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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.

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