Filipchuk M. Averaging method in the boundary value problems for differential equations with deviated argument

3. Investigation object: the boundary value problems for differential equations with deviated argument initial set of which contains one point. Investigation purpose: investigation of solvability. Investigation methods: averaging method, numerical-analytic. Theoretical and practical results, novelty: The sufficient conditions for existence of unique solutions of boundary value problems with twopoint, multipoint and integral boundary conditions in some -vicinities of unique solutions of respective averaging boundary value problems were obtained for periodic systems with variable delay. Assuming existence and uniqueness of solutions of original and averaging multipoint boundary value problems, the conditions which assure their closeness were obtained for nonperiodic systems with variable delay. The numerical-analytic method of investigation of boundary value problems with twopoint, multipoint and integral boundary conditions was substantiated for systems with transformed argument. Degree of applicatio n: it is planned. Sphere (area) of application: the theory of differential equations.