Asymptotically Antiperiodic Solutions for a Nonlinear Differential Equation with Piecewise Constant Argument in a Banach Space ()
1. Introduction
We are concerned with the differential Cauchy problem with piecewise constant argument:
(1)
where
is a bounded linear operator,
is the largest integer function, g is a continuous function on
and A is the infinitesimal generator of an exponentially semigroup
acting on the Banach space
. The main purpose of this work is to study, for the first time, the existence and the uniqueness of asymptotically w-anti- periodic solutions to (1) when w is an integer.
Differential equations with piecewise constant argument (EPCA) have the structure of continuous dynamical systems in intervals of constant length. Therefore they combine the properties of both differential and difference equations. They are used to model problems in biology, economy and in many other fields (see [1] - [7] ).
Recently, the authors of [18] introduced the concept of asymptotically antiperiodic functions and studied semilinear integrodifferential equations in this framework. In [19] , a new composition theorem for asymptotically antiperiodic functions is proved. This result is used to show the existence and the uniqueness of asymptotically antiperiodic mild solution to some fractional functional integro-differential equations in a Banach space. Motivated by [18] and [19] , we will show the existence and uniqueness of asymptotically antiperiodic mild solution for (1).
This work is organized as follows. In Section 2, we recall some fundamental properties of asymptotically antiperiodic functions. Section 3 is devoted to our main results. We illustrate our main result in Section 4, dealing with the existence and the uniqueness of asymptotically antiperiodic solution for a partial differential equation.
2 Preliminaries
Let
be a Banach space. The space
of the continuous bounded functions from
into
, endowed with the norm
, is a Banach space. The Banach subspace of functions f such that
is denoted by
. A positive number w being given,
will be the subset of
constituted of all w-periodic functions; it is also a Banach space. We recall the following properties of antiperiodic and asymptotically antiperiodic functions. We refer to [18] where they are proved.
Definition 2.1. A function
is said to be w-antiperiodic (or simply antiperiodic) if there exists
such that
for all
. The least such w will be called the antiperiod of f.
We will denote by
, the space of all w-antiperiodic functions
.
Theorem 2.1. Let
. Then the following are also in
.
i)
,
, c is an arbitrary real number.
ii)
, provided
on
. Here
.
iii)
, a is an arbitrary real number.
Theorem 2.2.
is a Banach space equipped with the supnorm.
Now we consider asymptotically w-antiperiodic function.
Definition 2.2. A function
is said to be asymptotically w-antiperiodic if there exist
and
, such that
![]()
g and h are called respectively the principal and corrective terms of f.
We will denote by
, the space of all asymptotically w-antiperiodic
- valued functions.
Remark 2.1.
is a Banach space equipped with the supnorm and the decomposition of an asymptotically antiperiodic is unique.
3. Main Results
We begin with the definition of a solution to (1).
Definition 3.1. A solution of Equation (1) on
is a function x(t) that satisfies the conditions:
1-x(t) is continuous on
.
2-The derivative
exists at each point
, with possible exception of the points
where one-sided derivatives exists.
3-Equation (1) is satisfied on each interval
with
.
Let
be the
semigroup generated by A and x a solution of (1). Then the function m defined by
is differentiable for
and we can write:
![]()
which leads to
(2)
The function
is a step function and
is a continuous function in the intervals
, where
. Therefore, the functions
and
are integrable over
with
. Integrating both sides of (2) over
, yields
![]()
Therefore, we give the following
Definition 3.2. Let
be the
semigroup generated by A. The function
given by
![]()
is the mild solution of the Equation (1).
Now we assume that:
(H.1) The operator A is the infinitesimal generator of an exponentially stable semigroup
such that there exist constants
and
with
![]()
The proof of the main result of this paper is based on the following two lemmas.
Lemma 3.1. Assume that (H.1) is satisfied and that
is a linear bounded operator. Let
, we define the nonlinear operator
by: for each ![]()
![]()
Then the operator
maps
into itself.
Proof. Define the function F by
![]()
Since
, it may be decomposed as
holds, where
and
. We note that
![]()
where
![]()
and
![]()
We claim that
. Since
, then
. Therefore:
, there exists a constant
such that
for all
. For all
, we have that
![]()
from which it follows that
![]()
Hence,
. Since H is clearly continuous, the claim is then proved. Now, we show that
:
![]()
Therefore
. It follows that
and
which proves that
. ,
Lemma 3.2. Assume that (H.1) is satisfied and also that
. Let
be such that:
i)
;
ii)
.
Define the nonlinear operator
by: for each ![]()
![]()
Then the operator
maps
into itself.
Proof. Let
. Then
with
and
. We have
![]()
with
. We have
![]()
Since
, we deduce that
.
We note also that
. In fact
![]()
We put
![]()
Since the function g is lipschitzian, then the function
is piecewise continuous. Therefore the function F is well defined. Since
with
and
, we observe that
![]()
where
![]()
and
![]()
The functions
and
are well defined because the function
and
are continuous on
where n is an integer. Since
and
, it follows that
and
. ,
Now we can state and prove the main result of this work.
Theorem 3.3. We assume that the hypothesis (H.1) is satisfied. We assume also that
. Let
such that:
i) ![]()
ii)
.
Then the Equation (1) has a unique asymptotically
antiperiodic solution if
![]()
Proof. Define the nonlinear operator
,
![]()
for every
, where
![]()
![]()
and
![]()
Since
we have
. Then, using Lemma 3.1 and Lemma 3.2, it follows that the operator
maps
into itself.
For every
,
![]()
Therefore, since
, using the Banach fixed point Theoren we conclude that Equation (1) has a unique asymptotically w-antiperiodic solution. ,
4. Application
As an application, consider for
and
, the Cauchy problem:
(3)
We take
and we define the linear operator A by
![]()
![]()
i)
,
ii)
.
Note that such a function exists. Take for instance
where f is a w-periodic function from
into
. Then we have
![]()
and
![]()
Theorem 4.1. We assume that
. Then System (3) has a unique asymptotically w-antiperiodic if ![]()
Proof. We have
,
,
and we apply Theorem 3.3. ,