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Generalized solutions of nonlinear integral-functional equations. (English) Zbl 1120.45002

Summary: A method of construction of generalized solutions with a point carrier in the singular part is proposed for nonlinear Volterra integral-functional equations \[ \int\limits_0^t K(t,s)(x(s)+ax(\alpha s)+g(s^lx(s),s))ds = f(t) \] with sufficiently smooth kernel and function \(f\); \(\alpha\) and \(a\) are constants, and \(0<|\alpha|<1\). The solution is constructed as a sum of singular and regular components. A special system of linear algebraic equations is used for the construction of the singular component. The regular part is constructed by the method of successive approximations combined with the method of undetermined coefficients. Theorems of existence and uniqueness of the generalized solutions are proved.

MSC:

45G05 Singular nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations
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