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On retarded integral inequalities and their applications. (English) Zbl 1057.26022
Summary: The main results of [O. Lipovan, J. Math. Anal. Appl. 285, 436–443 (2003; Zbl 1040.26007)] are here generalized to the following retarded integral inequalities: \[ u^m(t)\leq c^{m/(m-n)}+\frac {m}{m-n}\int^{\alpha(t)}_0 \biggl[f(s)u^n(s)w\bigl(u(s)\bigr)+ g(s) u^n(s)\biggr]ds \] and \[ u^m(t)\leq c^{m/(m-n)}+\frac{m}{m-n} \int^{\alpha(t)}_0 f(s)u^n(s)w\bigl(u(s)\bigr) ds+\frac {m}{m-n} \int^t_0 g(s)u^n(s)w\bigl(u(s)\bigr)ds, \] where \(m>n>0\) are constants and \(t\in R_+=[0,\infty)\). The results given here can be applied to the global existence of solutions to differential equations with time delay.

MSC:
26D15 Inequalities for sums, series and integrals
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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