A nonlinear inequality and evolution problems

Abstract

Assume that $g(t)\ge 0$, and $$\dot{g}(t)\le -\gamma (t)g(t)+\alpha (t,g(t))+\beta (t),t\ge 0;g(0)=g_0;\dot{g}:=\frac{dg}{dt},$$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right, $\dot{g}(t):={\lim }_{s\to +0}\frac{g(t+s)-g(t)}{s}$. It is assumed that $\gamma (t)$, and $\beta (t)$ are nonnegative continuous functions of $t$ defined on ${\mathbb{R}}_{+}:=[0,\infty )$, the function $\alpha (t,g)$ is defined for all $t\in {\mathbb{R}}_{+}$, locally Lipschitz with respect to $g$ uniformly with respect to $t$ on any compact subsets$[0,T]$, $T<\infty$, and non-decreasing with respect to $g$, $\alpha (t,g_1)\ge \alpha (t,g_2)$ if $g_1\ge g_2$. If there exists a function $\mu (t)>0$, $\mu (t)\in C^1({\mathbb{R}}_{+})$, such that $$\alpha \left(t,\frac{1}{\mu (t)}\right)+\beta (t)\le \frac{1}{\mu (t)}\left(\gamma (t)-\frac{\dot{\mu }(t)}{\mu (t)}\right),\forall t\ge 0;\mu (0)g(0)\le 1,$$ then $g(t)$ exists on all of ${\mathbb{R}}_{+}$, that is $T=\infty$, and the following estimate holds: $$0\le g(t)\le \frac{1}{\mu (t)},\forall t\ge 0.$$ If $\mu (0)g(0)<1$, then $0\le g(t)<\frac{1}{\mu (t)},\forall t\ge 0.$ A discrete version of this result is obtained. The nonlinear inequality, obtained in this paper, is used in a study of the Lyapunov stability and asymptotic stability of solutions to differential equations in finite and infinite-dimensional spaces.