Research Article | Open Access
Jaydev Dabas, Archana Chauhan, Mukesh Kumar, "Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay", International Journal of Differential Equations, vol. 2011, Article ID 793023, 20 pages, 2011. https://doi.org/10.1155/2011/793023
Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay
This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.
This paper is concerned with the existence and uniqueness of a mild solution of an impulsive fractional-order functional differential equation with the infinite delay of the form where is the infinitesimal generator of an -resolvent family , the solution operator is defined on a complex Banach space , is the Caputo fractional derivative, is a given function, and is a phase space defined in Section 2. Here, , , , are bounded functions, and represent the right and left limits of at , respectively.
We assume that , , , belongs to an abstract phase space . The term is given by , where is the set of all positive continuous functions on .
Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science. Many physical phenomena in evolution processes are modelled as impulsive differential equations and have been studied extensively by several authors, for instance, see [1–3], for more information on these topics. Impulsive integro-differential equations with delays represent mathematical models for problems in the areas such as population dynamics, biology, ecology, and epidemic and have been studied by many authors [2–7]. The study of fractional differential equations has emerged as a new branch of applied mathematics, which has been used for construction and analysis of mathematical models in science and engineering. In fact, the fractional differential equations are considered as models alternative to nonlinear differential equations. Many physical systems can be represented more accurately through fractional derivative formulation. For more detail, see, for instance, the papers [1, 3–5, 7–12] and references therein.
Recently, in , the author has established sufficient conditions for the existence of a mild solution for a fractional integro-differential equation with a state-dependent delay. Mophou and N’Guérékata  have investigated the existence and uniqueness of a mild solution for the fractional differential equation (1.1) without impulsive conditions. Authors of  have established the results assuming that generates an -resolvent family on a complex Banach space by means of classical fixed-point methods.
In , Benchohra et al. have considered the following nonlinear functional differential equation with infinite delay where is Riemann-Liouville fractional derivative, , with , and established the existence of a mild solution for the considered problem using the Banach fixed-point and the nonlinear alternative of Leray-Schauder theorems.
Motivated by the above-mentioned works, we consider the problem (1.1) to study the existence and uniqueness of a mild solution using the solution operator and fixed-point theorems. The paper is organized as follows: in Section 2, we introduce some function spaces and notations and present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. In the last section one example is presented.
In this section, we mention some definitions and properties required for establishing our results. Let be a complex Banach space with its norm denoted as , and represents the Banach space of all bounded linear operators from into , and the corresponding norm is denoted by . Let denote the space of all continuous functions from into with supremum norm denoted by . In addition, represents the closed ball in with the center at and the radius .
To describe a fractional-order functional differential equation with the infinite delay, we need to discuss the abstract phase space in a convenient way (for details see ). Let be a continuous function with . For any , we define
such that is bounded and measurable} and equip the space with the norm Let us define by
If is endowed with the norm then it is known that is a Banach space.
Now, we consider the space where is the restriction of to . The function to be a seminorm in , it is defined by If , is such that , then for all , the following conditions hold:(1), (2), (3), where is a constant and is continuous, is locally bounded, and are independent of . For more details, see .
A two parameter function of the Mittag-Lefller type is defined by the series expansion where is a contour which starts and ends at and encircles the disc counter clockwise. For short, . It is an entire function which provides a simple generalization of the exponent function: and the cosine function: , and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Lefller functions are associated with their Laplace integral see  for more details.
Definition 2.1. A closed and linear operator is said to be sectorial if there are constants , such that the following two conditions are satisfied: Sectorial operators are well studied in the literature. For details see .
Definition 2.2 (see Definition 2.3 in ). Let be a closed and linear operator with the domain defined in a Banach space . Let be the resolvent set of . We say that is the generator of an -resolvent family if there exist and a strongly continuous function such that and in this case, is called the -resolvent family generated by .
Definition 2.3 (see Definition 2.1 in ). Let be a closed linear operator with the domain defined in a Banach space and . We say that is the generator of a solution operator if there exist and a strongly continuous function such that and in this case, is called the solution operator generated by .
The concept of the solution operator is closely related to the concept of a resolvent family (see  Chapter 1). For more details on -resolvent family and solution operators, we refer to [14, 15] and the references therein.
Definition 2.4. The Riemann-Liouville fractional integral operator for order , of a function and , is defined by where is the Euler gamma function. The Laplace transform of a function is defined by provided the integral is absolutely convergent for .
Definition 2.5. Caputo’s derivative of order for a function is defined as for . If , then Obviously, Caputo’s derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order is given as
Lemma 2.6. If satisfies the uniform Holder condition with the exponent and is a sectorial operator, then the unique solution of the Cauchy problem is given by where denotes the Bromwich path. is called the -resolvent family, and is the solution operator, generated by .
