1Introduction AndreyZahariev AtanaskaGeorgieva , HristoKiskinov , StepanKostadinov and ψ -exponentialor ψ -ordinarydichotomouslinearpartinaBanachspace Existenceofsolutionsofnonlineardifferentialequationswith 2 ,1–10; ElectronicJournalofQualitativeTheoryof

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Existence of solutions of nonlinear differential equations with ψ-exponential or ψ-ordinary

dichotomous linear part in a Banach space

Atanaska Georgieva, Hristo Kiskinov, Stepan Kostadinov

^B

and Andrey Zahariev

Faculty of Mathematics and Informatics, Plovdiv University, 236 Bulgaria Blvd., Plovdiv, BG–4003, Bulgaria

Received 12 July 2013, appeared 13 March 2014 Communicated by László Hatvani

Abstract.In this article we consider nonlinear differential equations withψ-exponential and ψ-ordinary dichotomous linear part in a Banach space. By the help of the fixed point principle of Banach sufficient conditions are found for the existence ofψ-bounded solutions of these equations onRandR+.

Keywords: ψ-dichotomy for ordinary differential equations,ψ-boundedness.

2010 Mathematics Subject Classification:34G20, 34D09, 34C11.

1 Introduction

The problem ofψ-boundedness andψ-stability of the solutions of differential equations in finite dimensional Euclidean spaces has been studied by many auhtors, as e.g. Akinyele [1], Constantin [6]. In these papers, the functionψis a scalar continuous function (and increasing, differentiable and bounded in [1], nondecreasing and such that ψ(t) ≥ 1 on R+ in [6]). In Diamandescu [8–15] and Boi [2–4]ψis a nonnegative continuous diagonal matrix function.

Inspired by the famous monographs of Coppel [5], Daleckii and Krein [7] and Massera and Schaeffer [17], where the important notion of exponential and ordinary dichotomy is considered in detail, Diamandescu [8–12] and Boi [2–4] introduced and studied theψ-dichotomy for linear differential equations in finite dimensional Euclidean space.

In our paper [16] we introduced the concept ofψ-dichotomy for arbitrary Banach spaces, whereψis an arbitrary bounded invertible linear operator.

In this paper nonlinear perturbed differential equations withψ-dichotomous linear part are considered in an arbitrary Banach space. We will show that some properties of these equations will be influenced by the correspondingψ-dichotomous homogeneous linear equation. Suffi- cient conditions for the existence ofψ-bounded solutions of this equations onRandR+in case ofψ-exponential orψ-ordinary dichotomy are found.

BCorresponding author. Email: stkostadinov@uni-plovdiv.bg

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2 Preliminaries

Let X be an arbitrary Banach space with norm | · | and identity I. Let LB(X) be the space of all linear bounded operators acting in X with the norm k · k. By J we shall denoteR or R+= [0,∞).

We consider the nonlinear differential equation dx

dt = A(t)x+F(t,x), (2.1)

the corresponding linear homogenous equation dx

dt = A(t)x (2.2)

and the appropriate inhomogeneous equation dx

dt = A(t)x+ f(t), (2.3)

whereA(·): J → LB(X), f(·): J → Xare strongly measurable and Bochner integrable on the finite subintervals ofJandF(·,·): J×X→Xis a continuous function with respect tot.

By a solution of equation (2.1) (or (2.2) or (2.3)) we will understand a continuous function x(t)that is differentiable (in the sence that it is representable in the formx(t) =Rt

a y(τ)dτof a Bochner integral of a strongly measurable functiony) and satisfies (2.1) (or (2.2) or (2.3)) almost everywhere.

ByV(t)we will denote the Cauchy operator of (2.2).

Let RL(X) be the subspace of all invertible operators in LB(X)and ψ(·): J → RL(X)be continuous for anyt∈ J operator-function.

Definition 2.1 ( [16]). A function u(·): J → X is said to be ψ-bounded on J if ψ(t)u(t) is bounded onJ.

LetC_ψ(X)denote the Banach space of allψ-bounded and continuous functions with values inXwith the norm

|||f|||_C

ψ =sup

t∈J

|ψ(t)f(t)|.

