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

Authors: Hristo Kiskinov; Andrey Zahariev
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

We consider nonlinear impulsive differential equations withψ-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 R and R+.


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## Body

## 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 ofψ-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 ψ-dichotomy 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 aψ-dichotomous liner part in an arbitrary Banach space. We will show that some properties of these equations will be influenced by the corresponding ψ-dichotomous 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 ψ-ordinary dichotomy are found.


## 2. Preliminaries

LetX be an arbitrary Banach space with norm · and identity Id. By J we will denote R or R+=0,∞ and by I either Z or N∪{0}.We consider the nonlinear impulsive differential equation(1)dxdt=Atx+Ft,xt≠tn,(2)xtn+=Qnxtn+Hnxtnn∈I,where T={tn}n∈I is a finite or infinite sequence in J. We will say that condition (H1) is satisfied if the following conditions hold:(H 1.1)
A(t)(t∈J) is a continuous operator-valued function with values in the Banach space LB(X) of all linear bounded operators acting in X with the norm ·.(H 1.2)
Qn∈LB(X)(n∈I).(H 1.3)
The functionF(t,x):J×X→X is continuous with respect to t.(H 1.4)
Hn:X→X(n∈I) are continuous operators.(H 1.5)
tn<tn+1 and limn→±∞⁡tn=±∞(n∈I).Consider the corresponding linear impulsive equation(3)dxdt=Atxt≠tn,n∈I,(4)xtn+=Qnxtnn∈I.Definition 1.
By a solution of the impulsive equation (1), (2) (or (3), (4)) we will call a function x(t) which for t≠tn satisfies (1) (or (3)) and for t=tn satisfies condition (2) (or (4)) and is continuous from the left.As shown, for instance in [3, 4], if the operators Qn have bounded inverse ones, then for the impulsive linear equation (3), (4) there exists a Cauchy operator V(t),(t∈J) by means of which each solution x(t) of (3), (4) for which x(s)=ξ∈X has the form (5)xt=VtV-1sξt,s∈J.LetRL(X) be the subspace of all invertible operators in LB(X) whose inverse operators are bounded too. Let ψ(t):J→RL(X) be continuous with respect to t∈J operator function.Definition 2.
A functionu(t):J→X is said to be ψ-bounded on J if ψ(t)u(t) is bounded on J.LetCψ(X,T) be the space of all ψ-bounded on J functions with values in X which are continuous for t∉T, have discontinuities of the first kind for t∈T, and are continuous from the left, which is a Banach space with the norm (6)fCψ=supt∈Jψtft.Definition 3.
The linear impulsive equation (3), (4) is said to have a ψ-exponential dichotomy on J if there exist a pair of mutually complementary projections P1 and P2=Id-P1 and positive constants N1,N2,ν1,ν2 such that(7)ψtVtP1V-1sψ-1s≤N1e-ν1t-ss≤t,s,t∈J,ψtVtP2V-1sψ-1s≤N2e-ν2s-tt≤s,s,t∈J.Equation (3), (4) is said to have a ψ-ordinary dichotomy on J if (7) hold with ν1=ν2=0. In this case we denote N=max{N1,N2}.Remark 4.
Forψ(t)=Id for all t∈J 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 P1 and P2 from the definition for ψ-exponential dichotomy(8)Gt,s=VtP1V-1st>s,t,s∈J-VtP2V-1st<s,t,s∈J.Definition 5.
The nonnegative functionm(t) is said to be integrally bounded on J if for some l the following inequality holds: (9)Bmt=supt∈J∫tt+lmsds<∞.Definition 6.
The sequence of nonnegative numbers{mn}n∈I is said to be integrally bounded if for some l the following inequality holds: (10)Bmj=supt∈I⁡∑t≤tj<t+lmj<∞.For each integrable onJ function m(t) we introduce the notation (11)Lmt=∫Jmsdsand for each summable on I sequence of nonnegative numbers {mn}n∈I the notation (12)Lmj=∑j∈Imj.Definition 7.
Letr>0 be an arbitrary number. We will say that the function F(t,x)(t∈J,x∈X) and the operators Hj(j∈I) satisfy the condition (H2) with the operator function ψ(t):J→RL(X), if there exist positive functions m(t),k(t)(t∈J) and sequences of nonnegative numbers {mn}n∈I,{kn}n∈I such that (H 2.1)
ψtFt,x≤m(t)(ψtx≤r,t∈J);(H 2.2)
ψtFt,x1-Ft,x2≤ktψtx1-x2ψtx1,ψtx2≤r,t∈J;(H 2.3)
ψtjHjx≤mj(ψtjx≤r,j∈I);(H 2.4)
ψtjHjx1-Hjx2≤kjψtjx1-x2(ψtjx1,ψtjx2≤r,j∈I).Definition 8.
We say that the functionF(t,x) and the operators Hj belong to the class EDψ(a1,a2,a3,a4,r) if the condition (H2) is fulfilled with the operator function ψ(t), the functions m(t), k(t) are integrally bounded on J and B(m(t))≤a1, B(k(t))≤a2, and the sequences {mj} and {kj} are integrally bounded and B(mj)≤a3, B(kj)≤a4.Definition 9.
We say that the functionF(t,x) and the operators Hj belong to the class Dψ(a1,a2,a3,a4,r) if the condition (H2) is fulfilled with the operator function ψ(t), the functions m(t), k(t) are integrable on J and L(m(t))≤a1, L(k(t))≤a2, and the sequences {mj} and {kj} are summable on I and L(mj)≤a3, L(kj)≤a4.


