Geodesic
An ordinary integro-differential equation with~a~degenerate kernel and an integral condition
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 4, pp. 644-655

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We consider the questions of one value solvability of the nonlocal boundary value problem for a nonlinear ordinary integro-differential equation with a degenerate kernel and a reflective argument. The method of the degenerate kernel is developed for the case of considering ordinary integro-differential equation of the first order. After denoting the integro-differential equation is reduced to a system of algebraic equations with complex right-hand side. After some transformation we obtaine the nonlinear functional-integral equation, which one valued solvability is proved by the method of successive approximations combined with the method of compressing mapping. This paper advances the theory of nonlinear integro-differential equations with a degenerate kernel.
Keywords: integro-differential equation, degenerate kernel, reflective argument, integral form condition, one valued solvability. 
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     author = {T. K. Yuldashev},
     title = {An ordinary integro-differential equation with~a~degenerate kernel and an integral condition},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
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T. K. Yuldashev. An ordinary integro-differential equation with~a~degenerate kernel and an integral condition. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 4, pp. 644-655. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_4_a5/