Implicit Vector Integral Equations Associated with Discontinuous Operators

Abstract EN

Let I∶=[0,1].

We consider the vector integral equation h(u(t))=ft,∫Ig(t,z),u(z),dz for a.e.

t∈I, where f:I×J→R, g:I×I→ [0,+∞[, and h:X→R are given functions and X,J are suitable subsets of Rn.

We prove an existence result for solutions u∈Ls(I, Rn), where the continuity of f with respect to the second variable is not assumed.

More precisely, f is assumed to be a.e.

equal (with respect to second variable) to a function f*:I×J→R which is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry.

It is easily seen that such a function f can be discontinuous at each point x∈J.

Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case n=1.