Therefore, one can conclude that. That is,. This fact shows that all solutions of the equation considered are bounded.
We establish the following result. Then, there exists a finite positive constant K such that the solution x t of equation 1 defined by the initial functions. Now, in view of 4 , it follows that. Now, the estimates related to V 1 and V 2 yield. Now, let. Then, a direct computation along this solution shows that. In view of the assumptions of theorem and expression 5 , we have that.
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Making use of these inequalities, we get. Now, subject to the assumptions ii and iii of theorem, one easily finds that. Gathering the above discussion into 8 and making use of the assumption ii , it follows that.
In view of 7 , it follows that. Now, the inequality 7 and the last inequality together give that. This fact completes the proof of theorem.
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On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order
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Interval-valued functional differential equations under dissipative conditions
Translated by J. Liapunov, Stability of Motion. Academic Press, London Makay, On the asymptotic stability of the solutions of functional-differential equations with infinite delay. Differential Equations, 1 , Buy eBook. Buy Softcover. FAQ Policy. About this book Since the publication of my lecture notes, Functional Differential Equations in the Applied Mathematical Sciences series, many new developments have occurred.
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