The Ψ − asymptotic equivalence of the Lyapunov matrix differential equations with modiﬁed argument

. Using the notion of strict h − contraction, existence results for Ψ − asymptotic equivalence of two pairs of (Lyapunov) matrix di ﬀ erential equations with modiﬁed argument are given.


Introduction
The purpose of this paper is to provide sufficient conditions for Ψ−asymptotic equivalence of the Ψ−bounded solutions of two Lyapunov matrix differential equations with modified argument Z = A(t)Z + ZB(t) (1) Z = A(t)Z + ZB(t) + F(t, Z(g(t))), t ≥ t 0 .
These conditions can be expressed in the terms of fundamental matrices of the matrix differential equations and on the function F. Here, Ψ is a matrix function who allows obtaining a mixed asymptotic behavior for the components of solutions of the matrix differential equations.
In paper [11], the author consider the vector differential equation and the perturbed equation with modified argument y = A(t)y + f (t, y(g(t))).
The author demonstrated in Theorem 2.1, [11] that under some conditions, for every bounded solution x(t) of equation (5), there exists a unique bounded solution y(t) of equation (6) such that lim t→∞ | y(t) − x(t) |= 0.
The present paper extend these results in three directions: from systems of differential equations to Lyapunov matrix differential equations with modified argument, the introduction of the matrix function Ψ that permits to obtaining a mixed asymptotic behavior for the components of solutions and in connection with the belonging to space L r of solutions.
The main tools used in this paper are a fixed point theorem via strict h-contraction and the technique of variation of constants formula combined with Kronecker product of matrices and Hölder's inequality, which has been successfully applied in various fields of matrix theory. See, for example, the cited papers and the references cited therein.

Preliminaries
In this section we present some basic notations, definitions, hypotheses and results which are useful later on. Let Let M d×d be the linear space of all real d × d matrices.
In this paper, we assume that A and B are continuous d × d matrices on R + and g : R + → R + and F : R + × M d×d −→ M d×d are continuous functions.
By a solution of equation (2) we mean a continuous differentiable d × d matrix function Z(t) satisfying the equation (2) for all t ∈ R + = [0, ∞) Let Ψ i : R + −→ (0, ∞), i = 1, 2, ..., d, be continuous functions and A matrix P is said to be a projection if P 2 = P.
Otherwise, is said that the function ϕ is Ψ−unbounded on R + .
Otherwise, is said that the matrix function M is Ψ− unbounded on R + . Remark 2.1 1. The Definitions extend the definitions of boundedness of vector and matrix functions. 2. For Ψ = I d , one obtain the notion of classical boundedness (see [4]). 3. It is easy to see that if Ψ and Ψ −1 are bounded on R + , then the Ψ−boundedness is equivalent with the classical boundedness.
For important properties and rules of calculation of the Vec operator, see Lemmas 2.2, 2.3, 2.5, [7].
For "corresponding Kronecker product system associated with (2)", see Lemma 2.4, [7]. The Lemmas 2.6 and 2.7, [7], play an important role in the proofs of main results of present paper. Now, we remember the notions of strict h-contractions and strict comparison functions which are useful in the proof of our main results.  Let (X, d) be a metric space and f : X → X an operator. Definition 2.6 ( [12]) The operator f is called a strict h-contraction if satisfies the following conditions: At the end of this section, we recall Lemma 2.1 ( [8]) which is useful in the proofs of our main results. This Lemma is a generalization of Lemma 10, [6], Lemma 1, [4], (p. 68) and a Lemma from [13].
Lemma 2.1 Let U(t) be an invertible d × d matrix which is a continuous function of t on R + and let P a projection, P ∈ M d×d . Suppose that there exist a continuous function ϕ : R + → (0, ∞) and the constants M > 0 and p > 1 such that

Main results
The purpose of this section is to give sufficient conditions for Ψ − asymptotic equivalence of the Ψ − bounded solutions of two pairs of matrix differential equations, namely (1) - (2) and The first result is motivated by the Theorem 2.1, [11]. For t 0 ≥ 0, we consider the equation (3) and perturbed equation Theorem 3.1 Suppose that: 1). There are supplementary projections P 1 , P 2 ∈ M d×d , a continuous function ϕ : R + → (0, ∞) that satisfies the condition ∞ 0 (ϕ(s)) q ds = +∞, q > 1, and a constant K > 0 such that the fundamental matrix X(t) for the linear matrix differential equation (3) satisfies the condition where t 0 is sufficiently large; 2). There exist a strict comparison function h : R + → R + and a function λ(t) ∈ L p ([t 0 , ∞)), Conversely, to each Ψ− bounded solution Z(t) of equation (8) for which Ψ(t)Z(t) belongs to L r ([t 0 , ∞)), there corresponds a unique Ψ− bounded solution Z 0 (t) of equation (3) for which (9) holds. Proof. We prove this theorem by means of the above Theorem 2.1.

Remark 3.1 If we put
and Ψ = I d , we obtain Theorem 2.1 from [11]. Thus, Theorem 3.1 generalizes and extends Theorem 2.1 from [11] in three directions: from systems of differential equations with modified argument to matrix differential equations with modified argument, the introduction of the matrix function Ψ that permits to obtaining a mixed asymptotic behavior for the components of solutions of the above equations and in connection with the belonging to space L r of solutions.
The object of the next Theorem is to obtain new results in connection with Ψ− asymptotic equivalence of two Lyapunov matrix differential equations with modified argument, namely (1) and (2).
With the help of Lemma 2.1, [7], we have for t, s ∈ R + and i = 1, 2. It follows that the hypothesis 1) of Theorem 3.1 is satisfied with I d ⊗Ψ in role of Ψ, Y T (t)⊗X(t) in role of X(t) and I d ⊗ P i in role of P i . b). With the help of Lemma 2.5, [7], we have, for t ≥ t 0 and z 1 , Consequently, the hypothesis 2) of Theorem 3.1 is satisfied with I d ⊗ Ψ in role of Ψ and f in role of F. c). With the help of Lemma 2.5, [7], we have, for t ≥ t 0 , It follows that the hypothesis 3) of Theorem 3.1 is satisfied with I d ⊗ Ψ in role of Ψ and f in role of F. d). With the help of Lemmas 2.1 and 2.7, [7], we have, for t ≥ t 0 , Consequently, the hypothesis 5) of Theorem 3.1 is satisfied with I d ⊗ Ψ in role of Ψ, Y T (t) ⊗ X(t) in role of X(t) and I d ⊗ P i in role of P i . We observe that the hypotheses of Theorem 3.1 are satisfied for in case of (11), in particular case U(t), I d ⊗ Ψ and I d ⊗ P i respectively. Now, we finish the proof. Let Z 0 (t) be a Ψ− bounded solution of equation (1) for which Ψ(t)Z 0 (t) belongs to L r ([t 0 , ∞)). From Lemmas 2.5 and 2.6, [7], the function z 0 (t) = Vec(Z 0 (t)) is a I d ⊗ Ψ− bounded solution of equation (12) and belongs to L r ([t 0 , ∞)). From Theorem 3.1, variant for systems, there exists a unique I d ⊗ Ψ− bounded solution z(t) of equation (11) for which (I d ⊗ Ψ(t)) z belongs to L r ([t 0 , ∞)) such that lim t→∞ |(I d ⊗ Ψ(t)) (z(t) − z 0 (t))| = 0.

Remark 3.2
The introduction of the function ϕ does not alter conditions (9) and (10) of the theorems; this function ϕ can only serve to weaken the required hypotheses on F. Instead, the introduction of matrix Ψ allows to obtain mixed asymptotic behaviors for the components of the solutions of matrix differential equations.