Abstract
In this paper, we consider the global existence and boundedness of solutions for a certain nonlinear integro-differential equation of second order with multiple constant delays. We obtain some new sufficient conditions which guarantee the global existence and boundedness of solutions to the considered equation. The obtained result complements some recent ones in the literature. An example is given of the applicability of the obtained result. The main tool employed is an appropriate Lyapunov-Krasovskii type functional.
Highlights
It should be noted that Napoles Valdes [ ] dealt with the ordinary integro-differential equation of second order: t x + a(t)f t, x, x x + g t, x + h(x) = C(t, τ )x (τ ) dτ
It is worth mentioning that the global existence and boundedness of solutions of equation ( ) have not yet been discussed in the literature
Since the Lyapunov-Krasovskii type functional W (t) is positive definite and decreasing, W (t) ≤, along the trajectories of system ( ), we can say that W (t) is bounded [t, T]
Summary
It should be noted that Napoles Valdes [ ] dealt with the ordinary integro-differential equation of second order: t x + a(t)f t, x, x x + g t, x + h(x) = C(t, τ )x (τ ) dτ . The author investigated extendibility, boundedness, stability, and square integrability of solutions to the considered equation. Graef and Tunç [ ] discussed the continuability, boundedness, and square integrability of solutions to the second order functional integro-differential equation with multiple delays: n x + a(t)f t, x, x x + g t, x, x + hi x(t – τi)
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