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Boundary value problems for differential equations with reflection of the argument. (English) Zbl 0583.34055

The article studies existence and uniqueness of certain boundary value problems for ordinary differential equations with reflection of the argument. Such equations represent a particular case of functional differential equations whose arguments are involutions. Important in their own right, they have applications in the investigation of stability of differential-difference equations. Differential equations with involutions can be transformed to higher order ordinary differential equations and admit of point data initial or boundary conditions. Initial value problems for such equations have been studied in numerous works. However, boundary value problems even for differential equations with reflection of the argument have not been considered yet.

The present paper initiates studies in this direction. For nonlinear equations the method is based on the use of Schauder’s fixed point theorem, and linear equations with reflection of the argument are changed to higher order equations without reflection.

34K10Boundary value problems for functional-differential equations