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

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 ψ-boundedness and ψ-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 ψ-dichotomy for linear differential equations in a finite dimensional Euclidean space, where ψ is a nonnegative continuous diagonal matrix function. The concept of ψdichotomy for arbitrary Banach spaces was introduced and studied in [16, 17]. In this case ψ(t) 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 ψ-exponential dichotomy and in [19] for the particular case of ψ-ordinary dichotomy. This paper considers nonlinear perturbed impulsive differential equations with a ψ-dichotomous liner part in an arbitrary Banach space. We will show that some properties of these equations will be influenced by the corresponding ψdichotomous impulsive homogeneous linear equation. Sufficient conditions for existence of ψ-bounded solutions of this equations on R and R + in case of ψ-exponential or ψordinary dichotomy are found.


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][3][4][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][12][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 R and R + in case of -exponential or ordinary dichotomy are found.

Preliminaries
Let  be an arbitrary Banach space with norm |⋅| and identity Id.By  we will denote R or R + = [0, ∞) and by  either Z or N ∪ {0}.
Remark 4. For () = Id 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.
(4) The operators   have bounded inverse ones.
Proof.Let  = R.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 Thus the operator  maps the ball  , into it self.Now we will prove that the operator  is a contraction in the ball  , .Let  1 ,  2 ∈  , .Using the same technique as above we obtain Hence Thus by 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.
(4) The operators   have bounded inverse ones.
Proof.Let  = R.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  1 and  3 the operator  maps the ball  , into it self.Now we will prove that the operator  is a contraction in the ball  , .Let  1 ,  2 ∈  , .We obtain Thus by sufficiently small  2 and  4 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 .
Let  ∈  1 and |(0)| ≤  < .We consider in the space   (, ) the operator  :   (, ) →   (, ) defined by the formula Thus the operator  maps the ball  , into it self.Now we will prove that the operator  is a contraction in the ball  , .Let  1 ,  2 ∈  , .We obtain as in the proof of Theorem 10 the estimate 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 R with projections  1 and  2 .(2) Conditions (H1) and (H2) hold.
First we will prove that the operator  maps the ball  , into it self.One has Hence by  2 +  4 <  −1 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  () =  () (37) and vice versa.

2
International Journal of Differential Equationswhere  = {  } ∈ 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 ‖ ⋅ ‖.   <  +1 and lim  → ±∞   = ±∞ ( ∈ ).