International Journal of Differential Equations
Volume 2015 (2015), Article ID 748607, 7 pages
http://dx.doi.org/10.1155/2015/748607

Research Article

Nonlinear Impulsive Differential Equations with Weighted Exponential or Ordinary Dichotomous Linear Part in a Banach Space

Hristo Kiskinov and Andrey Zahariev

Faculty of Mathematics and Informatics, University of Plovdiv, 236 Bulgaria Boulevard, 4003 Plovdiv, Bulgaria

Received 7 July 2015; Accepted 15 September 2015

Academic Editor: Elena Braverman

Copyright © 2015 Hristo Kiskinov and Andrey Zahariev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider nonlinear impulsive differential equations with ψ-exponential and ψ-ordinary dichotomous linear part in a Banach space. By the help of Banach’s fixed-point principle sufficient conditions are found for the existence of ψ-bounded solutions of these equations on and .

1. Introduction

Impulsive differential equations are an adequate mathematical apparatus for simulation of numerous processes and phenomena in biology, physics, chemistry, control theory, and so forth which during their evolutionary development are subject to short time perturbations in the form of impulses. The qualitative investigation of these processes began with the work of Mil’man and Myshkis [1]. For the first time such equations were considered in an arbitrary Banach space in [2–5].

The problem of -boundedness and -stability of the solutions of differential equations in finite dimensional Euclidean spaces, introduced for the first time by Akinyele [6], has been studied since then by many authors. A beautiful explanation about the benefits of such a use of weighted stability and boundedness can be found, for example, in [7]. Inspired by the famous monographs of Coppel [8], Daleckii and Krein [9], and Massera and Schaeffer [10], where the important notion of exponential and ordinary dichotomy for ordinary differential equations is considered in detail, Diamandescu [11–13] and Boi [14, 15] introduced and studied the -dichotomy for linear differential equations in a finite dimensional Euclidean space, where is a nonnegative continuous diagonal matrix function. The concept of -dichotomy for arbitrary Banach spaces was introduced and studied in [16, 17]. In this case is an arbitrary bounded invertible linear operator.

A weighted dichotomy for linear differential equations with impulse effect in arbitrary Banach spaces is considered in [18] for a -exponential dichotomy and in [19] for the particular case of -ordinary dichotomy.

This paper considers nonlinear perturbed impulsive differential equations with a -dichotomous liner part in an arbitrary Banach space. We will show that some properties of these equations will be influenced by the corresponding -dichotomous impulsive homogeneous linear equation. Sufficient conditions for existence of -bounded solutions of this equations on and in case of -exponential or -ordinary dichotomy are found.

2. Preliminaries

Let be an arbitrary Banach space with norm and identity . By we will denote or and by either or .

We consider the nonlinear impulsive differential equation

where is a finite or infinite sequence in . We will say that condition (H1) is satisfied if the following conditions hold:

(H 1.1) is a continuous operator-valued function with values in the Banach space of all linear bounded operators acting in with the norm .(H 1.2).(H 1.3)The function is continuous with respect to .(H 1.4) are continuous operators.(H 1.5) and .

Consider the corresponding linear impulsive equation

Definition 1. By a solution of the impulsive equation (1), (2) (or (3), (4)) we will call a function which for satisfies (1) (or (3)) and for satisfies condition (2) (or (4)) and is continuous from the left.

As shown, for instance in [3, 4], if the operators have bounded inverse ones, then for the impulsive linear equation (3), (4) there exists a Cauchy operator by means of which each solution of (3), (4) for which has the form

Let be the subspace of all invertible operators in whose inverse operators are bounded too. Let be continuous with respect to operator function.

Definition 2. A function is said to be -bounded on if is bounded on .

Let be the space of all -bounded on functions with values in which are continuous for , have discontinuities of the first kind for , and are continuous from the left, which is a Banach space with the norm

Definition 3. The linear impulsive equation (3), (4) is said to have a -exponential dichotomy on if there exist a pair of mutually complementary projections and and positive constants such that

Equation (3), (4) is said to have a -ordinary dichotomy on if (7) hold with . In this case we denote .

Remark 4. For for all we obtain the notion exponential and ordinary dichotomy for impulsive differential equations considered in [3, 20, 21]. That is why our main results in this paper appear as a generalization of some results there.

Let us introduce the principal Green function of the nonhomogeneous equation corresponding to (3), (4) with the projections and from the definition for -exponential dichotomy

Definition 5. The nonnegative function is said to be integrally bounded on if for some the following inequality holds:

Definition 6. The sequence of nonnegative numbers is said to be integrally bounded if for some the following inequality holds:

For each integrable on function we introduce the notation

and for each summable on sequence of nonnegative numbers the notation

Definition 7. Let be an arbitrary number. We will say that the function and the operators satisfy the condition (H2) with the operator function , if there exist positive functions and sequences of nonnegative numbers such that

(H 2.1);(H 2.2);(H 2.3);(H 2.4).

