Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021View this Special Issue
Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions
This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.
The fractional evolution equation has been applied to many fields, and scholars have obtained abundant research achievements [1–13]. Impulsive fractional integrodifferential equations can describe some phenomena which often occur in physics, geology, and economics, for instance, earthquake, the closing of the switch in the circuit, and so on. Many scholars are committed to this subject and have achieved plentiful results [1–7]. Based on the fact that nonlocal initial conditions are more effective than classical initial conditions in applied physics, the study of differential equations with nonlocal conditions has attracted more and more researchers’ attention [8–13].
Ji and Li  studied the following impulsive differential evolution equations with nonlocal conditions: where is the generator of a strongly continuous semigroup ; sufficient conditions for the existence of mild solutions have been established by the Hausdorff measure of noncompactness and fixed point theorems.
Zhu et al.  investigated the fractional semilinear integrodifferential equations of mixed type with nonlocal conditions: where , is a closed linear operator with domain defined on a Banach space ; the existence and uniqueness of mild solutions have been established by -set contraction and -resolvent family.
Gou and Li  studied the fractional impulsive integrodifferential equations in Banach space ; local and global existences of mild solutions have been proved by measure of noncompactness and Sadovskii’s fixed point theorem: where , is a closed linear operator and generates a uniformly bounded -semigroup .
Inspired by these contributions, we consider the following impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions: where is the Caputo’s fractional derivative of order , , is a closed linear operator with domain defined on a Banach space , and two integral operators and are defined by
, are continuous and nonlinear functions, , , and are to be specified later, are continuous impulsive functions, the prefixed numbers satisfy , , and represent the left limit of at .
In this paper, we demonstrate the existence of PC-mild solutions for problem (4) via the theory of semigroup and fixed point theorem under the condition of compact resolvent operator. Meanwhile, the uniqueness of PC-mild solutions is proved in the case of noncompact resolvent operator. The kernels and of the integral operators and are nonlinear functions; the function of the nonlocal conditions is noncompact. In addition, the closed linear operator is dependent on . The rest of this paper is organized as follows. In Section 2, some basic definitions and lemmas are collected that will be needed throughout the remaining sections. The existence and uniqueness of PC-mild solutions are shown in Section 3 via the theories of resolvent operators and various fixed point theorems. Finally, the summary of our results comes in Section 4.
Let be a Banach space, and . The collection of all continuous functions from into , denoted , is a Banach space equipped with the norm for . Let endowed with the PC-norm , , .
Lemma 1 (nonlinear alternative for single-valued maps). Let be a Banach space, be a closed convex set, be an open subset of , and . Suppose that is completely continuous, then either (i) has a fixed point in or(ii)there is a and with
Lemma 2 (see ). Let , , denote where . Then, for any fixed constant and any real number , we get
Definition 3 (see [18, 19]). The Caputo fractional derivative of order of a function is defined as where , , denotes the Gamma function. The Laplace transform of the Caputo fractional derivative of order is given as where is the Laplace transform of the function .
Definition 4 (see [20, 21]). Let be a closed and linear operator with domain defined on a Banach space and . Let be the resolvent set of ; is called the generator of a -resolvent family if there exist and a strongly continuous function such that and In this case, is called the -resolvent family generated by .
Definition 6. A function is said to be a PC-mild solution of problem (4) if satisfies the integral equation:
3. Existence and Uniqueness of Mild Solution
Theorem 7. Assume that the conditions - hold true and the resolvent operator is compact.
The function is continuous, and there exist nonnegative Lebesgue integrable functions , , for every , , such that There exist nonnegative Lebesgue integrable functions , , , for all , such that The functions and are continuous, and there exist constants , , , , such that Then, problem (4) has at least one PC-mild solution in .
Proof. Let us consider the operator as follows:
It is easy to see that the operator is well defined in .
At first, we claim that is a continuous operator. Let be a sequence such that in . Since for all , where . Using the fact that , , and are continuous, we obtain Therefore, is continuous.
Furthermore, for any , we prove that is equicontinuous in . For all and , by the condition , we have For all , , we get , by the condition , meanwhile, According to the condition and the above inequalities, for all , we get where Obviously, and are nonnegative Lebesgue integrable functions, then In view of Lemma 5, the compactness of the resolvent operator implies the continuity in the uniform operator topology. As a result, from the above inequalities, we deduce that independently of as . That is, is equicontinuous in .
In the end, we demonstrate that is precompact.
For any , , and , the operator is defined by Since is compact resolvent operator, the set is relatively compact in for every ().
Moreover, for any , , one can show that Thus, is totally bounded. Hence, is relatively compact in , and so, with the help of the Arzelà-Ascoli theorem, is completely continuous.
For , let , we get Then, using the conditions -, it follows that That is, for , then there exists a constant such that . Let , obviously, there is no such that for . Therefore, thanks to Lemma 1, one gets that has at least one fixed point in , which is a PC-mild solution of problem (4). This completes the proof. ☐
Remark 8. Theorem 7 is proved under the condition that is compact for and the functions meet corresponding conditions; in the case that the resolvent operator is noncompact, we would obtain Theorem 9 and Theorem 10.
Theorem 9. Suppose that the conditions - are satisfied, , and .
The function is continuous, and there exist nonnegative Lebesgue integrable functions , for any , , , such that There exist nonnegative Lebesgue integrable functions , for each , , such that The functions and are continuous, and there exist nonnegative constants , such that Then, problem (4) has a unique PC-mild solution in .
Proof. It follows from the conditions -, for any , , one can derive Based on the assumption, we have , which means that the operator is a contraction mapping. Hence, the operator has a unique fixed point , which implies that problem (4) has a unique PC-mild solution. This completes the proof. ☐
Theorem 10. Assume that the conditions and hold, , , .
The function is continuous, and there exist nonnegative Lebesgue integrable functions , for all , , , such that There exist constants , for all , satisfying Then, problem (4) has a unique PC-mild solution in . For all , , iterative sequence are defined by uniformly converge to the unique PC-mild solution in , and for any ,
Proof. Combining the conditions and , for all , , we get
where . It is easy to see that . Notice that , then there is such that . For the above , there exists a continuous function such that
where . We next prove the following inequalities, for every positive integer and ,
Assume that, for any positive integer , we have By the formula , for all ,