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Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions
Advances in Difference Equations volume 2011, Article number: 634701 (2011)
Abstract
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.
1. Introduction
Let denote a complex Banach space and denote
by the space of all bounded linear operators from
into
with the usual operator norm
. Let us recall the following definitions.
Definition 1.1 (see [1]).
Let be a continuous function and
. Then the expression

is called the Riemann-Liouville integral of order .
Definition 1.2 (see [2]).
Let be a linear and closed operator with domain
defined on
. By a solution operator associated with
in
, we mean a family
of strongly continuous operators satisfying
-
(1)
and
-
(2)
(1.2)
where
is a constant and
stands for the resolvent of
. In this case, we also say that
is a solution operator generated by
.
Remark 1.3.
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 [3], one can find that solution operators are a particular case of
-regularized families and a solution operator
corresponds to a
-regularized family.
Remark 1.4.
Note that solution operator does not satisfy the semigroup property.
Remark 1.5.
Various solution operators are usually key tools in dealing with the abstract Cauchy problems and related issues. For more information, please see, for example, [4–11] and references therein.
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.
In the present paper we study the existence and uniqueness of mild solutions for the Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions in the form

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.
As usual, the solution with the points of discontinuity at the moments
follows that
, that is, at which it is continuous from the left.
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).
Interest in impulsive nonlocal Cauchy problems stems mainly from the observation that on one side, nonlocal initial conditions have better effects in treating physical problems than the usual ones (see [21, 22, 24–27] and the references therein for more detailed information about the importance of nonlocal initial conditions in applications); on the other side, the dynamics of many evolutionary processes from some research fields are subject to abrupt changes of states at certain moments of time between intervals of continuous evolution, such changes can be well approximated as being instantaneous changes as state, that is, in the form of "impulses" (cf. [20, 28] and the references therein). This class of equations has been the object of extensive study in recent years, see [29–31] and the references therein for more comments and citations. It is worth mentioning that in [31], Liang et al. considered the following impulsive nonlocal Cauchy problem

where is the generator of a strongly continuous semigroup in a Banach space and the existence and uniqueness of mild and classical solutions for the Cauchy problem, under various criterions, are proved. Also, Wang et al. [32] proved the existence and uniqueness of mild and classical solutions for the nonlocal Cauchy problem in the form

where ,
is a
-almost sectorial operator (not necessarily densely defined).
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.
2. Preliminaries
Throughout this paper, we take to be the Banach space of all
-valued continuous functions from
into
endowed with the uniform norm topology

Put

with ,
, and let
be the restriction of a function
to
.
Consider the set of functions

endowed with the norm

It is easy to see is a Banach space.
Let . It follows from [33] that if
is sectorial of type
, that is,
is a closed linear operator, and there exist constants
and
such that
and

then is a generator of a solution operator
, which is given by

provided that , where
is a suitable path lying outside the sector
. And Cuesta [18, Theorem 1], has proved that if
is a sectorial operator of type
and there is a positive constant
which depends on
such that the estimate

holds for all .
We recall that the Laplace transform of a abstract function is defined by

We first treat the following problem:

Formally applying the Laplace transform in (2.9), we obtain

which establishes the following result:

This means that

Motivated by the above consideration, we give the following definition.
Definition 2.1.
Let . A solution
of the integral equation

is called a mild solution of the following problem:

where is the solution operator generated by
.
We list the following basic assumptions of this paper.
is continuous in
on
and there exists a constant
such that

for all .
is continuous and there exists a function
such that

for all .
is completely continuous and there exists a continuous nondecreasing function
such that for each
,

is Lipschitz continuous with Lipschitz constant
.
For ,
is Lipschitz continuous with Lipschitz constant
.
For ,
is completely continuous and there exists a continuous nondecreasing function
such that for each
,

The following fixed-point theorem plays a key role in the proof of our main results.
Lemma 2.2 (see [34]).
Let be a convex, bounded, and closed subset of a Banach space
and let
be a condensing map. Then,
has a fixed point in
.
3. Main Results
To set the framework for our main existence results, we will make use of the following lemma.
Lemma 3.1.
Let . Assume that
is a sectorial operator of type
and
is a solution operator generated by
. Suppose in addition that
is a continuous function. If
is a mild solution of the Cauchy problem (2.14) in the sense of Definition 2.1, then,
satisfying the following impulsive integral equation:

is a mild solution of problem (1.3), where


Proof.
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
.
By Definition 2.1, note that

for all . Taking
, then we get

Hence, it follows form that

If , then combining Definition 2.1 and the result above, we deduce that

This proves, for the case , that the conclusion of theorem holds.
Now taking in (3.7), one has

