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An application of Darbo fixed-point theorem to a class of functional integral equations. (English) Zbl 1316.45006

The authors consider the functional integral equation \[ x(t) = g(t, x(\beta_1(t)), \dots, x(\beta_s(t))) + f(t, x(\alpha_1(t)), \dots, x(\alpha_m(t))) \int_0^{\phi(t)} u(t, \tau, x(\gamma_1(\tau)), \dots, x(\gamma_n(\tau))) d\tau \] and find sufficient conditions for a continuous solution to exist. These conditions include the continuity of all given functions, Lipschitz properties of \(f\) and \(g\), and a growth condition on \(u\).

MSC:

45G10 Other nonlinear integral equations
45D05 Volterra integral equations
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
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