Study on existence of solutions for some nonlinear functional-integral equations with applications (Q2862107)

In this paper, the authors consider the functional-integral equation NEWLINE\[NEWLINE x(t)=\left(u(t,x(t))+f\left(t, \int_0^t p(t,s,x(s))\, ds, x(\alpha(t))\right)\right) \times g\left(t, \int_0^a q(t,s,x(s))\, ds, x(\beta(t))\right) NEWLINE\]NEWLINE for \(t\in [0,a]\), where \(u, f, g\) satisfy certain Lipschitz conditions, \(p, q\) satisfy some sublinearity conditions, and \(\alpha, \beta\) are deviating arguments. By using measures of noncompactness in Banach algebras, the Darbo condition, and a fixed-point theorem for products of operators given by \textit{J. Banas} and \textit{M. Lecko} [Panam. Math. J. 12, No.~2, 101--109 (2002; Zbl 1009.47048)], they establish the existence of at least one solution of the equation.