Perturbing Lyapunov functions and stability criteria for initial time difference (Q1400697)

Consider the differential equation \[ x'=f(t,x) \tag{1} \] with \(f\in C(\mathbb{R}_{+}\times \mathbb{R}^{n},\mathbb{R}^{n})\). \(x(t,t_{0},x_{0})\) is the solution of (1) through \((t_{0},x_{0}).\) Let \(\eta=\tau_{0}-t_{0}.\) \newline Definition: The solution \(x(t,t_{0},x_{0})\) is said to be \newline (i) equistable if to given \(\varepsilon >0\) and \(t_{0}\in \mathbb{R}_{+},\) there exist \(\delta= \delta(t_{0},\varepsilon)>0\) and \(\sigma=\sigma(t_{0},\varepsilon) >0\) such that \(\| x_{0}-y_{0}\| <\delta, | \eta| <\sigma\) implies \(\| x(t+\tau,\tau_{0},y_{0})-x(t,t_{0},x_{0})\| <\varepsilon\) for \(t\geq t_{0};\) (ii) uniformly stable if \(\delta\) and \(\sigma\) in (i) are independent of \(t_{0}.\) \newline Let \(S(\rho):=\{x\in \mathbb{R}^{n}:\| x\| <\rho\}; S^{c}(\rho):=\{x\in \mathbb{R}^{n}:\| x\| \leq\rho\};\) \(K:=\{\phi\in C((0,\rho),\mathbb{R}_{+}): \phi(t)\) is increasing in \(t\) and \(\phi(t)\to 0\) as \(t\to 0\}\). One of the main results of this paper is the following: Assume that (i) \(V_{1}\in C(\mathbb{R}_{+}\times S(\rho),\mathbb{R}_{+})\), \(V_{1}(t,x)\) is locally Lipschitzian in \(x\) and \(V_{1}(t,0)\equiv 0\) and \[ D^{+}V_{1}(t,x)\leq g_{1}(t,V_{1}(t,x),| \eta| ),\quad (t,x)\in \mathbb{R}_{+}\times S(\rho), \] where \(g_{1}\in C(\mathbb{R}^{3},\mathbb{R});\) (ii) for every \(\gamma >0,\) there exists a \(V_{2}\in C(\mathbb{R}_{+}\times S(\rho)\cap S^{c}(\gamma)),V_{2}(t,x)\) is locally Lipschitzian in \(x,\) \[ b(\| x\| )\leq V_{2}(t,x)\leq a(\| x\| ),\quad (t,x)\in \mathbb{R}_{+}\times S(\rho)\cap S^{c}(\gamma), \] where \(a,b\in K\) and \[ D^{+}V_{1}(t,x)+ D^{+}V_{2}(t,x)\leq g_{2}(t,V_{1}(t,x)+ D^{+}V_{2}(t,x),| \eta| ), \quad (t,x)\in \mathbb{R}_{+}\times S(\rho)\cap S^{c}(\gamma), \] where \(g_{2}\in C(\mathbb{R}^{3}_{+},\mathbb{R});\) (iii) the scalar differential equation \[ w_{1}'=g_{1}(t,w_{1},| \eta| ),\quad w_{1}(t_{0})=w_{10}\geq 0, \] is equistable and the scalar differential equation \[ w_{2}'=g_{2}(t,w_{2},| \eta| ),\quad w_{2}(t_{0})=w_{20}\geq 0, \] is uniformly stable. Then the solution \(x(t,t_{0},x_{0})\) of equation (1) is equistable.