega }}_R}$. Therefore, from the Lemma 3, the map $\mathcal{F}$ exists as a fixed point in $\overline{{\mathsf{\Omega }}_R}$, which is a mild solution of (4). □

Remark 4.

From the paper [5], we know that the solution $\left\{{\mathcal{S}}_{\alpha }\left(t\right)\right\}(t\ge 0)$ about the (4) is defined, which is not the generality. However, if $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$, a solution $\left\{{\mathcal{S}}_{\alpha }\left(t\right)\right\}(t\ge 0)$ exists for problem (4) in which the conditions are more general than [5].

Remark 5.

In our results, there is no similar restricted condition to the condition about coefficient restriction (5.7) in reference [5].

Theorem 2.

Let E be a Banach space and $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$, $u_0,u_1\in D\left(A\right)$. Suppose ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology, if the functions $f:\varpi \times E\to E$ satisfy $\left(F3\right)$. Then problem (4) has a unique mild solution.

Proof. 

From assumption $\left(F3\right)$ and for any bounded set $D\subset C(\varpi ,E)$, we obtain that

$$\parallel f(t,u)\parallel \le L\parallel u\parallel ,t\in \varpi ,u\in E.$$

$$\kappa \left(f\right(t,D\left(t\right)\left)\right)\le L\kappa \left(D\right(t\left)\right),t\in \varpi .$$

By above two inequality, we easily see that the conditions $\left(F1\right)$ and $\left(F2\right)$ of Theorem 1 are satisfied. Then problem (4) has at least one mild solution. Nextly, we show uniqueness of solution. Let $u_1$, $u_2\in \overline{{\mathsf{\Omega }}_R}$ be two fixed points of the operator $\mathcal{F}$ defined by (11). For $t\in \varpi$, we get that

$$\begin{array}{ccc}\hfill \parallel u_1\left(t\right)-u_2\left(t\right)\parallel & =& \parallel \left(\mathcal{F}{u}_1\right)\left(t\right)-\left(\mathcal{F}{u}_2\right)\left(t\right)\parallel \hfill \\ & \le & {\int }_0^t\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)\parallel \parallel f(s,u_1\left(s\right))-f(s,u_2\left(s\right))\left|\right|ds\hfill \\ & \le & \frac{ML{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t\parallel u_1\left(s\right)-u_2\left(s\right)\parallel ds.\hfill \end{array}$$

From the Bellman inequality, we obtain $u_1\left(t\right)=u_2\left(t\right),t\in \varpi ,$ i.e., (4) has a unique mild solution. □

4. Stability

In this section, we prove the stability of solutions for problems (4).

Theorem 3.

Let E be a Banach space and $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. Suppose ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology, if the functions $f:\varpi \times E\to E$ satisfy $\left(F3\right)$ and $\left(F4\right)$ and $\left(F5\right)$ are satisfied. Then problem (4) is Ulam-Hyers-Rassias stable with respect to ψ.

Proof. 

Let $x\in C^1(\varpi ,E)\cap C((0,a],D\left(A\right))\cap C^2((0,a],E)$ be a solution of (9). Let u be the unique mild solution of the following evolution equations:

$$\left\{\begin{array}{c}{D}_t^{\alpha }u\left(t\right)+Au\left(t\right)=f(t,u\left(t\right)),t\in \varpi ,\hfill \\ u\left(0\right)=x\left(0\right),u^{\prime }\left(0\right)=x^{\prime }\left(0\right),\hfill \end{array}\right.$$

and it satisfies

$$u\left(t\right)={\mathcal{S}}_{\alpha }\left(t\right)x\left(0\right)+(1\ast {\mathcal{S}}_{\alpha })\left(t\right)x^{\prime }\left(0\right)+(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,u\left(t\right)),t\in \varpi .$$

By inequality (10) and $\left(F5\right)$, it follows that

$$\begin{array}{c}\parallel x\left(t\right)-{\mathcal{S}}_{\alpha }\left(t\right)x\left(0\right)-(1\ast {\mathcal{S}}_{\alpha })\left(t\right)x^{\prime }\left(0\right)-(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,x\left(t\right))\parallel \hfill \\ \le \epsilon M{\int }_0^t{e}^{\omega (t-s)}{g}_{\alpha }(t-s)\psi \left(s\right)ds\hfill \\ \le \frac{\epsilon M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\zeta }_{\psi }\psi \left(t\right).\hfill \end{array}$$

Hence, for $t\in \varpi$ with conditions $\left(F4\right)$ and $\left(F5\right)$ we can obtain that

$$\begin{array}{ccc}& & \parallel x\left(t\right)-u\left(t\right)\parallel \hfill \\ & =& \parallel x\left(t\right)-{\mathcal{S}}_{\alpha }\left(t\right)x\left(0\right)-(1\ast {\mathcal{S}}_{\alpha })\left(t\right)x^{\prime }\left(0\right)-(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,u\left(t\right))\parallel \hfill \\ & \le & \parallel x\left(t\right)-{\mathcal{S}}_{\alpha }\left(t\right)x\left(0\right)-(1\ast {\mathcal{S}}_{\alpha })\left(t\right)x^{\prime }\left(0\right)-(g_{\alpha -1}\ast {\mathcal{S}}_{\alpha }\ast f)(t,x\left(t\right))\parallel \hfill \\ & & +{\int }_0^t\parallel (g_{\alpha -1}\ast {\mathcal{S}}_{\alpha })(t-s)\parallel \parallel f(s,x\left(s\right))-f(s,u\left(s\right))\parallel ds\hfill \\ & \le & \frac{\epsilon M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\zeta }_{\psi }\psi \left(t\right)+\frac{M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{L}_f\left(s\right)\parallel x\left(s\right)-u\left(s\right)\parallel ds.\hfill \end{array}$$

