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Some new integral inequalities and their applications in studying the properties of integral and integro-differential equations. (English) Zbl 1231.45006

The authors study integral inequalities of the form \[ u(t) \leq \left (c_1+\int_0^{\alpha(t)}f(t,s)u(s)\,d s\right ) \left (c_2+\int_0^{\beta(t)}g(t,s)u(s)\,d s\right ), \] where \(\alpha(t)\leq t\) and either \(\beta(t)=\alpha(t)\) or \(\beta(t)=t\) and show that under certain conditions one has \[ u(t)\leq \frac{c_1c_2Q(t)}{1-c_1c_2\int_0^t R(s)Q(s)\, ds},\quad t\geq 0, \] where \[ R(t) =\frac{d}{dt}\left(\left(\int_0^{\alpha(t)}f(t,s)\,ds\right) \left(\int_0^{\beta(t)} g(t,s)\,ds\right)\right) \] and \(Q(t)= exp\left(\int_0^{\alpha(t)} c_2f(t,s)\,ds+\int_0^{\beta(t)}c_1 g(t,s)\,ds\right)\). Similar results are established for the inequality \[ u(t) \leq \left (c_1+\int_0^{\alpha(t)}f(t,s)\omega(u(s))\,d s\right ) \left (c_2+\int_0^{\beta(t)}g(t,s)\omega(u(s))\,d s\right ), \] and these results are used to prove results on the boundedness and asymptotic behavior of solutions to related integral equations.

MSC:

45D05 Volterra integral equations
26D15 Inequalities for sums, series and integrals
45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
45J05 Integro-ordinary differential equations
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