A condition for the unique solvability of nonlocal boundary value problems for systems of functional-differential equations
Keywords:
parametrization method, parameter, boundary condition, unambiguous solvability, kernel.Abstract
When considering non-local boundary value problems for functional-differential
equations, when the derivative of the desired function is contained in the right side, one could use
the resolvent of the integral equation. But, as is known, the resolvent of an integral equation of the
second kind of the Fredholm type cannot always be uniquely determined. In some cases, you can
use the properties of the kernel of the integro-differential equation. In this paper, we consider a nonlocal
boundary value problem for systems of integro-differential equations with involution, when
the kernel of the integral term containing the derivative has a partial derivative. Using the properties
of an involutive transformation, the problem is reduced to the study of a multipoint boundary value
problem for systems of integro-differential equations. The parameterization method proposed by
Professor D. Dzhumabaev was applied to this problem. New parameters are introduced, and based
on these parameters, we pass to new variables. When passing to new variables, we obtain the initial
conditions for the initial equation. With the help of this condition, it is possible to determine the
solution of the resulting Cauchy problem, as well as the system of linear equations. Applying the
Fredholm theory to solve the obtained systems of integral equations, i.e. the unique solvability of
the problem under study, we reduce to the reversibility of the matrix, which depends on the initial
data. An example was shown as an illustration of the proposed method.
References
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