By applying the Bellman inequality for the above inequality, we get that

$$\begin{array}{ccc}\hfill \parallel x\left(t\right)-u\left(t\right)\parallel & \le & \frac{\epsilon M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\zeta }_{\psi }\psi \left(t\right)\exp \left(\frac{M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{L}_f\left(s\right)ds\right)\hfill \\ & \le & c_{L_f}\epsilon \psi \left(t\right)\hfill \end{array}$$

for any $t\in \varpi$, where $c_{L_f}:=\frac{M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\zeta }_{\psi }\exp \left(\frac{M{e}^{\omega a}{a}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{L}_f\left(s\right)ds\right)>0.$ Therefore, problem (4) is Ulam-Hyers-Rassias stable with respect to $\psi$. □

With the proof of Theorem 3 and by combining it with reference [17], we easily obtain following Corollaries.

Corollary 1.

Let E be a Banach space and $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. Suppose ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology, if the functions $f:\varpi \times E\to E$ satisfy $\left(F3\right)$ and $\left(F4\right)$ and $\left(F5\right)$ are satisfied. Then problem (4) is generalized Ulam-Hyers-Rassias stable with respect to ψ.

Corollary 2.

Let E be a Banach space and $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. Suppose ${\mathcal{S}}_{\alpha }\left(t\right)(t\ge 0)$ is continuous in the uniform operator topology, if the functions $f:\varpi \times E\to E$ satisfy $\left(F3\right)$, and $\left(F4\right)$ and $\left(F5\right)$ are satisfied. Then problem (4) is Ulam-Hyers stable.

Remark 6.

To the best of our knowledge, rare results are available for the stability of problems (4).

5. Application

In order to illustrate our results of existence, uniqueness and stability, we consider the following supdiffusion equations

$$\left\{\begin{array}{c}{D}_t^{\alpha }u(t,x)=\Delta u(t,x)+\frac{t}{2}sinu(t,x),t\in (0,1],x\in {\mathbb{R}}^n,\hfill \\ u(0,x)={\phi }_1\left(x\right),u_t^{\prime }(0,x)={\phi }_2\left(x\right),x\in {\mathbb{R}}^n,\hfill \end{array}\right.$$

(15)

where ${\phi }_1$, ${\phi }_1\in L^2\left({\mathbb{R}}^n\right)$, $\Delta$ is the Laplace operator.

Let $E=L^2\left({\mathbb{R}}^n\right)$ with the norm ${\parallel ·\parallel }_2$. We define an operator A in Hilbert space E by

$$D\left(A\right)=\{u\in E\mid \Delta u\in E\},Au:=-\Delta u.$$

We can know that A is a positive definite operator, namely, $A\in {\mathcal{A}}^{\alpha }({\vartheta }_0,{\omega }_0)$. Let $u\left(t\right)=u(t,·)$, $f(t,u\left(t\right))=\frac{t}{2}sinu(t,·)$, $u_0={\phi }_1(·)$, $u_1={\phi }_2(·)\in L^2\left({\mathbb{R}}^n\right)$; then supdiffusion Equation (15) can be transformed into the abstract form of problem (4). Additionally, since

$${\parallel f(t,u)\parallel }_2={\left({\int }_{{\mathbb{R}}^n}\frac{t^2}{4}{sin}^{2}u(t,x)dx\right)}^{\frac{1}{2}}\le \frac{t}{2}{\parallel u\parallel }_2.$$

$${\parallel f(t,u)-f(t,v)\parallel }_2={\left({\int }_{{\mathbb{R}}^n}\frac{t^2}{4}{(sinu(t,x)-sinv(t,x))}^{2}dx\right)}^{\frac{1}{2}}\le \frac{t}{2}{\parallel u-v\parallel }_2\le \frac{1}{2}{\parallel u-v\parallel }_2.$$

Let $\mathsf{\Omega }\subset {\mathbb{R}}^n$ be a bounded domain; for $u\in \mathsf{\Omega }$, we have $\kappa \left(f(t,\mathsf{\Omega }\left(t\right))\right)\le \frac{t}{2}\kappa \left(\mathsf{\Omega }\left(t\right)\right)$. By the above analysis, we can easily verify that the assumptions $\left(F1\right)$, $\left(F2\right)$ and $\left(F3\right)$ are satisfied with $\xi \left(t\right)=\eta \left(t\right)=\frac{t}{2}$; $L=\frac{1}{2}$. Therefore, by Theorems 1 and 2, we have the following existence and uniqueness results.

Theorem 4.

The initial value problem of supdiffusion Equation (15) has a solution $u^{\star }\in ((0,a],{\mathbb{R}}^n)$.

Theorem 5.

The initial value problem of supdiffusion Equation (15) has a uniqueness solution $u^{\star \star }\in ((0,a],{\mathbb{R}}^n)$.

Let $\psi \left(t\right)=\frac{t}{2}$ and ${\zeta }_{\psi }=1$; by Theorem 3, Corollarys 1 and 2, we have the following stability results.

Theorem 6.

The solution of the initial value problem of supdiffusion Equation (15) is Ulam-Hyers-Rassias stable with respect to $\frac{t}{2}$.

Corollary 3.

The solution of initial value problem of supdiffusion Equation (15) is generalized Ulam-Hyers-Rassias stable with respect to $\frac{t}{2}$.

Corollary 4.

The solution of initial value problem of supdiffusion Equation (15) is Ulam-Hyers stable.