Existence of solutions of nonlinear differential equations with ψ-exponential or ψ-ordinary dichotomous linear part in a Banach space

In this article we consider nonlinear differential equations with ψ-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 of ψ-bounded solutions of these equations on R and R+.

In our paper [16] we introduced the concept of ψ-dichotomy for arbitrary Banach spaces, where ψ is an arbitrary bounded invertible linear operator.
In this paper nonlinear perturbed differential equations with ψ-dichotomous linear part are considered in an arbitrary Banach space.We will show that some properties of these equations will be influenced by the corresponding ψ-dichotomous homogeneous linear equation.Sufficient conditions for the existence of ψ-bounded solutions of this equations on R and R + in case of ψ-exponential or ψ-ordinary dichotomy are found.

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 • .By J we shall denote R or We consider the nonlinear differential equation the corresponding linear homogenous equation and the appropriate inhomogeneous equation where A(•) : J → LB(X), f (•) : J → X are strongly measurable and Bochner integrable on the finite subintervals of J and F(•, •) : J × X → X is a continuous function with respect to t.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 form x(t) = t a y(τ)dτ of a Bochner integral of a strongly measurable function y) and satisfies (2.1) (or (2.2) or (2.3)) almost everywhere.
By V(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 any t ∈ J operator-function.
Let C ψ (X) denote the Banach space of all ψ-bounded and continuous functions with values in X with the norm

Definition 2.2 ( [16]
).The equation (2.2) is said to have a ψ-exponential dichotomy on J if there exist a pair of mutually complementary projections P 1 and P 2 = I − P 1 and positive constants The equation (2.2) is said to have a ψ-ordinary dichotomy on J if (2.4) and (2.5) hold with Remark 2.3.For ψ(t) = I for all t ∈ J we obtain the notion of exponential and ordinary dichotomy in [5,7,17].
Nonlinear equations with ψ-dichotomous linear part 3 Let us introduce the principal Green function of (2.3) with the projections P 1 and P 2 from the definition for ψ-exponential dichotomy Clearly G is continuous except at t = s where it has a jump discontinuity.
Definition 2.4.Let r > 0 be an arbitrary number.We say that the conditions (H) are fulfilled if there exist positive functions m(t), k(t) such that Definition 2.5.The nonnegative function m(t) is said to be integrally bounded on J if the following inequality holds: Definition 2.6.We say that the function F(t, x) belongs to the class ED ψ (a 1 , a 2 , r) if the conditions (H) are fulfilled, the functions m(t), k(t) are integrally bounded on J and B(m For each integrable on J function m(t) we introduce the notation Definition 2.7.We say that the function F(t, x) belongs to the class D ψ (a 1 , a 2 , r) if the conditions (H) are fulfilled, the functions m(t), k(t) are integrable on J and L(m(t)) ≤ a 1 , L(k(t)) ≤ a 2 .