## 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 ψ-exponential dichotomy on R with projections P1 and P2.(2)
Conditions (H1) and (H2) hold.(3)
The functionF(t,x) and the operators Hj belong to the class EDψ(a1,a2,a3,a4,r).(4)
The operatorsQn have bounded inverse ones.
Then for an arbitraryr>0, for sufficient small values of a1, a2, a3, a4 the impulsive equation (1), (2) has a unique solution x(t), which is defined for t∈R and for which ψtxt≤r(t∈R).Proof.
LetJ=R. Consider in the space Cψ(X,T) the operator Q:Cψ(X,T)→Cψ(X,T) defined by the formula(13)Qxt=∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj,where G is defined by (8).
Now we will show that the ball(14)Sψ,r=x:xCψ≤ris invariant with respect to Q and the operator Q is contracting.
First we will prove that the operatorQ maps the ball Sψ,r into itself. One has(15)ψtQxt≤∫-∞tψtGt,τFτ,xτdτ+∫t∞ψtGt,τFτ,xτdτ+∑tj<tψtGt,tj+Hjxtj+∑t≤tjψtGt,tj+Hjxtj.We will estimate the addends in (15). For x∈Sψ,r we obtain(16)∫-∞tψtGt,τFτ,xτdτ=∫-∞tψtVtP1V-1τFτ,xτdτ=∫-∞tψtVtP1V-1τψ-1τψτFτ,xτdτ≤∫-∞tψtVtP1V-1τψ-1τψτFτ,xτdτ≤N1∫-∞te-ν1t-τmτdτ≤N1∫s≤0eν1smt+sds≤N1a1∑k=0∞e-ν1k≤N1a11-e-ν1,∫t∞ψtGt,τFτ,xτdτ=∫t∞ψtVtP2V-1τFτ,xτdτ=∫t∞ψtVtP2V-1τψ-1τψτFτ,xτdτ≤∫t∞ψtVtP2V-1τψ-1τψτFτ,xτdτ≤N2∫t∞e-ν2τ-tmτdτ≤N2∫s≥0e-ν2smt+sds≤N2a1∑k=0∞e-ν2k≤N2a11-e-ν2,∑tj<tψtGt,tj+Hjxtj=∑tj<tψtVtP1V-1tj+Hjxtj=∑tj<tψtVtP1V-1tj+ψ-1tj+ψtj+Hjxtj≤∑tj<tψtVtP1V-1tj+ψ-1tj+ψtjHjxtj≤N1∑tj<teν1tj-tmj≤N1a31-e-ν1l,∑t≤tjψtGt,tj+Hjxtj=∑t≤tjψtVtP2V-1tj+Hjxtj=∑t≤tjψtVtP2V-1tj+ψ-1tj+ψtj+Hjxtj≤∑t≤tjψtVtP2V-1tj+ψ-1tj+ψtjHjxtj≤N2∑t≤tjeν2t-tjmj≤N2a31-e-ν2l.
From the estimates (16) it follows that (17)ψtQxt≤N1a11-e-ν1+N2a11-e-ν2+N1a31-e-ν1l+N2a31-e-ν2l.
Hence bya1≤r(N2/(1-e-ν2)+N1/(1-e-ν1))-1 and a3≤(N2/(1-e-ν2l)+N1/(1-e-ν1l))-1 we obtain (18)ψtQxt≤r.Thus the operator Q maps the ball Sψ,r into it self.
Now we will prove that the operatorQ is a contraction in the ball Sψ,r. Let x1,x2∈Sψ,r. Using the same technique as above we obtain(19)ψtQx1t-ψtQx2t≤∫-∞∞ψtGt,τFτ,x1τ-Fτ,x2τdτ+∑j=-∞∞ψtGt,tj+Hjx1tj-Hjx2tj≤∫-∞∞ψtGt,τψ-1τψτFτ,x1τ-Fτ,x2τdτ+∑j=-∞∞ψtVtP1V-1tj+ψ-1tj+ψtjHjx1tj-Hjx2tj≤supτ∈J⁡ψτx1τ-x2τ∫-∞∞ψtGt,τψ-1τkτdτ+supτ∈J⁡ψτx1τ-x2τ∑j=-∞∞ψtVtP1V-1tj+ψ-1tj+kj≤supτ∈J⁡ψτx1τ-x2τN2a21-e-ν2+N1a21-e-ν1+N2a41-e-ν2l+N1a41-e-ν1l.Hence(20)Qx1-Qx2Cψ≤N2a21-e-ν2+N1a21-e-ν1+N2a41-e-ν2l+N1a41-e-ν1lx1-x2Cψ.Thus by a2<(N2/(1-e-ν2)+N1/(1-e-ν1))-1 and a4<(N2/(1-e-ν2l)+N1/(1-e-ν1l))-1 the operator Q is a contraction in the ball Sψ,r.
From Banach’s fixed point principle, the existence of a unique fixed point of the operatorQ follows.
It is not hard to verify that each solution of the impulsive differential equation (1), (2) which lies in the ball Sψ,r is also a solution of the equation(21)xt=Qxtand vice versa.Corollary 11.
