Uniqueness for retarded delay differential equations without Lipschitz condition

M. Bartha, Bolyai Institute, University of Szeged, Hungary
J. Terjéki, Bolyai Institute, University of Szeged, Hungary

E. J. Qualitative Theory of Diff. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ., No. 2. (2008), pp. 1-6.

Abstract: Consider the equation $\dot x(t)=f(t, x(t), x(t-r(t)))$ with the initial condition $x_{0}=\phi$. Here $f$ is a continuous real function, but it does not satisfy other regularity conditions. We prove that the initial value problem has a unique solution under the following monotonicity conditions:
$(x-y)f(t,x,y)\leq 0$ for all $t,\,x,\,y\in\R$, $f(t,x_1,y)\geq f(t,x_2,y)$ for all $t,\,y\in\R$, and $x_1<x_2$, and if there is $t_0\geq 0$ such that $r(t_0)=0$, then the function $t_0-t+r(t)$ does not change sign on an interval $[t_0, t_0+\delta)$.

We show an example that the result cannot be applied in the state dependent case.

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