Main results
Theorem 3.1.Let the following conditions be fulfilled: 1.The linear part of (2.1) has ψ-exponential dichotomy on R with projections P 1 and P 2 .
2. The function F(t, x) belongs to the class ED ψ (a 1 , a 2 , r).
Then for an arbitrary r > 0 for sufficient small values of a 1 , a 2 the equation (2.1) has a unique solution x(t), which is defined for t ∈ R and for which |ψ(t)x(t)| ≤ r (t ∈ R).
where G is defined by (2.6).Let x(t) be a solution of equation (2.1) that remains for t ∈ J in the ball A. Georgieva, H. Kiskinov, S. Kostadinov, A. Zahariev Then the function F(t, x(t)) is ψ-bounded on J and it follows (see [16,Theorem 3.6]) that such solution satisfies the integral equation The converse is also true: a solution of the integral equation (3.2) which remains for t ∈ J in the ball S ψ,r satisfies the differential equation (2.1) for t ∈ J. Now we shall show that the ball S ψ,r is invariant with respect to Q and the operator Q is contracting.
First we shall prove that the operator Q maps the ball S ψ,r into itself.Indeed we have We have Hence by we obtain Thus the operator Q maps the ball S ψ,r into itself.Now we shall prove that the operator Q is a contraction in the ball S ψ,r .Let x 1 , x 2 ∈ S ψ,r .We obtain Nonlinear equations with ψ-dichotomous linear part 5 Hence the operator Q is a contraction in the ball S ψ,r .From Banach's fixed point principle the existence of a unique fixed point of the operator Q follows.
Theorem 3.3.Let the following conditions be fulfilled: 1.The linear part of (2.1) has ψ-ordinary dichotomy on R with projections P 1 and P 2 .
2. The function F(t, x) belongs to the class D ψ (a 1 , a 2 , r).Then for each r > 0 for sufficient small values of a 1 , a 2 the equation (2.1) has a unique solution x(t), which is defined for t ∈ R and 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 for t ∈ J in the ball S ψ,r satisfies the integral equation and vice versa.
We consider again in the space C ψ (X) the operator Q : C ψ (X) → C ψ (X) defined in (3.1).
For |ψ(t)Qx(t)| we obtain the following estimate: Thus the operator Q maps the ball S ψ,r into itself.
A. Georgieva, H. Kiskinov, S. Kostadinov, A. Zahariev Now we shall prove that the operator Q is a contraction in the ball S ψ,r .Let x 1 , x 2 ∈ S ψ,r .We obtain Thus by a 2 < (max{N 1 , N 2 }) −1 the operator Q is a contraction in the ball S ψ,r .From Banach's fixed point principle the existence of a unique fixed point of the operator Q follows.
Theorem 3.4.Let the following conditions be fulfilled: 1.The linear part of (2.1) has ψ-exponential dichotomy on R + with projections P 1 and P 2 .
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 such x(t) satisfies the integral equation where ξ = P 1 x(0).The converse is also true: a solution of the integral equation (3.3) satisfies the differential equation (2.1) for t ∈ J.
Let ξ ∈ X 1 and |ψ(0)ξ| ≤ ρ < r.We consider in the space C ψ (X) the operator Q : C ψ (X) → C ψ (X) defined by the formula First we shall prove, that the operator Q maps the ball S ψ,r into itself.Indeed we have For the first addend with ρ ≤ r 2N 1 we obtain Using the same technique and notations as in the proof of Theorem 3.1 we obtain for the second addend the estimate Hence by we obtain Thus the operator Q maps the ball S ψ,r into itself.Now we shall prove that the operator Q is a contraction in the ball S ψ,r .Let x 1 , x 2 ∈ S ψ,r .We obtain as in the proof of Theorem 3.1 the estimate the operator Q is a contraction in the ball S ψ,r .From Banach's fixed point principle the existence of a unique fixed point of the operator Q follows.
Theorem 3.5.Let the following conditions be fulfilled: 1.The linear part of (2.1) has ψ-ordinary dichotomy on R + with projections P 1 and P 2 .
First we shall prove, that the operator Q maps the ball S ψ,r into itself.We have For the first addend with ρ ≤ r 2N 1 we obtain For the second addend with a 1 ≤ r 2 max{N 1 ,N 2 } as in the proof of Theorem 3.3 we have Thus the operator Q maps the ball S ψ,r into itself.Let x 1 , x 2 ∈ S ψ,r .As in the proof of Theorem 3.3 we obtain the estimate Hence by a 2 < (max{N 1 , N 2 }) −1 the operator Q is a contraction in the ball S ψ,r .
From the fixed point principle of Banach it follows the existence of a unique fixed point of the operator Q.
In the proof of Theorem 3.4 it was already mentioned that every solution of the differential equation (2.1) which lies in the ball S ψ,r fulfil the equality and vice versa.Corollary 3.6.Let the conditions of Theorem 3.5 hold and let x 1 (t) and x 2 (t) be two solutions whose initial values fulfil P 1 x 1 (0) = ξ and P 1 x 2 (0) = η.Let N = max{N 1 , N 2 }.
Then for Na 2 < 1 the following estimate holds Proof.Applying the presentation (3.3) for the solutions x 1 and x 2 we obtain For the norm of Φ we obtain the estimate For sufficiently small a 2 we have Φ ≤