which implies that

provided that . Then, again making use of Definition 2.1, we get for all
,

here and
are given by (3.2) and (3.3) with
, respectively. A continuation of the same process shows that for any
, the assertion of theorem holds.
In this work, we adopt the following concept of mild solution for the problem (1.3).
Definition 3.2.
Let . Assume that
is a sectorial operator of type
,
is a solution operator generated by
, and
and
are given by (3.2) and (3.3), respectively. A solution
of the integral equation

here , is called a mild solution of the Cauchy problem (1.3).
Remark 3.3.
Note that if there is no discontinuity, that is, if ,
, then Definition 2.1 is equivalent to Definition 3.2.
Now we present and prove our main results.
Theorem 3.4.
Let . Assume that
is a sectorial operator of type
and
is a solution operator generated by
. Suppose in addition that assumptions
are fulfilled. Then the Cauchy problem (1.3) admits at least one mild solution, provided

Proof.
Consider the mapping , which is defined for each
by

Then it is clear that is well defined.
To prove the theorem, it is sufficient to prove that has a fixed point in
.
Put

for as selected below.
We first show that there exists an integer such that
maps
into
. For the case
, by assumption
and the estimate (2.7), a straightforward calculation yields that

We claim that there exists an integer such that
provided that
. In fact, if this is not the case, then for each
, there would exist
and
such that
. Thus, by (3.15) and assumption
we obtain

Dividing on both sides by and taking the lower limit as
, we get

which contradicts (3.12).
Since the interval is divided into finite subintervals by
, we only need to prove that for a fixed
,

maps into
, here
is a positive number yet to be determined, as the cases for other subintervals are the same.
From the Hypotheses , we infer for any
,

Now, an application of the same idea with above discussion yields that there exists a such that
. Indeed, if this is not the case, then we would deduce that

This is a contradiction to (3.12). Thus, we prove that there exists an integer such that
.
For , we decompose the mapping
as follows:

Next, we show that for each ,
is completely continuous, while
is a contraction. In fact, it follows from assumption
and the estimate (2.7) that
,
is completely continuous. Note also that

For the case , it is clear that the conclusion holds in view of (3.12). For
, by
,
and (2.7) we get

provided that . Hence, we deduce that

which means that is a contraction due to (3.12).
Thus, is a condensing map on
. Then, it follows from Lemma 2.2 that the Cauchy problem (1.3) admits at least one mild solution. This completes the proof.
Theorem 3.5.
Let . Assume that
is a sectorial operator of type
,
is a solution operator generated by
, and the Hypotheses
are satisfied. Then the Cauchy problem (1.3) admits at least one mild solution, provided

Proof.
Assume that the map and the set
are defined the same as in Theorem 3.4. First we claim that there exists an positive number
such that
. For the case
, the proof of the assertion follows from Theorem 3.4. For the case
, if the conclusion is not true, then for each positive integer
, there would exist
and
such that
with
, where
denotes
depending upon
. Thus, by assumptions
,
,
, we have

Dividing on both sides by and taking the lower limit as
, we have

This is a contradiction to (3.25).
For , decompose the mapping
as follows:

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.
Theorem 3.6.
Let . Assume that
is a sectorial operator of type
and
is a solution operator generated by
. Then, under assumptions
,
,
, the Cauchy problem (1.3) has a unique mild solution, provided

Proof.
Assume that the map is defined the same as in Theorem 3.4. Now, we prove that
is a contraction. Take any
. For the case
, the conclusion follows from assumptions
,
, and (3.29). For
, a direct calculation yields

in view of assumptions ,
,
. Hence, we deduce that

which implies is a contractive mapping on
due to (3.29). Thus
has a unique fixed point
, this means that
is a mild solution of (1.3). This completes the proof of the theorem.
4. Example
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.
Consider the BVP of partial differential equation in the form

where ,
is a constant in
,
is a constant yet to be determined,
stands for the operator with respect to the spatial variable
which is given by

In what follows we consider the space with norm
and the operator
with domain

Clearly is densely defined in
and is sectorial of type
. Hence
is a generator of a solution operator satisfying the estimate (2.7) on
. Here, without lost of generality, we take
.
Set

Then we have

Note that the problem (4.1) also can be reformulated as the abstract problem (1.3), and due to (4.5), it is not difficult to see that assumptions ,
, and
hold with

which implies that one can choose large enough such that the first inequality of (3.29) is satisfied. Hence, according to Theorem 3.6, the Cauchy problem (4.1) has a unique mild solution.
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Acknowledgments
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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Wang, RN., Xia, J. Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions. Adv Differ Equ 2011, 634701 (2011). https://doi.org/10.1155/2011/634701
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DOI: https://doi.org/10.1155/2011/634701
Keywords
- Banach Space
- Cauchy Problem
- Mild Solution
- Fractional Brownian Motion
- Sectorial Operator