Measurable solutions of implicit integral equations with discontinuous right-hand side

We prove an existence theorem for measurable solutions $u:[a,b] o Y$ of the integral equation $psi(t,u(t))= phiig(t,int_a^th(s,u(s)),dsig)$, where $Y$ is a compact, connected and locally connected metric space, and $h:[a,b] imes Y o R^n$, $psi:[a,b] imes Y o R$ and $phi:[a,b] imes R^n o R$ are given functions. Our result extends and improves a previous result, valid for the case where $n=1$ and $psi$ does not depend on $t$ explicitly. A function $phi:[a,b] imes R^n o R$ satisfying our assumptions can be discontinuous (with respect to the second variable) even at all points $xin R^n$.