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On stability properties for one-dimensional functional differential equations. (English) Zbl 0746.34045
The author considers the equation (1) $x'(t)=F(t,x\sb t)$, where $F: [0,\infty)\times BC\to R$ $(BC$ denotes the set of bounded, continuous functions mapping $(\infty,0]$ into $R)$, $F(\cdot,0)\equiv 0$. If $a\in R$, $\psi\in C((-\infty,a],R)$ and $t\le a$, then $\psi\sb t\in BC$ is defined by $\psi\sb t(s)=\psi(t+s)$, $s\le 0$. For any $\psi\in C(R,R)$ with $\psi\sb t\in BC$ the function $t\to F(t,\psi\sb t)$ is continuous on $[0,\infty)$. For given $t\sb 0\ge 0$, $\varphi\in BC$, the function $x(\cdot)=x(\cdot;t\sb 0,\varphi)\in C((-\infty,t\sb 0+\omega),R)$ is a solution of (1) through $(t\sb 0,\varphi)$ on $[t\sb 0,t\sb 0+\omega)$, $\omega>0$, if $x\sb{t\sb 0}=\varphi$ and (1) holds on $[t\sb 0,t\sb 0+\omega)$. It is assumed that additional conditions are satisfied for $F$ such that $x(\cdot;t\sb 0,\varphi)$ uniquely exists on $[t\sb 0,\infty)$ for all $t\sb 0\ge 0$ and $\varphi\in BC$. The main results of the paper give sufficient conditions for the uniform stability and for the uniform asymptotic stability of the zero solution of (1).

34K20Stability theory of functional-differential equations