Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions
© R.-N. Wang and J. Xia. 2011
Received: 23 November 2010
Accepted: 7 March 2011
Published: 15 March 2011
We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of the paper, we construct an example to illustrate the feasibility of our results.
Definition 1.1 (see ).
Definition 1.2 (see ).
It is to be noted that in the border case , the family corresponds to a classical strongly continuous semigroup, whereas in the case a solution operator corresponds to the concept of a cosine family. Moreover, according to , one can find that solution operators are a particular case of -regularized families and a solution operator corresponds to a -regularized family.
Starting from some speculations of Leibniz and Euler, the fractional calculus (such as the Riemann-Liouville fractional integral) which allows us to consider integration and differentiation of any order, not necessarily integer, have been the object of extensive study for analyzing not only stochastic processes driven by fractional Brownian motion, but also nonrandom fractional phenomena in physics and optimal control (cf. e.g., [1, 12, 13]). One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional integration or fractional differentiation (see, e.g., [1, 14–17]). Let us point out that many phenomena in engineering, physics, economy, chemistry, aerodynamics, and electrodynamics of complex medium can be modeled by this class of equations.
where , is a generator of a solution operator , , and stand for the right and left limits of at , respectively, and , are appropriate functions to be specified later. As can be seen, the convolution integral in (1.3) is the Riemann-Liouville fractional integral, and the function constitutes a nonlocal condition.
We mention that in recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established (see, e.g., [2, 18–23] and references therein).
In this work, motivated by the above contributions, we shall combine these earlier work and extend the study to the Cauchy problem (1.3). New existence and uniqueness results in the case when is a generator of a solution operator, under various criterions, are proved. In the last part of paper, we construct an example to illustrate the feasibility of our results.
Motivated by the above consideration, we give the following definition.
We list the following basic assumptions of this paper.
The following fixed-point theorem plays a key role in the proof of our main results.
Lemma 2.2 (see ).
3. Main Results
To set the framework for our main existence results, we will make use of the following lemma.
Assume that is a mild solution of (2.14) in the sense of Definition 2.1. Obviously, if , then one sees from Definition 2.1, that the assertion of theorem remains true. Thus, the rest proof of the theorem is done under .
In this work, we adopt the following concept of mild solution for the problem (1.3).
Now we present and prove our main results.
which contradicts (3.12).
This is a contradiction to (3.25).
Next, we will verify that for each , is a completely continuous operator, while, is a contraction. Obviously, by assumptions , it easily seen that is a completely continuous operator. Moreover, by a similar proof with that in Theorem 3.4, we can prove that is a contraction.
As a consequence of the above discussion and Lemma 2.2, we can conclude that the problem (1.3) admits at least one mild solution. The proof is completed.
In this section, we present an example to illustrate the abstract results of this paper, which do not aim at generality but indicate how our theorems can be applied to concrete problems.
This research was supported in part by the NSF of JiangXi Province of China (2009GQS0018) and the Youth Foundation of JiangXi Provincial Education Department of China (GJJ10051).
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