Existence of mild solutions for fractional evolution equations with nonlocal conditions

The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering and science. Indeed, we can find numerous applications in electrochemistry, control, porous media, electromagnetic processing etc. (see [1–5]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [6–11] and references therein.

On the other hand, the nonlocal initial condition, in many cases, has much better effect in applications than the traditional initial condition. As remarked by Byszewski and Lakshmikantham (see [12, 13]), the nonlocal initial value problems can be more useful than the standard initial value problems to describe many physical phenomena.

where $D^q$ is the Caputo fractional derivative of order $q\in (0,1)$, the linear operator −A is the infinitesimal generator of an analytic semigroup ${\{S(t)\}}_{t\ge 0}$ in X, the functions f, h and g will be specified later. $K\in C(\mathrm{\Delta },{\mathbb{R}}^{+})$, where $\mathrm{\Delta }=\{(t,s)|0\le s\le t\le T\}$, ${\mathbb{R}}^{+}=[0,+\mathrm{\infty })$. Throughout this paper, we always assume that $K^{\ast }={max}_{(t,s)\in \mathrm{\Delta }}K(t,s)$.

In some existing articles, the fractional differential equations with nonlocal initial conditions were treated under the hypothesis that the nonlocal term is completely continuous or global Lipschitz continuous. It is obvious that these conditions are not easy to verify in many cases. To make the things more applicable, in [6] the authors studied the existence and uniqueness of mild solutions of Eq. (1) under the case $K(t,s)\equiv 0$. In their main results, they did not assume the complete continuity of the nonlocal term, but they needed the following assumptions:

(F1) there exist a constant $q_1\in [0,q)$ and $m\in L^{\frac{1}{q_1}}(J,{\mathbb{R}}^{+})$ such that $\parallel f(t,x)\parallel \le m(t)$ for all $x\in X$ and almost all $t\in J$;

(F2) there exists a constant $L>0$ such that $\parallel g(u)-g(v)\parallel \le L\parallel u-v\parallel$ for $u,v\in C(J,X)$;

and some other conditions.

In this paper, we will improve the conditions (F1) and (F2). We only assume that f and h satisfy local growth conditions (see assumption (${\mathrm{H}}_3$)) and g is local Lipschitz continuous (see assumption (${\mathrm{H}}_4$)). We will carry out our investigation in the Banach space $X_{\alpha }:=(D(A^{\alpha }),{\parallel \cdot \parallel }_{\alpha })$, $0<\alpha <1$, where $D(A^{\alpha })$ is the domain of the fractional power of A. Finally, an example is given to illustrate the applicability of our main results. We can see that the main results in [6] cannot be applied to our example.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of an analytic semigroup and on the mild solutions of Eq. (1). In Section 3, we study the existence and uniqueness of mild solutions of Eq. (1). In Section 4, we give an example to illustrate the applicability of our results.

$A^{-\alpha }\in \mathcal{L}(X)$ is injective, and $A^{\alpha }$ can be defined by $A^{\alpha }={(A^{-\alpha })}^{-1}$ with the domain $D(A^{\alpha })=A^{-\alpha }(X)$. For $\alpha =0$, let $A^{\alpha }=I$.

Lemma 1 ([14])

as $t\to 0$. Therefore, $S(t)$ ($t\ge 0$) is a strongly continuous semigroup in $X_{\alpha }$, and ${\parallel S_{\alpha }(t)\parallel }_{\alpha }\le \parallel S(t)\parallel$ for all $t\ge 0$. To prove our main results, the following lemma is needed.

Lemma 2 ([15])

If $S(t)$ ($t\ge 0$) is a compact semigroup in X, then $S_{\alpha }(t)$ ($t\ge 0$) is an immediately compact semigroup in $X_{\alpha }$, and hence it is immediately norm-continuous.

The following lemma follows from the results in [6–8, 10].

In this paper, we adopt the following definition of a mild solution of Eq. (1).

for all $t\in J$.

To prove our main results, we also need the following two lemmas.

