On the Ψ-asymptotic equivalence of the Ψ- bounded solu- tions of two Lyapunov matrix differential equations

Using Schauder Tychonoff fixed point theorem and the technique of Kronecker product of matrices, we prove existence results for Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations. 2010 Mathematics Subject Classification: 34C11, 34D05, 34D10.


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 These conditions can be written in terms of fundamental matrices of the matrix differential equations and of the function F. Here, Ψ is a matrix function who allows obtaining a mixed asymptotic behavior for the components of solutions of the differential equations.
History of the problem. A classical result in connection with boundedness of solutions of systems of ordinary differential equations was given by Coppel [5]. The problems of Ψ−bounded solutions for systems of ordinary differential equations or for Lyapunov matrix differential equations have been studied by many authors. See, for instance [2], [6], [7], [8], [9], [11], and the references therein. The results of this paper extend known results of W. A. Coppel [5], F. Brauer and J. S. W. Wong [3], [4], T. G. Hallam [9].
The main tools used in this paper are Schauder -Tychonoff fixed point theorem and the technique of Kronecker product of matrices (combined with the variation of constants formula), 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.
∥Ax∥ . It is well- In the previous equations, we assume that A and B are continuous d × d matrices on R + and F : R + × M d×d −→ M d×d is a continuous function.
A solution of the equation (2) is a continuous differentiable d×d matrix function satisfying the equation (2) for all t ∈ R + = [0, ∞).
For important properties and rules of calculation of the Vec operator, see Lemmas 2.2, 2.3, 2.5 in [8].
For the corresponding Kronecker product system associated with (2), see Lemma 2.4 in [8].
The Lemmas 2.6 and 2.7 in [8], play an important role in the proofs of main results of present paper.

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 (3)-(6) and (1)- (2). The first result is motivated by a Theorem of Hallam [9]. 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)ds = ∞ and a constant K > 0 such that the fundamental matrix X(t) for the matrix differential equation (3) satisfies the inequality 2) The continuous matrix function F : for all t ≥ t 0 and Z ∈ M d×d , where ω(t, r) : R + × R + → R + is a continuous nonnegative function and also nondecreasing in r, for each fixed t ≥ t 0 . Furthermore, let γ λ (t) = sup s≥t ω(s, λ) and assume that lim Then, corresponding to each Ψ−bounded solution Z 0 (t) of (3), there exists a Ψ−bounded solution Z(t) of the matrix differential equation such that lim Conversely, to each Ψ−bounded solution Z(t) of (6), there exists a Ψ−bounded solution Z 0 (t) of (3) such that (7) holds.
Proof. We prove this Theorem by means of the fixed point theorem of Schauder-Tychonoff (Coppel [5], Chapter I, section 2).
Let C Ψ denote the set of all matrix functions Z(t) which are contiuous and Ψ−bounded on [t 0 , ∞) and let S ρ be the subset of those functions For Z ∈ S 2ρ , we define the operator T by

From hypotheses 1) and 2) , T Z exists and is continuous differentiable on
From the first assumption of Theorem 3.1, it follows that the integral is convergent for all Z ∈ S 2ρ and t ≥ t 0 .

