Comparison of solutions to boundary-value problems for linear functional-differential equations

E. S. Zhukovskiy ; Kh. M. T. Takhir

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 3, pp. 284-292

Résumé

We consider the issues of solvability of boundary value problems for linear functional-differential equations. Statements allowing one to obtain conditions for the existence of a unique solution and for non-negativity of the Green's function, and to obtain a fundamental solution to the homogeneous equation are suggested. In order to apply these statements, one needs to define a “reference” boundary value problem that possesses the corresponding properties and to define an operator by means of the operators generated by the problem under study and the “reference” problem according to the given rule. If the spectral radius of this operator is less than 1, then the boundary value problem under consideration is uniquely solvable. Similarly, in order to obtain conditions for the nonnegativity of the Green's function and the fundamental solution, it is required to determine a special operator by the rule given in the paper and to verify its positivity. An example of application of the statements obtained to a particular boundary value problem with an integral boundary condition for the equation containing argument deviations to the unknown function and to its derivative is considered.

Détail

Keywords: linear functional-differential equation, boundary value problem, green's function, fundamental solution of a homogeneous equation, positive operator.