Lemma 4 A measurable function $H:[0,T]\to X$ is Bochner integrable if $|H|$ is Lebesgue integrable.

Lemma 5 (Krasnoselskii’s fixed point theorem)

Let X be a Banach space, let B be a bounded closed and convex subset of X and let $Q_1$ and $Q_2$ be mappings from B into X such that $Q_{1}x+Q_{2}y\in B$ for every pair $x,y\in B$. If $Q_1$ is a contraction and $Q_2$ is completely continuous, then the operator equation $Q_{1}x+Q_{2}x=x$ has a solution on B.

Lemmas 4 and 5, which can be found in many books, are classical.

The following are the basic assumptions of this paper.

(${\mathrm{H}}_1$) $S(t)$ ($t\ge 0$) is a compact operator semigroup in X.

for all $u,v\in B_r:=\{u\in C(J,X_{\alpha }):{\parallel u(t)\parallel }_{\alpha }\le r,t\in J\}$.

In this section, we introduce the existence and uniqueness theorems of mild solutions of Eq. (1). The discussions are based on fractional calculus and Krasnoselskii’s fixed point theorem. Our main results are as follows.

then Eq. (1) has at least one mild solution on J.

which contradicts (3). Hence, for some $r>0$, $Q_{1}u+Q_{2}v\subset B_r$ for every pair $u,v\in B_r$.

The next proof will be given in two steps.

Step 1. $Q_1$ is a contraction on $B_r$.

which implies that ${\parallel Q_{1}u-Q_{1}v\parallel }_C\le ML{\parallel u-v\parallel }_C$. It follows from (3) that $ML<1$, hence $Q_1$ is a contraction on $B_r$.

Step 2. $Q_2$ is a completely continuous operator on $B_r$.

as $n\to \mathrm{\infty }$. Hence, ${lim}_{n\to \mathrm{\infty }}{\parallel Q_2{u}_n-Q_{2}u\parallel }_C=0$. This means that $Q_2$ is continuous on $B_r$.

Next, we will show that the set $\{Q_{2}u:u\in B_r\}$ is relatively compact. It suffices to show that the family of functions $\{Q_{2}u:u\in B_r\}$ is uniformly bounded and equicontinuous, and for any $t\in J$, the set $\{(Q_{2}u)(t):u\in B_r\}$ is relatively compact.

It follows from Lemma 3 that $I_1\to 0$ as $t_2\to t_1$ and $ϵ\to 0$ independently of $u\in B_r$. From the expressions of $I_2$ and $I_3$, it is clear that $I_2\to 0$ and $I_3\to 0$ as $t_2\to t_1$ independently of $u\in B_r$. Therefore, we prove that $\{Q_{2}u:u\in B_r\}$ is a family of equicontinuous functions.

It remains to prove that for any $t\in J$, the set $W(t):=\{(Q_{2}u)(t):u\in B_r\}$ is relatively compact.

Therefore, there are relatively compact sets arbitrarily close to the set $W(t)$ for $t\in (0,T]$ and since it is compact at $t=0$, we have the relative compactness of $W(t)$ in $X_{\alpha }$ for all $t\in J$.

Therefore, the set $\{Q_{2}u:u\in B_r\}$ is relatively compact by the Ascoli-Arzela theorem. Thus, the continuity of $Q_2$ and relative compactness of the set $\{Q_{2}u:u\in B_r\}$ imply that $Q_2$ is a completely continuous operator. Hence, Krasnoselskii’s fixed point theorem shows that the operator equation $u=Q_{1}u+Q_{2}u$ has a solution on $B_r$. Therefore, Eq. (1) has at least one mild solution. The proof is completed. □

The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.

(${\mathrm{H}}_5$) There exists a constant $\beta \in [\alpha ,1]$ such that the functions $f,h:J\times X_{\alpha }\to X_{\beta }$ are strongly measurable.

for any $x,y\in B_r$ and $t\in J$.

hold, then Eq. (1) has a unique mild solution.