Solvability of Volterra-Stieltjes operator-integral equations and their applications. (English) Zbl 0986.45006

The authors investigate continuous solutions of a certain operator-integral equation of Volterra-Stieltjes type \[ x(t)= h(t), (Tx) (t)\int_{0}^{t} u(t,s, x(s)) d_{s} g(t,s), \] where \(t\in [0,1 ]= I\) and \(T: C(I)\to C(I)\) is a continuous operator which satisfies classical Darbo condition formulated in terms of the Hausdorff measure of noncompactness. The main result of the paper states that under suitable assumptions imposed on \(h\), \(T\), \(u\) and \(g\), the given equation has at least one solution. Its proof is based on a suitable application of the Darbo fixed point theorem. In the second part of the paper the authors discuss some special cases of the equation in question. In particular, the famous Chandrasekhar quadratic integral equation \[ x(t)=1,x(t)\int_{0}^{t}{t\over t,s}\phi(s)x(s) ds \] is considered. The results of the paper generalize some previous ones obtained by themselves and other authors.


45N05 Abstract integral equations, integral equations in abstract spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
45G10 Other nonlinear integral equations
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