Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay.

Deng, Shengfu; Prather, Carl

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This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $\{\begin{matrix} x^{'} (t) = f (t, x (t), x (α_{1} (t)), \dots, x (α_{n} (t))) for a.e. t \in [0, T], Δ x (t_{k}) = I_{k} (x (t_{k})), k = 1, \dots, m, x (0) = x (T) . \end{matrix}$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.