Abstract
LetI∶=[0,1]. We consider the vector integral equationh(u(t))=ft,∫Ig(t,z),u(z),dzfor a.e.t∈I,wheref:I×J→R, g:I×I→ [0,+∞[,andh:X→Rare given functions andX,Jare suitable subsets ofRn. We prove an existence result for solutionsu∈Ls(I, Rn), where the continuity offwith respect to the second variable is not assumed. More precisely,fis assumed to be a.e. equal (with respect to second variable) to a functionf*:I×J→Rwhich is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a functionfcan be discontinuous at each pointx∈J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar casen=1.
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