Delay equations and nonuniform exponential stability (Q947017)

The authors consider the delay linear equation \[ v'=L(t)v_t, \tag{1} \] where \(L(t): C([-r,0],\mathbb{R}^n)\to\mathbb{R}^n\), \(v_t(\theta):=v(t+\theta)\), \(\theta\in[-r,0]\). There exists a \(k>0\) such that for any \(t\in\mathbb{R}\) \(\int_t^{t+r}\| L(\tau)\| \,d\tau\leq k(1+| t| )\). Equation (1) admits a \textit{nonuniform exponential contraction} if there exist constants \(a<0\), \(D>0\), \(\varepsilon\geq0\) such that \(\| T(t,s)\| \leq D\exp(a(t-s)+\varepsilon| s| )\), \(t\geq s\), where \(T(t,s)\) is the evolution operator of the equation. The authors obtain two theorems on stability of the zero solution of the nonlinear equation \[ v'=L(t)v_t+f(t,v_t) \tag{2} \] assuming that (1) admits the nonuniform exponential contraction. Let \(\varphi\) be an initial function of the solution \(v(\cdot,s,\varphi)\) to the perturbed equation (2). Theorem 1: \(\| v_t\| \leq 2\bar{D}\| \varphi\| \exp[a(t-s)+(k+\varepsilon)| s| ]\) for every \(t\geq s\), if \[ \| f(t,x)-f(t,y)\| \leq\delta\| x-y\| (\| x\| ^q+\| y\| ^q)\quad \text{for every \(t\in\mathbb{R}\), and \(x,y\in C\)}, \] for some \(q>0\) such that \(qa+\varepsilon+k<0\) and sufficiently small \(\delta>0\), and if \(\| \varphi\| \leq e^{-(k+\varepsilon)(1+2/q)| s| }\). The second theorem gives the analogous estimate of the solution's norm with arbitrary initial function from C but at the expense of assuming an exponential decay of the perturbation in time.