### Integral equations in the theory of boundary value problems of dynamical systems

#### Abstract

The solvability and construction of the general solution of the first kind Fredholm integral equation are among the insufficiently explored problems in mathematics. There are various approaches for solving the problem. Note the following methods for solving ill-posed problem: regularization method, the method of successive approximations, the method of undetermined coefficients. The purpose of this work is to create a new method for solvability

and construction of solution for integral equation of the first kind. It follows from the foregoing, the study of the solvability and construction of solution of the first kind Fredholm integral equation is topical. In this paper the solvability and construction of the solution matrix Fredholm integral equation of the first kind is considered. Construction of an approximate solution of Fredholm integral equation of the first kind. The results for the matrix Fredholm integral equation of the first kind similarly to asymmetric and symmetric kernels are valid. A new method for studying the solvability and construction of solution for the first kind Fredholm integral equation is proposed. Necessary and sufficient conditions for existence of solutions for a given right-hand side are obtained in two cases: when the origin function belongs to the space L2; the origin function belongs to a given set of L2: Solvability conditions and the method of construction an approximate solution of the first kind integral Fredholm equation are obtained.

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