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

Cubiotti, Paolo; Yao, Jen-Chih

Abstr. Appl. Anal. 2014, Article ID 301675, 6 p. (2014).

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

References:

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