Definition 8. We say that the function and the operators belong to the class if the condition (H2) is fulfilled with the operator function , the functions , are integrally bounded on and , , and the sequences and are integrally bounded and , .

Definition 9. We say that the function and the operators belong to the class if the condition (H2) is fulfilled with the operator function , the functions , are integrable on and , , and the sequences and are summable on and , .

3. Main Results

Theorem 10. Let the following conditions be fulfilled:

(1)The linear impulsive differential equation (3), (4) (i.e., the linear part of (1), (2)) has -exponential dichotomy on with projections and .(2)Conditions (H1) and (H2) hold.(3)The function and the operators belong to the class .(4)The operators have bounded inverse ones.

Then for an arbitrary , for sufficient small values of , , , the impulsive equation (1), (2) has a unique solution , which is defined for and for which .

Proof. Let . Consider in the space the operator defined by the formula

where is defined by (8).
Now we will show that the ball

is invariant with respect to and the operator is contracting.
First we will prove that the operator maps the ball into itself. One has

We will estimate the addends in (15). For we obtain

From the estimates (16) it follows that

Hence by and we obtain

Thus the operator maps the ball into it self.
Now we will prove that the operator is a contraction in the ball . Let . Using the same technique as above we obtain

Hence

Thus by and the operator is a contraction in the ball .
From Banach’s fixed point principle, the existence of a unique fixed point of the operator follows.
It is not hard to verify that each solution of the impulsive differential equation (1), (2) which lies in the ball is also a solution of the equation

and vice versa.

Corollary 11. If the conditions of Theorem 10 are fulfilled and if, moreover, , then is a unique solution of (1), (2) in .

Theorem 12. Let the following conditions be fulfilled:

(1)The linear impulsive differential equation (3), (4) (i.e., the linear part of (1), (2)) has -exponential dichotomy on with projections and .(2)Conditions (H1) and (H2) hold.(3)The function and the operators belong to the class .(4)The operators have bounded inverse ones.

Then for an arbitrary , for sufficient small values of , , , the impulsive equation (1), (2) has a unique solution , which is defined for and for which .

Proof. Let . In the proof of Theorem 10 it was mentioned that each solution of the impulsive differential equation (1), (2) that remains for in the ball satisfies the equation

and vice versa.
We consider again in the space the operator defined in (13). For we obtain the following estimate:

Thus by sufficiently small and the operator maps the ball into it self.
Now we will prove that the operator is a contraction in the ball . Let . We obtain

Hence

Thus by sufficiently small and the operator is a contraction in the ball .
From Banach’s fixed point principle follows the existence of a unique fixed point of the operator .

Theorem 13. Let the following conditions be fulfilled:

(1)The linear impulsive differential equation (3), (4) (i.e., the linear part of (1), (2)) has -exponential dichotomy on with projections and .(2)Conditions (H1) and (H2) hold.(3)The function and the operators belong to the class .(4)The operators have bounded inverse ones.

Then for any by sufficient small there exists such that the impulsive equation (1), (2) has for each with a unique solution on for which and .

Proof. Let and be a solution of (1), (2) that remains for in the ball From the results obtained in [18, Theorem 1 and Remark 2] it follows that such satisfies the integral equation

where . The converse is also true: a solution of the (26) satisfies the differential equation (1), (2) for .
Let and . We consider in the space the operator defined by the formula

First we will prove that the operator maps the ball into it self. Indeed one has

For the first addend with we obtain

Using the same technique as in the proof of Theorem 10 we obtain for the second addend the estimate

Hence by and we obtain

Thus the operator maps the ball into it self.
Now we will prove that the operator is a contraction in the ball . Let . We obtain as in the proof of Theorem 10 the estimate

By and the operator is a contraction in the ball .
From Banach’s fixed point principle the existence of a unique fixed point of the operator follows.

Theorem 14. Let the following conditions be fulfilled:

(1)The linear impulsive differential equation (3), (4) (i.e., the linear part of (1), (2)) has -exponential dichotomy on with projections and .(2)Conditions (H1) and (H2) hold.(3)The function and the operators belong to the class .(4)The operators have bounded inverse ones.

Then for any by sufficient small , , , there exists such that the impulsive equation (1), (2) has for each with a unique solution on for which and .

Proof. Let , , and . We consider again in the space the operator defined by formula (27).
First we will prove that the operator maps the ball into it self. One has

For the first addend with we obtain

For the second addend with as in the proof of Theorem 12 one has

Thus the operator maps the ball into it self.
Let . As in the proof of Theorem 12 we obtain the estimate

Hence by the operator is a contraction in the ball .
From Banach’s fixed point principle the existence of a unique fixed point of the operator follows.
In the proof of Theorem 13 it was already mentioned that every solution of the impulsive differential equation (1), (2) which lies in the ball fulfills the equality

and vice versa.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research has been partially supported by Plovdiv University NPD Grant NI15-FMI-004.

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