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Solvability of a linear boundary value problem for a fredholm integro-differential equation with impulsive inputs

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Abstract

Abstract—We suggest a method for the study and solution of a linear boundary value problem for a Fredholm integro-differential equation with impulsive inputs at given times. The method is based on a partition of the interval and the introduction of auxiliary parameters as the values of the solution at the initial points of subintervals. For each partition containing impulsive points, we construct a Fredholm integral equation of the second kind. We introduce the definition of a regular partition and show that the set of regular mappings is nonempty. By using the resolvent of the constructed integral equation, the fundamental matrix of the differential part, and the original data of the problem, we form a system of linear equations for the introduced parameters. The equivalence of the solvability of that system and the considered linear boundary value problem are proved. We obtain necessary and sufficient conditions for the solvability and unique solvability.

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Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Almaty, Kazakhstan

    D. S. Dzhumabaev

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  1. D. S. Dzhumabaev
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Correspondence to D. S. Dzhumabaev.

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Original Russian Text © D.S. Dzhumabaev, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 9, pp. 1189–1205.

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Dzhumabaev, D.S. Solvability of a linear boundary value problem for a fredholm integro-differential equation with impulsive inputs. Diff Equat 51, 1180–1196 (2015). https://doi.org/10.1134/S0012266115090086

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  • DOI: https://doi.org/10.1134/S0012266115090086

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