Definition 2.2( [16]). The equation (2.2) is said to have aψ-exponential dichotomy onJif there exist a pair of mutually complementary projectionsP₁ andP₂ = I−P₁and positive constants N₁,N₂,ν₁,ν₂such that

||ψ(t)V(t)P₁V⁻¹(s)ψ⁻¹(s)|| ≤N₁e⁻^ν¹⁽^t⁻^s⁾ (s≤t; s,t∈ J) _(2.4)

||ψ(t)V(t)P₂V⁻¹(s)ψ⁻¹(s)|| ≤N₂e⁻^ν²⁽^s⁻^t⁾ (t≤s; s,t∈ J) (2.5) The equation (2.2) is said to have a ψ-ordinary dichotomy on J if (2.4) and (2.5) hold with ν₁ =ν₂=0.

Remark 2.3. For ψ(t) = I for all t ∈ J we obtain the notion of exponential and ordinary dichotomy in [5,7,17].

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Let us introduce the principal Green function of (2.3) with the projections P₁ andP₂ from the definition forψ-exponential dichotomy

G(t,s) =

(V(t)P₁V⁻¹(s) (t>s; t,s∈ J)

−V(t)P2V⁻¹(s) (t< s; t,s∈ J). (2.6) ClearlyGis continuous except att= swhere it has a jump discontinuity.

Definition 2.4. Letr >0 be an arbitrary number. We say that the conditions (H) are fulfilled if there exist positive functionsm(t),k(t)such that

H1. |ψ(t)F(t,x)| ≤m(t) (|ψ(t)x| ≤r, t∈ J)

H2. |ψ(t)(F(t,x₁)−F(t,x₂))| ≤k(t)|ψ(t)(x₁−x₂)| (|ψ(t)x₁|,|ψ(t)x₂| ≤r, t∈ J)

Definition 2.5. The nonnegative functionm(t)is said to be integrally bounded onJ if the following inequality holds:

B(m(t)) =sup

t∈J

Z _t₊₁

t m(s)ds <_∞.

Definition 2.6. We say that the function F(t,x)belongs to the class ED_ψ(a₁,a₂,r)if the conditions (H) are fulfilled, the functions m(t),k(t) are integrally bounded on J and B(m(t)) ≤ a₁,B(k(t))≤a₂.

For each integrable onJfunctionm(t)we introduce the notation L(m(t)) =

Z

Jm(s)ds.

Definition 2.7. We say that the functionF(t,x)belongs to the classD_ψ(a₁,a₂,r)if the conditions (H) are fulfilled, the functionsm(t)_,k(t)are integrable onJandL(m(t))≤a₁,L(k(t))≤a₂.

3 Main results

Theorem 3.1. Let the following conditions be fulfilled:

1. The linear part of (2.1)hasψ-exponential dichotomy onRwith projections P₁and P2. 2. The function F(t,x)belongs to the class EDψ(a₁,a2,r).

Then for an arbitrary r > 0 for sufficient small values of a₁,a2 the equation(2.1) has a unique solution x(t), which is defined for t∈ _Rand for which|ψ(t)x(t)| ≤r(t ∈_R).

Proof. LetJ =_R. We consider in the spaceC_ψ(X)the operatorQ: C_ψ(X)→C_ψ(X)defined by the formula

Qx(t) =

Z

JG(t,τ)F(τ,x(τ))dτ (3.1) whereGis defined by (2.6).

Letx(t)be a solution of equation (2.1) that remains fort∈ J in the ball S_ψ,r={x :|||x|||_C

ψ ≤r}_.

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Then the functionF(t,x(t))isψ-bounded onJ and it follows (see [16, Theorem 3.6]) that such solution satisfies the integral equation

x(t) =Qx(t). (3.2)

The converse is also true: a solution of the integral equation (3.2) which remains fort ∈ Jin the ballS_ψ,rsatisfies the differential equation (2.1) fort ∈ J.

Now we shall show that the ballS_ψ,r is invariant with respect to Qand the operatorQis contracting.

First we shall prove that the operatorQmaps the ballS_ψ,rinto itself. Indeed we have

|ψ(t)Qx(t)| ≤

ψ(t)

Z

JG(t,τ)F(τ,x(τ))dτ . We have

|ψ(t)Qx(t)| ≤

ψ(t)

