## On the solution of differential equations with delayed and advanced arguments.(English)Zbl 1092.34549

The paper studies existence and uniqueness of solutions on the interval $$[-1,+\infty)$$ (and similarly on the interval $$(-\infty,1]$$) of the differential-difference equation with both delay and advanced argument $x'(t)=x(t-1)+x(t+1)$ with initial condition $$x(t)=\varphi(t)$$, $$-1\leq t\leq 1$$, where $$\varphi\in C^\infty([-1,1])$$. It is proved that a solution exists if and only if $$\varphi^{(n+1)}(0)=\varphi^{(n)}(-1)+\varphi^{(n)}(1)$$, $$n=0,1,\dots\,$$. In this case, applying a step-inductive method, the solution is determined constructively and on this base its uniqueness is also shown.

### MSC:

 34K06 Linear functional-differential equations 34K05 General theory of functional-differential equations
Full Text: