Bounds of Solutions of Integrodifferential Equations

Abstract

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: $x\text{'}\left(t\right)-ℱ(t,x(t),{{\displaystyle \int }}_0^{t}k(t,s,x\left(t\right),x\left(s\right))ds)=h\left(t\right)$, $x\left(0\right)=x_0$, where $h:R_{+}\to R$, $k:R_{+}^2\times R^2\to R$, $ℱ:R_{+}\times R^2\to R$ are continuous functions, $R_{+}=[0,\infty )$.