A nonlinear inequality and evolution problems

Assume that $g(t)\geq 0$, and $\dot{g}(t)\leq -γ(t)g(t)+α(t,g(t))+β(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt},$ 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 $γ(t)$, and $β(t)$ are nonnegative continuous functions of $t$ defined on $\R_+:=[0,\infty)$, the function $α(t,g)$ is defined for all $t\in \R_+$, 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 $g$, $α(t,g_1)\geq α(t,g_2)$ if $g_1\ge g_2$. If there exists a function $μ(t)>0$, $μ(t)\in C^1(\R_+)$, such that $α\left(t,\frac{1}{μ(t)}\right)+β(t)\leq \frac{1}{μ(t)}\left(γ(t)-\frac{\dotμ(t)}{μ(t)}\right),\quad \forall t\ge 0;\quad μ(0)g(0)\leq 1,$ then $g(t)$ exists on all of $\R_+$, that is $T=\infty$, and the following estimate holds: $0\leq g(t)\le \frac 1{μ(t)},\quad \forall t\geq 0.$ If $μ(0)g(0)< 1$, then $0\leq g(t)< \frac 1{μ(t)},\quad \forall t\geq 0.$ 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.

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