On smoothness of solutions

to mixed type differential difference equations

E.P.Ivanova

Moscow State Aviation Institute

Moscow, 125871, Volokolamskoe shosse 4, Russia

 

The paper is concerned with the problem of solutions smoothness of initial value problems (IVP) for mixed type differential-difference equations:

 

                                                                    

                                               

Here  is real constant,  There are given the functions:     is  a  continuous nonincreasing function;  are piecewise continuous functions,  where C _t^m (`E) is the space of measurable functions on , belonging to

C ^m  with respect to  t  for almost all s such that  and piecewise continuous with  respect to  for all such that    

            In this paper there were received the consistency conditions that are necessary

and sufficient conditions for smooth continuation initial function to solution  

There was proved the theorem of existence and smoothness of solutions of  IVP (1)-(3).

 

            Theorem.  Let  the function  satisfies the solvability condition,  and the consistency conditions are fulfilled for

            Then the solution of  IVP (1)-(3) exists and is unique on  and

 

                   

               

 

There was proved the theorem of smoothness of solutions  with respect to the variables