Abstract
This paper deals with some existence of mild solutions for two classes of impulsive integrodifferential equations in Banach spaces. Our results are based on the fixed point theory and the concept of measure of noncompactness with the help of the resolvent operator. Two illustrative examples are given in the last section.
1. Introduction
The existence of mild solutions is developed in [1, 2] for some semilinear functional differential equations. There has been a significant development in functional evolution equations in recent years (see the monographs [3–5], the papers [6–12], and the references therein).
The study of an abstract nonlocal Cauchy problem was initiated by Byszewski [13] in 1991. Evolution equations with nonlocal initial conditions were motivated by physical problems. As a matter of fact, it is demonstrated that the evolution equations with nonlocal initial conditions have better effects in applications than the classical Cauchy problems. For example, it was pointed in [14, 15] that the nonlocal problems are used to represent mathematical models for evolution of various phenomena, such as nonlocal neural networks, nonlocal pharmacokinetics, nonlocal pollution, and nonlocal combustion. Due to nonlocal problems having a wide range of applications in real-world applications, evolution equations with nonlocal initial conditions were studied by many authors. Xue [16] studied the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in separable Banach spaces. Xue discussed the semilinear nonlocal differential equations when the semigroup generated by the coefficient operator is compact and the nonlocal term is not compact. Fan and Li [4] discussed the existence for impulsive semilinear differential equations with nonlocal conditions by using Sadovskii’s fixed point theorem and Schauder’s fixed point theorem.
Recently, several researchers obtained other results by application of the technique of measure of noncompactness (see [17–19] and the references therein).
Impulsive differential equations have become more important in recent years in some mathematical models of real phenomena, especially in biological or medical domains, and in control theory (see, for example, the monographs [20–22] and the papers [23–25]). In this paper, we first discuss the existence of mild solutions for the following nonlocal problem of impulsive integrodifferential equations where are given functions, the set is given later, is a real (or complex) Banach space with norm , generates a -semigroup on the Banach space , and is a closed linear operator on with
In [26–29] the authors initially offered to study some classes of impulsive differential equations with noninstantaneous impulses. Motivated by the above papers, we next discuss the existence of mild solutions for the following nonlocal problem of noninstantaneous impulsive integrodifferential equations: where are given functions such that is a given function, the set is given later, and .
2. Preliminaries
By we denote the space of the bounded linear operator from into itself. Let be the Banach space of continuous functions from into . Let be the Banach space of measurable functions that are essentially bounded and equipped with the norm
Consider the Banach space with the norm
A semigroup of bounded linear operators is uniformly continuous if
Here, denotes the identity operator in .
We note that if a semigroup is of class , then it satisfies the growth condition
, for with some constants and .
If, in particular, and , i.e, , for then the semigroup is called a contraction semigroup. For more details on strongly continuous operators, we refer the reader to the books [30, 31].
Let denote the class of all bounded subsets of a metric space
Definition 1. Letbe a complete metric space. A mapis called a measure of noncompactness onif it satisfies the following properties for all. (a) if and only if is precompact (regularity)(b) (invariance under closure)(c) (semiadditivity)
Definition 2 [32]. Letbe a Banach space and letbe the family of bounded subsets of. The Kuratowski measure of noncompactness is the mapdefined bywhere .
For our purpose, we will need the following fixed point theorem:
Theorem 3. (Monch’s fixed point theorem [33]). Let be a bounded, closed, and convex subset of a Banach space such thatand letbe a continuous mapping ofinto itself. If the implicationholds for every subset of then has a fixed point.
3. Mild Solutions with Instantaneous Impulses
In this section, we are concerned with the existence results of the problem (1).
Definition 4 [10]. A resolvent operator for the Cauchy problemis a bounded linear operator-valued function verifying the following conditions: (i) (the identity map of ) and for some constants and (ii)For each is strongly continuous for (iii) is bounded for . For and
Let us introduce the following hypotheses:
The operator is the infinitesimal generator of a uniformly continuous semigroup
For all is a closed linear operator from to and For any , the map is bounded differentiable and the derivative is bounded uniformly continuous on .
Theorem 5 [10, 34]. Assume that and hold. Then, there exists a unique uniformly continuous resolvent operator for the Cauchy problem (9).
Definition 6 [34]. By a mild solution of the problem (1), we mean a function that satisfies
The following hypotheses will be used in the sequel.
The function is measurable on for each , and the function is continuous on for a.e. ,
There exists a function such that
There exist positive constants , such that and
For each bounded set , we have and for each bounded set we have where .
Set
Theorem 7. Assume that the hypotheses – hold. If then problem (1) has at least one mild solution defined on .
Proof. Transform problem (1) into a fixed point problem. Consider the operator defined by
Let such that
and consider the ball .
For any and each we have
Thus,
This proves that transforms the ball into itself. We shall show that the operator satisfies all the assumptions of Theorem 3. The proof will be given in three steps.
Step 1. is continuous.
Let be a sequence such that as in . Then, for each we have
Since as and are continuous, the Lebesgue-dominated convergence theorem implies that
Step 2. is bounded and equicontinuous.
Since and is bounded, then is bounded.
Next, let , and let . Thus, we have
Hence, we get
As the resolvent operator is uniformly continuous, the right-hand side of the above inequality tends to zero as .
Step 3. The implication (8) holds.
Now let be a subset of such that . is bounded and equicontinuous, and therefore, the function is continuous on . By and the properties of the measure , for each , we have
Hence,
From (17), we get , that is, , for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 3, we conclude that has a fixed point which is a mild solution of our problem (1).
4. Mild Solutions with Noninstantaneous Impulses
In this section, we are concerned with the existence results of problem (2). Denote by and there exist , and with and , the Banach space equipped with the standard supremum norm.
Definition 8 [34]. By a mild solution of problem (2), we mean a function that satisfies
The following hypotheses will be used in the sequel:
The functions and are measurable on and , respectively, for each , and the functions and are continuous on for a.e. in and , respectively.
There exist functions , such that
There exists a positive constant , such that
For each bounded set and for each we have and for each bounded set we have where
Set
Theorem 9. Assume that the hypotheses–hold. Ifthen problem (2) has at least one mild solution defined on .
Proof. Transform problem (2) into a fixed point problem. Consider the operator defined by
Let , such that
For any and each , we have
Thus,
Next, for each it is clear that
Hence,
This proves that transforms the ball into itself.
We shall show that the operator satisfies all the assumptions of Theorem 3. The proof will be given in three steps.
Step 1. is continuous.
Let be a sequence such that as in Then, for each we have
and for each we have
Since as and are continuous, the Lebesgue-dominated convergence theorem implies that
Step 2. is bounded and equicontinuous.
Since and is bounded, then is bounded.
Next, let , and let . Thus, we have
Hence, we get
As the right-hand side of the above inequality tends to zero.
Step 3. The implication (8) holds.
Now let be a subset of such that . is bounded and equicontinuous, and therefore, the function is continuous on . By and the properties of the measure , for each , we have
Next, for each , we have
Thus, for each , we get
Hence,
From (36), we get ; that is, , for each , and then is relatively compact in . In view of the Ascoli-Arzelà theorem, is relatively compact in . Applying now Theorem 3, we conclude that has a fixed point which is a mild solution of problem (2).
5. Examples
Let be the Hilbert space with the scalar product . It is known that is a Banach space with the norm
Example 1. Consider the following problem of impulsive integrodifferential equations
where ,
and
with .
We define the strongly elliptic operator by
where and .
It is well known (see [31]) that generates a uniformly continuous semigroup in the Hilbert space .
For , we have
Thus, under the above definitions of , , and , system (54) can be represented by problem (1). Furthermore, more appropriate conditions on ensure the hypotheses Consequently, Theorem 7 implies that problem (54) has at least one mild solution on
Example 2. Consider now the following problem of impulsive integrodifferential equations
where ,
with .
Again, as the above example, simple computations show that all conditions of Theorem 9 are satisfied. It follows that problem (59) has at least one mild solution on
Data Availability
There is no data used in this work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.