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

Iakovleva, Valentina; Vanegas, Carmen Judith

Electron. J. Differ. Equ. 2005, Conf. 13, 57-63 (2005).

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.

Reviewer: Ivan Ginchev (Varna)

MSC:

34K06	Linear functional-differential equations
34K05	General theory of functional-differential equations

Keywords:

differential-difference equations; functional-differential equations