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

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


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


Zbl 0482.47027
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