A nonlinear inequality and evolution problems

TL;DR

This paper develops a nonlinear inequality for differential inequalities involving functions with certain growth conditions, providing criteria for global existence and bounds of solutions, with applications to stability analysis in differential equations.

Contribution

It introduces a new nonlinear inequality and its discrete version, useful for analyzing the stability of solutions to differential equations in various spaces.

Findings

Established conditions for global existence of solutions.

Derived bounds for solutions using a function ur(t).

Applied results to stability analysis of differential equations.

Abstract

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