If the conditions of Theorem10 are fulfilled and if, moreover, F(t,0)=0(t∈R), then x=0 is a unique solution of (1), (2) in Cψ(X,T).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 ψ-exponential dichotomy on R with projections P1 and P2.(2)
Conditions (H1) and (H2) hold.(3)
The functionF(t,x) and the operators Hj belong to the class Dψ(a1,a2,a3,a4,r).(4)
The operatorsQn have bounded inverse ones.
Then for an arbitraryr>0, for sufficient small values of a1, a2, a3, a4 the impulsive equation (1), (2) has a unique solution x(t), which is defined for t∈R and for which ψtxt≤r(t∈R).Proof.
LetJ=R. In the proof of Theorem 10 it was mentioned that each solution x(t) of the impulsive differential equation (1), (2) that remains for t∈J in the ball Sψ,r satisfies the equation (22)xt=∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtjand vice versa.
We consider again in the spaceCψ(X,T) the operator Q:Cψ(X,T)→Cψ(X,T) defined in (13). For ψtQxt we obtain the following estimate:(23)ψtQxt≤∫-∞∞ψtGt,τFτ,xτdτ+∑j=-∞∞ψtGt,tj+Hjxtj≤∫-∞∞ψtGt,τψ-1τψτFτ,xτdτ+∑j=-∞∞ψtVtP1V-1tj+ψ-1tj+ψtj+Hjxtj≤∫-∞∞ψtGt,τψ-1τψτFτ,xτdτ+∑j=-∞∞ψtVtP1V-1tj+ψ-1tj+ψtjHjxtj≤N∫-∞∞ψτFτ,xτdτ+N∑j=-∞∞ψtjHjxtj=N∫-∞∞mτdτ+N∑j=-∞∞mj=NLmt+Lmj≤Na1+a3.
Thus by sufficiently smalla1 and a3 the operator Q maps the ball Sψ,r into it self.
Now we will prove that the operatorQ is a contraction in the ball Sψ,r. Let x1,x2∈Sψ,r. We obtain(24)ψtQx1t-ψtQx2t≤∫-∞∞ψtGt,τFτ,x1τ-Fτ,x2τdτ+∑j=-∞∞ψtGt,tj+Hjx1tj-Hjx2tj≤∫-∞∞ψtGt,τψ-1τψτFτ,x1τ-Fτ,x2τdτ+∑j=-∞∞ψtVtP1V-1tj+ψ-1tj+ψtjHjx1tj-Hjx2tj≤Nsupτ∈J⁡ψτx1τ-x2τ∫-∞∞kτdτ+∑j=-∞∞kj≤Nsupτ∈J⁡ψτx1τ-x2τa2+a4.Hence (25)Qx1-Qx2Cψ≤Na2+a4x1-x2Cψ.Thus by sufficiently small a2 and a4 the operator Q is a contraction in the ball Sψ,r.
From Banach’s fixed point principle follows the existence of a unique fixed point of the operatorQ.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 ψ-exponential dichotomy on R with projections P1 and P2.(2)
Conditions (H1) and (H2) hold.(3)
The functionF(t,x) and the operators Hj belong to the class EDψ(a1,a2,a3,a4,r).(4)
The operatorsQn have bounded inverse ones.
Then for anyr>0 by sufficient small a1,a2,a3,a4 there exists ρ<r such that the impulsive equation (1), (2) has for each ξ∈X1=P1X with ψ0ξ≤ρ a unique solution x(t) on R+ for which P1x(0)=ξ and ψtxt≤r(t∈R+).Proof.
LetJ=R+ and x(t) be a solution of (1), (2) that remains for t∈J in the ball Sψ,r={x:xCψ≤r}. From the results obtained in [18, Theorem 1 and Remark 2] it follows that such x(t) satisfies the integral equation(26)xt=Vtξ+∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj,where ξ=P1x(0). The converse is also true: a solution of the (26) satisfies the differential equation (1), (2) for t∈J.