Z

JG(t,τ)F(τ,x(τ))dτ

≤

Z

J

kψ(t)G(t,τ)ψ⁻¹(τ)k |ψ(τ)F(τ,x(τ))|dτ

=

Z

t≤τ

kψ(t)G(t,τ)ψ⁻¹(τ)k |ψ(τ)F(τ,x(τ))|dτ +

Z

t≥τ

kψ(t)G(t,τ)ψ⁻¹(τ)k |ψ(τ)F(τ,x(τ))|dτ

≤ N₂ Z

t≤τ

e⁻^ν²⁽^τ⁻^t⁾m(τ)dτ+N₁ Z

t≥τ

e⁻^ν¹⁽^t⁻^τ⁾m(τ)dτ

≤ N₂ Z

s≥0e⁻^ν²^sm(t+s)ds+N₁ Z

s≤0e^ν¹^sm(t+s)ds

≤ N2a₁

∑

∞ k=₀

e⁻^ν²^k+N₁a₁

∑

∞ k=₀

e⁻^ν¹^k = ^N²^a¹

1−e⁻^ν² + ^N¹^a¹ 1−e⁻^ν¹. Hence bya₁≤r

N2

1−e⁻^ν² + ^N¹

1−e⁻^ν¹

−1

we obtain

ψ(t)

Z

J

G(t,τ)F(_τ,x(τ))_dτ

≤r.

Thus the operatorQmaps the ballS_ψ,rinto itself.

Now we shall prove that the operator Qis a contraction in the ballS_ψ,r. Let x₁,x₂ ∈ S_ψ,r. We obtain

|ψ(t)Qx₁(t)−ψ(t)Qx₂(t)| ≤

ψ(t)

Z

JG(t,τ)(F(τ,x₁(τ))−F(τ,x₂(τ))dτ

≤

Z

J

ψ(t)G(t,τ)ψ⁻¹(τ)

|ψ(τ)(F(τ,x₁(τ))−F(τ,x₂(τ))|dτ

≤

Z

J

k(τ)|ψ(τ)(x₁(τ))−x₂(τ))|dτ

≤

Z

J

k(τ)dτ sup

τ∈J

|ψ(τ)(x₁(τ))−x2(τ))|

≤

N₂a₂

1−e⁻^ν² + ^N¹^a² 1−e⁻^ν¹

sup

τ∈J

|ψ(τ)(x₁(τ))−x₂(τ))|.

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Hence

|||Qx₁−Qx₂|||_C

ψ ≤

N2a2

1−e⁻^ν² + ^N¹^a² 1−e⁻^ν¹

|||x₁−x₂|||_C

ψ. Thus bya₂< ^N²

1−e⁻^ν² + ^N¹

1−e⁻^ν¹

−1

the operatorQis a contraction in the ballS_ψ,r.

From Banach’s fixed point principle the existence of a unique fixed point of the operator Q follows.

Corollary 3.2. If the conditions of Theorem3.1are fulfilled and if, moreover, F(t, 0) =0(t ∈_R)then x=₀is a unique solution of (2.1)in Cψ(_X)_.

Proof. LetF(t, 0) =0(t ∈_R). Then from H2 it follows

|ψ(t)F(t,x(t))| ≤k(t)|ψ(t)x(t)| (t∈_R)_.

Thus every solution x(t)exceptx(t)≡ 0(t ∈_R)will leave any ballSψ,r₁ (r₁ <r)byt→ _∞or t→ −_∞.

1. The linear part of (2.1)hasψ-ordinary dichotomy onRwith projections P₁and P₂. 2. The function F(t,x)belongs to the class D_ψ(a₁,a₂,r).

Then for each r>0for sufficient small values of a₁,a₂the equation(2.1)has a unique solution x(t), which is defined for t ∈_Rand for which|ψ(t)x(t)| ≤r(t ∈_R).

Proof. Let J = R. In the proof of Theorem 3.1 it was mentioned that each solution x(t) of equation (2.1) that remains fort∈ Jin the ballS_ψ,rsatisfies the integral equation

x(t) =

Z

JG(t,τ)F(τ,x(τ))dτ and vice versa.

We consider again in the spaceCψ(X)the operatorQ: Cψ(X)→Cψ(X)defined in (3.1).

For|ψ(t)Qx(t)|we obtain the following estimate:

|ψ(t)Qx(t)| ≤

ψ(t)

Z

JG(t,τ)F(τ,x(τ))dτ . Witha₁ ≤rmax{N₁,N₂}we have

|ψ(t)Qx(t)| ≤

ψ(t)

Z

≤

Z

J

ψ(t)G(t,τ)ψ⁻¹(τ) |ψ(τ)F(τ,x(τ))|dτ

=

Z

t≤τ

|ψ(τ)F(τ,x(τ))|dτ +

Z

t≥τ

|ψ(τ)F(τ,x(τ))|dτ

≤ N₂ Z

t≤τ

m(τ)dτ+N₁ Z

t≥τ

m(τ)dτ

≤ max{N₁,N₂}

Z

Jm(τ)dτ≤max{N₁,N₂}a₁ ≤r.

