Implicit integral equations with discontinuous right-hand side. (English) Zbl 0886.47031

Cammaroto, F.; Cubiotti, P.

Commentat. Math. Univ. Carol. 38, No. 2, 241-246 (1997).

A theorem concerning the existence of a weak bounded solution on the half-line is proved for the integral equation \[ h(u(t)) = f (\int_I g(t,x)u(x)dx), \] where \(g:I\times I \rightarrow [0,\infty)\), \(h:[\alpha,\beta]\rightarrow R\) and \(f:[0,\sigma]\rightarrow R\) \((0<\alpha <\beta,\sigma >0)\).
The main point is to overcome difficulties related to the discontinuity of the nonlinear function \(f\). For this, the results for the associated inclusions of the type \[ \psi(u)(t) \in F(t,\phi(u)(t)), \] due to O. N. Ricceri and B. Ricceri [Appl. Anal. 38, No. 4, 259-270 (1990; Zbl 0706.47038)], have been employed.

Reviewer: J.Andres (Olomouc)

MSC:

47H04

Set-valued operators

Keywords:

bounded solutions; integral equations; discontinuous right-hand side

Citations:

Zbl 0706.47038