Letξ∈X1 and ψ0ξ≤ρ<r. We consider in the space Cψ(X,T) the operator Q:Cψ(X,T)→Cψ(X,T) defined by the formula(27)Qxt=Vtξ+∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj.
First we will prove that the operatorQ maps the ball Sψ,r into it self. Indeed one has(28)ψtQxt≤ψtVtξ+ψt∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj.For the first addend with ρ≤r/2N1 we obtain (29)ψtVtξ≤N1e-ν1tψ0ξ≤N1e-ν1tρ≤r2.Using the same technique as in the proof of Theorem 10 we obtain for the second addend the estimate(30)ψt∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj≤N1a11-e-ν1+N2a11-e-ν2+N1a31-e-ν1l+N2a31-e-ν2l.
Hence bya1≤(r/2)(N2/(1-e-ν2)+N1/(1-e-ν1))-1 and a3≤(r/2)(N2/(1-e-ν2l)+N1/(1-e-ν1l))-1 we obtain (31)ψtQxt≤r.Thus the operator Q maps the ball Sψ,r into it self.
Now we will prove that the operatorQ is a contraction in the ball Sψ,r. Let x1,x2∈Sψ,r. We obtain as in the proof of Theorem 10 the estimate(32)Qx1-Qx2Cψ≤N2a21-e-ν2+N1a21-e-ν1+N2a41-e-ν2l+N1a41-e-ν1lx1-x2Cψ.By a2<(N2/(1-e-ν2)+N1/(1-e-ν1))-1 and a4<(N2/(1-e-ν2l)+N1/(1-e-ν1l))-1 the operator Q is a contraction in the ball Sψ,r.
From Banach’s fixed point principle the existence of a unique fixed point of the operatorQ 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 ψ-exponential dichotomy on R with projections P1 and P2.(2)
Conditions (H1) and (H2) hold.(3)
The functionF(t,x) and the operators Hj belong to the class Dψ(a1,a2,a3,a4,r).(4)
The operatorsQn have bounded inverse ones.
Then for anyr>0 by sufficient small a1, a2, a3, a4 there exists ρ<r such that the impulsive equation (1), (2) has for each ξ∈X1=P1X with ψ0ξ≤ρ a unique solution x(t) on R+ for which P1x(0)=ξ and ψtxt≤r(t∈R+).Proof.
LetJ=R+, ξ∈X1, and ψ0ξ≤ρ<r. We consider again in the space Cψ(X,T) the operator Q:Cψ(X,T)→Cψ(X,T) defined by formula (27).
First we will prove that the operatorQ maps the ball Sψ,r into it self. One has(33)ψtQxt≤ψtVtξ+ψt∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj.For the first addend with ρ≤r/2N1 we obtain (34)ψtVtξ≤N1e-ν1tψ0ξ≤N1e-ν1tρ≤r2.For the second addend with a1+a3≤r/2N as in the proof of Theorem 12 one has(35)ψt∫-∞∞Gt,τFτ,xτdτ+∑j=-∞∞Gt,tj+Hjxtj≤Na1+a3≤r2.Thus the operator Q maps the ball Sψ,r into it self.
Letx1,x2∈Sψ,r. As in the proof of Theorem 12 we obtain the estimate (36)Qx1-Qx2Cψ≤Na2+a4x1-x2Cψ.Hence by a2+a4<N-1 the operator Q is a contraction in the ball Sψ,r.
From Banach’s fixed point principle the existence of a unique fixed point of the operatorQ follows.
In the proof of Theorem13 it was already mentioned that every solution of the impulsive differential equation (1), (2) which lies in the ball Sψ,r fulfills the equality (37)xt=Qxtand vice versa.



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