Proof. Let , then we get Taking the Laplace transform of (2.19), we have Since exists, that is, , from (2.20), we obtain By the inverse Laplace transform of (2.21), we get Set , in (2.22), we have On simplification, we obtain Set and in (2.24). We have This completes the proof of the lemma.
Definition 2.7. A function is called a mild solution of (1.1) if the following holds: on with , the restriction of to the interval is continuous and satisfies the following integral equation:
Now, we introduce the following assumptions: (H1)there exist such that (H2)for each , there exists such that (H3) where and and
If and , then for any and , we have and . Hence, we have . See  for details.
3. The Main Results
Our first result is based on the Banach contraction principle.
Theorem 3.1. Assume that the assumptions (H1)–(H3) are satisfied. If , then the system (1.1) has a unique mild solution.
Proof. Consider the operator defined by Let be the function defined by then . For each with , we denote by the function defined by If satisfies (2.26), then we can decompose as for , which implies for , and the function satisfies Set such that and let be the seminorm in defined by thus is a Banach space. We define the operator by It is clear that the operator has a unique fixed-point if and only if has a unique fixed-point. To prove that has a unique fixed-point, let , then for all . We have For , we have Similarly, when , we get Thus, for all , we have Hence, is a contraction map, and therefore it has an unique fixed-point , which is a mild solution of (1.1) on . This completes the proof of the theorem.
The second result is established using the following Krasnoselkii’s fixed-point theorem.
Theorem 3.2. Let be a closed-convex and nonempty subset of a Banach space . Let and be two operators such that whenever , is compact and continuous; is a contraction mapping, then there exists such that .
Now, we make the following assumptions: (H4) is continuous, and there exist two continuous functions such that (H5)the function is continuous, and there exists such that
Before going further, we need the following lemma.
Lemma 3.3 (see Lemma 3.2 in ). Let then for any , If , then
Theorem 3.4. Suppose that the assumptions (H1), (H4), (H5) are satisfied with then the impulsive problem (1.1) has at least one mild solution on .
Proof. Choose and consider , then is a bounded, closed-convex subset in .
Let and be defined as
Step 1. Let , then show that , for , we have and by using Lemma 3.3, we conclude that Similarly, when , , we have the estimate which implies that .Step 2. We will show that the mapping is continuous on . For this purpose, let be a sequence in with , then for , , we have Since the functions are continuous, hence in which implies that the mapping is continuous on .Step 3. Uniform boundedness of the map is an implication of the following inequality: for , , we have Step 4. To show that the map (3.17) is equicontinuous, we proceed as follows. Let , , , , then we obtain Since is strongly continuous, the continuity of the function allows us to conclude that , which implies that is equicontinuous. Finally, combining Step 1 to Step 4 together with Ascoli’s theorem, we conclude that the operator is compact.
Now, it only remains to show that the map is a contraction mapping. Let and , , then we have since , which implies that is a contraction mapping. Hence, by the Krasnoselkii fixed-point theorem, we can conclude that the problem (1.1) has at least one solution on . This completes the proof of the theorem.
Our last result is based on the following Schaefer’s fixed-point theorem.
Theorem 3.5. Let be a continuous and compact mapping on a Banach space into itself, such that the set is bounded, then has a fixed-point.
Lemma 3.6 (see ). Let be a real function, is nonnegative and locally integrable function on , and there are constants and such that Then there exists a constant such that
Theorem 3.7. Assume that the assumptions (H4)-(H5) are satisfied, and if and , then the impulsive problem (1.1) has at least one mild solution on .
Proof. We define the operator as in Theorem 3.3. Note that is well defined in .We complete the proof in the following steps.Step 1. For the continuity of the map , let be a sequence in such that in . Since the function is continuous on , This implies that Now, for every , we get where as . Moreover, we have where as , for all . The impulsive functions are continuous, then we get This implies that is continuous.Step 2. maps bounded sets into bounded sets in . To prove that for any , there exists a such that for each , then we have , then for any , we have Using Lemma 3.3, we obtain . Similarly, we have This implies that Step 3. We will prove that is equicontinuous. Let , with , we have where Since for and as , is strongly continuous. This implies that , Hence, . Similarly, for , with , we have Since is also strongly continuous, so as . Thus, from the above inequalities, we have . So, is equicontinuous. Finally, combining Step 1 to Step 3 with Ascoli’s theorem, we conclude that the operator is compact.Step 4. We show that the set is bounded. Let , then for some . Then for each , we have for , we get then for all , we have where and . Let be such that . If , then (3.43) can be written as Using Lemma 3.6, there exists a constant , and (3.43) becomes As a consequence of Schaefer’s fixed-point theorem, we deduce that has a fixed-point on . This completes the proof of the theorem.
To illustrate the application of the theory, we consider the following partial integro-differential equation with fractional derivative of the form where is Caputo’s fractional derivative of order , are prefixed numbers, and . Let , and define the operator by with the domain , are absolutely continuous, , then where is the orthogonal set of eigenvectors of