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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].

It is proved (necessary and) sufficient conditions for Ψ − conditional asymptotic stability of the trivial solution of linear or nonlinear Lyapunov matrix differential equations.

It is proved (necessary and) sufficient conditions for Ψ– conditional exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations

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