Integrable solutions of a functional-integral equation. (English) Zbl 0679.45003

Banaś, Józef; Knap, Zygmunt

Rev. Mat. Univ. Complutense Madr. 2, No. 1, 31-38 (1989).

Under certain assumptions on the functions f,g,k the authors prove that the functional-integral equation \[ x(t)=g(t)+f(t,\int^{1}_{0}k(t,s)x(\phi (s))ds), \] \(t\in [0,1)\) has at least one solution \(x\in L^ 1[0,1]\), which is a.e. nonincreasing on \(L^ 1[0,1]\). The method of proof is based on the notion of measure of weak noncompactness and the fixed point theorem due to G. Emmanuele [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 25, 353- 358 (1981; Zbl 0482.47027)].

Reviewer: J.Kolomý

MSC:

45G10	Other nonlinear integral equations
47J25	Iterative procedures involving nonlinear operators

Keywords:

integrable solutions; functional-integral equation; measure of weak noncompactness; fixed point theorem

Citations:

Zbl 0482.47027