From hypotheses, T Z exists and is continuous differentiable on [t 0 , ∞).
This operator has the following properties: a) T maps S 2ρ into itself; Indeed, for any Z ∈ S 2ρ , and for t ≥ t 0 , we have This proves the assertion. b). T is continuous, in the sense that if Z n ∈ S 2ρ , (n = 1, 2, ...) and Z n → Z uniformly on every compact subinterval J of [t 0 , ∞), then T Z n → T Z uniformly on every compact subinterval J of [t 0 , ∞); Indeed, from (8) we have, for any t ∈ J ⊂ [t 0 , ∞), . For a fixed ε > 0, we choose t 1 ≥ t 0 sufficiently large (t 1 ≥ β) such that Since F, Ψ, φ are continuous and Z n → Z uniformly on [t 0 , t 1 ], there exists n 0 ∈ N such that for s ∈ [t 0 , t 1 ] and n ≥ n 0 . Thus, for t ∈ J and n ≥ n 0 , the first integral term of (9) is strictly smaller then ε/4. For the second integral term of (9), we write ... and have two cases: i) For t ∈ J, n ≥ n 0 and for s ∈ [t, t 1 ], similarly, we also deduce that the first integral term is strictly smaller then ε/4. ii) For t ∈ J, n ∈ N and s ≥ t 1 , we have and then, the second integral term is strictly smaller then ε/2.
From (9) and these results, we get the sequence (T Z n ) n converges uniformly to T Z on compact subintervals of [t 0 , ∞).
c) The functions in the image set T S 2ρ are echicontinuous and uniformly bounded at every point of every compact subinterval J of [t 0 , ∞).
Indeed, from a), T S 2ρ ⊂ S 2ρ . This shows the functions in the image set T S 2ρ are uniformly bounded at every point of every compact subinterval J of [t 0 , ∞).
On the other hand, for V = T Z and for t ≥ t 0 , we have for t ≥ t 0 , and the matrices A(t)Ψ −1 (t), Ψ(t)V(t), φ(t)Ψ −1 (t), φ −1 (t)Ψ(t)F(t, Z(t)) are uniformly bounded on every compact subinterval J of [t 0 , ∞), the derivatives of the functions in T S 2ρ are uniformly bounded on every compact subinterval J of [t 0 , ∞). This shows the functions in T S 2ρ are echicontinuous on every compact subinterval J of R + . Thus, all the conditions of the Schauder -Tychonoff theorem are satisfied. We conclude the operator T has at least one fixed point Z in S 2ρ . This fixed point Z is clearly a Ψ− bounded solution of (6).
To prove that (7) holds, we use the relation lim t→∞ |Ψ(t)X(t)P 1 | = 0, which is an immediate consequence of the inequality where N is a positive constant (see Theorem 4, [6]; actually, the proof of this result needs to be modified to include the function φ). For a fixed ε > 0, we choose t 2 > t 1 so that From the choice of t 1 , for t ≥ t 1 we can write From the above mentioned results, we obtain This shows that (7) is satisfied.
To prove the last statement of Theorem 3.1, consider a Ψ−bounded solution Z(t) of equation (6). Define With the previous arguments, we can show that Z 0 (t) is a Ψ− bounded solution of equation (3) that satisfies (7).
The proof of Theorem 3.1 is complete.
we get a version of Theorem 3.1 for systems of differential equations. In addition, putting Ψ = diag [ψ, ψ, · · · ψ], where ψ : R + → (0, ∞) is a continuous function, equation (6) becomes equation (2) from [9]. Thus, Theorem 3.1 generalizes Theorem 1 in [9], in two directions: from systems of differential equations to matrix differential equations and the introduction of the matrix function Ψ which allows obtaining a mixed asymptotic behavior for the components of solutions of the above equations. In addition, the function φ satisfies the better condition ∞ 0 φ(s)ds = ∞.
The goal of the next theorem is to obtain a new result in connection with Ψ−asymptotic equivalence of the Ψ−bounded solutions of two Lyapunov matrix differential equations, namely (1) and (2).

Theorem 3.2 Suppose that:
1) There exist supplementary projections P 1 , P 2 ∈ M d×d , a continuous function φ : R + → (0, ∞) that satisfies the condition ∞ 0 φ(s)ds = ∞ and a constant K > 0 such that the fundamental matrices X(t) and Y(t) for the linear matrix differential equations (3) and (4) respectively, satisfy the inequality 2) The continuous matrix function F : R + × M d×d −→ M d×d satisfies the inequality for all t ≥ t 0 and Z ∈ M d×d , where ω(t, r) : R + × R + → R + is a continuous function and is nondecreasing in r, for each fixed t ≥ t 0 . Furthermore, let γ λ (t) = sup s≥t ω(s, λ) and assume that lim t→∞ γ λ (t) = 0 for each λ ∈ (0, ∞).
Proof. We will use the version of Theorem 3.1 for systems of differential equations and some results from [8]. From Lemma 2.6, [8], we know that Z(t) is a Ψ−bounded solution on R + of (2) iff z(t) = Vec(Z (t)) is a I ⊗ Ψ(t)−bounded solution of the corresponding Kronecker product system associated with (2), i.e. the system where f (t, z) = Vec(F(t, Z)).
We verify the hypotheses of Theorem 3.1 (version for systems of differential equations). a) From Lemma 2.7, [8], we know that Y T (t) ⊗ X(t) is a fundamental matrix for the homogeneous system associated to (12), i.e. the system From Lemmas 2.1, 2.3, [8], we obtain that hypothesis 1) in Theorem 3.2 implies hypothesis 1) in Theorem 3.1 (with I ⊗ Ψ(t) in the role of Ψ(t) and I ⊗ P i in role of P i ). b) Similar arguments to those above show that the from the hypothesis 2) in Theorem 3.2 it follows that all other hypotheses of Theorem 3.1 hold as well.
We find the general solutions of (1) and (2)  .
Then, the general solution of equation is Z g 0 = X(t)C, where C is a real 2 × 2 constant matrix.

Z.
A fundamental matrix for this equation is and then, Since f (t) is unbounded (see in [5], pp. 71) and g(t) is bounded on R + , the solution Z(t) is Ψ− bounded on R + iff c 1 = c 2 = 0. In this case, Because c 0, it is impossible to make lim t→∞ |Ψ(t) (Z(t) − Z 0 (t))| = 0. This proves the assertion.