Thus the operatorQmaps the ballS_ψ,rinto itself.

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Now we shall prove that the operator Qis a contraction in the ballS_ψ,r. Let x₁,x₂ ∈ S_ψ,r. We obtain

|ψ(t)Qx₁(t)−ψ(t)Qx₂(t)| ≤

ψ(t)

Z

JG(t,τ)(F(τ,x₁(τ))−F(τ,x₂(τ))dτ

≤

Z

J

|ψ(τ)(F(τ,x₁(τ))−F(τ,x₂(τ))|dτ

≤

Z

J

k(τ)|ψ(τ)(x₁(τ))−x₂(τ))|dτ

≤

Z

J

ψ(t)G(t,τ)ψ⁻¹(τ) k(τ)_dτ _sup

τ∈J

|ψ(τ)(x₁(τ))−x₂(τ))|

≤(max{N₁,N₂}a₂)sup

τ∈J

|ψ(τ)(x₁(τ))−x₂(τ))|. Hence

|||Qx₁−Qx₂|||_C

ψ ≤(a₂max{N₁,N₂})|||x₁−x₂|||_C

ψ. Thus bya₂ <(max{N₁,N₂})⁻¹the operatorQis a contraction in the ballSψ,r.

1. The linear part of (2.1)hasψ-exponential dichotomy onR+with projections P₁and P₂. 2. The function F(t,x)belongs to the class ED_ψ(a₁,a₂,r).

Then for any r > 0by sufficient small a₁,a₂ there exists ρ < r such that the equation(2.1) has for eachξ ∈ X₁ = P₁X with|ψ(0)ξ| ≤ ρ a unique solution x(t)onR+ for which P₁x(0) = ξ and

|ψ(t)x(t)| ≤r (t∈_R+).

Proof. Let J = _R₊ and x(t) be a solution of equation (2.1) that remains for t ∈ J in the ball S_ψ,r = {x : |||x|||_C

ψ ≤ r}. From the results obtained in [16, Theorem 3.6 and Remark 3.8] it follows that suchx(t)satisfies the integral equation

x(t) =V(t)ξ+

Z

JG(t,τ)F(τ,x(τ))dτ (3.3) whereξ = P₁x(0). The converse is also true: a solution of the integral equation (3.3) satisfies the differential equation (2.1) fort∈ J.

Letξ ∈X₁and|ψ(0)ξ| ≤ρ <r. We consider in the spaceCψ(X)the operatorQ: Cψ(X)→ Cψ(X)defined by the formula

Qx(t) =V(t)ξ+

Z

JG(t,τ)F(τ,x(τ))dτ (3.4) First we shall prove, that the operatorQmaps the ballS_ψ,rinto itself. Indeed we have

|ψ(t)Qx(t)| ≤ |ψ(t)V(t)ξ|+

ψ(t)

Z

JG(t,τ)F(τ,x(τ))dτ . For the first addend withρ≤ _2N^r

1 we obtain

|ψ(t)V(t)ξ| ≤ N₁e⁻^ν¹^t|ψ(0)ξ| ≤ N₁e⁻^ν¹^tρ≤ ^r 2.

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Using the same technique and notations as in the proof of Theorem3.1we obtain for the second addend the estimate

ψ(t)

Z

≤ ^N²^a¹

1−e⁻^ν² + ^N¹^a¹ 1−e⁻^ν¹. Hence bya₁≤ ^r₂ ^N²

1−e⁻^ν² + ^N¹

1−e⁻^ν¹

−1

we obtain

ψ(t)

Z

≤ ^r 2. Thus the operatorQmaps the ballSψ,rinto itself.

Now we shall prove that the operatorQis a contraction in the ball S_ψ,r. Let x₁,x₂ ∈ S_ψ,r. We obtain as in the proof of Theorem3.1the estimate

|||Qx₁−Qx₂|||_C

ψ ≤

N₂a₂

1−e⁻^ν² + ^N¹^a² 1−e⁻^ν¹

|||x₁−x₂|||_C

ψ

Bya₂ < ^N²

1−e⁻^ν² + ^N¹

1−e⁻^ν¹

−1

the operatorQis a contraction in the ballS_ψ,r.

