Non-autonomous implicit integral equations with discontinuous right-hand side. (English) Zbl 1099.45004

The authors study the problem of existence of solutions to the implicit integral equation \[ h(u(t))=f(t,\int _I g(t,z)u(z)\,dz) \] for almost all \(t\in I=[0,1]\), and where \(f\: I\times [0,\lambda ]\to \mathbb R\), \(g\: I\times I\to [0,\infty )\) and \(h\: (0,\infty )\to \mathbb R\). F. Cammaroto and P. Cubiotti [Commentat. Math. Univ. Carol. 38, No. 2, 241–246 (1997; Zbl 0886.47031)] established an existence theorem for solutions \(u\in L^ {\infty }(I)\) to a less general equation under assumptions on \(f\) considerably weaker than continuity. In the paper under review the existence of solutions \(u\in L^ s(I)\) is established, again without a-priori assuming that \(f\) is continuous.


45G10 Other nonlinear integral equations


Zbl 0886.47031
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