On ψ Bounded Solutions for a Nonlinear Lyapunov Matrix Differential Equation on 𝕉

Abstract

Using Banach and Schauder - Tychono fixed point theorems, existence results for a nonlinear Lyapunov matrix differential equation on 𝕉 are given. The obtained results generalize and extend the results from [5] and [18].

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

“…For Z ∈ S 2ρ , we define the operator T by (8) for t ≥ t 0 , where Z 0 (t) is a Ψ−bounded solution of (3) such that Z 0 ∈ S ρ and t 0 is such that γ 2ρ (t 0 ) ≤ ρ/K.…”

“…The Lemmas 2.6 and 2.7 in [8], play an important role in the proofs of main results of present paper.…”

“…The important rules of calculation of the Kronecker product are given in [1], [10] and Lemma 2.1 [8].…”

“…For important properties and rules of calculation of the Vec operator, see Lemmas 2.2, 2.3, 2.5 in [8].…”

“…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].…”