1. The linear part of (2.1)hasψ-ordinary dichotomy onR+with projections P₁and P₂. 2. The function F(t,x)belongs to the class Dψ(a₁,a2,r).

Then for any r > 0by sufficiently small a₁,a₂there existsρ < r such that the equation(2.1) has for eachξ ∈ X₁ = P₁X with |ψ(0)ξ| ≤ ρ a unique solution x(t)on R+ for which P₁x(0) = ξ and

|ψ(t)x(t)| ≤r (t ∈_R+).

Proof. Let J = _R+, ξ ∈ X₁ and|ψ(0)ξ| ≤ ρ < r. We consider again in the space Cψ(X)the operatorQ: C_ψ(X)→C_ψ(X)defined by the formula (3.4).

First we shall prove, that the operatorQmaps the ballS_ψ,rinto itself. We have

|ψ(t)Qx(t)| ≤ |ψ(t)V(t)ξ|+

ψ(t)

Z

JG(t,τ)F(τ,x(τ))dτ . For the first addend withρ≤ _2N^r

1 we obtain

|ψ(t)V(t)ξ| ≤N₁|ψ(0)ξ| ≤ N₁ρ≤ ^r 2. For the second addend witha₁ ≤ ^r

2 max{N1,N2} as in the proof of Theorem3.3we have

ψ(t)

Z

JG(t,τ)F(_τ,x(τ))_dτ

≤_max{N₁,N₂}a₁≤ ^r 2. Thus the operatorQmaps the ballS_ψ,rinto itself.

Letx₁,x₂ ∈S_ψ,r. As in the proof of Theorem3.3we obtain the estimate

|||Qx₁−Qx₂|||_C

ψ ≤(a₂max{N₁,N₂})|||x₁−x₂|||_C

ψ. Hence bya₂<(_max{N₁,N₂})⁻¹the operatorQis a contraction in the ballS_ψ,r.

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From the fixed point principle of Banach it follows the existence of a unique fixed point of the operatorQ.

In the proof of Theorem3.4it was already mentioned that every solution of the differential equation (2.1) which lies in the ballS_ψ,rfulfil the equality

x(t) =Qx(t) and vice versa.

Corollary 3.6. Let the conditions of Theorem3.5hold and let x₁(t)and x₂(t)be two solutions whose initial values fulfil P₁x₁(0) =ξand P₁x₂(0) =η. Let N=max{N₁,N₂}.

Then for Na₂ <₁the following estimate holds

|ψ(t)(x₁(t)−x₂(t))| ≤ ^N

1−Na₂|ψ(0)(ξ−η)| (t ∈_R₊). Proof. Applying the presentation (3.3) for the solutionsx₁andx₂we obtain

x₁(t)−x₂(t) =V(t)(ξ−η) +

Z _∞

0 G(t,τ)(F(τ,x₁(τ))−F(τ,x₂(τ)))dτ.

From here and the conditions of Theorem3.5foru(t) =ψ(t)(x₁(t)−x2(t))we obtain

|u(t)| ≤ N|ψ(0)(ξ−η)|+N Z _∞

0 k(τ)u(τ)dτ.

Let us consider the equation

u(t) =α+N Z _∞

0 k(τ)u(τ)dτ, (3.5)

whereα = N|ψ(0)(ξ−η)|. Let us introduce the functionalΦ: C → _R₊, whereCis the space of all bounded functions onR+with values inR+by the formula

(_Φu)(t) =N Z _∞

0 k(τ)u(τ)dτ.

For the norm ofΦwe obtain the estimate k_Φk ≤N

Z _∞

0 k(τ)dτ≤ Na₂. For sufficiently smalla₂we havek_Φk ≤1.

Let I_C be the identity of the space C. Then the equation (I_C−_Φ)u = α has a bounded solutionu(t), i.e. there exists a constantc=sup_t_∈_R₊|u(t)|<∞. We shall estimate the constant cfrom equation (3.5):

c≤α+Nc Z _∞

0

k(τ)_dτ≤α+Nca₂, i.e.

c≤ ^α

1−Na₂. Finally we obtain

|ψ(t)(x₁(t)−x₂(t))| ≤ ^N|ψ(0)(ξ−η)|

1−Na₂ .

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Acknowledgements

This research has been partially supported by Plovdiv University NPD grant NI13 FMI-002.

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