## Implicit vector integral equations associated with discontinuous operators.(English)Zbl 1463.45026

Summary: Let $$I : = [0,1]$$. We consider the vector integral equation $$h(u(t)) = f \left(t, \int_I g(t, z), u(z), d z\right)$$ for a.e.$$t \in I$$, where $$f : I \times J \rightarrow \mathbb{R}, g : I \times I \rightarrow [0, + \infty [$$, and $$h : X \rightarrow \mathbb{R}$$ are given functions and $$X, J$$ are suitable subsets of $$\mathbb{R}^n$$. We prove an existence result for solutions $$u \in L^s(I, \mathbb{R}^n)$$, 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 $$f^* : I \times J \rightarrow \mathbb{R}$$ 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 $$x \in J$$. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar case $$n = 1$$.

### MSC:

 45G10 Other nonlinear integral equations
Full Text:

### References:

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