Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

TL;DR

This paper investigates the stability of solutions to abstract nonlinear evolution equations in Banach spaces under nonclassical assumptions, providing conditions for stability and convergence, and extending results to Hilbert spaces with unbounded operators.

Contribution

It introduces new stability criteria for nonlinear evolution equations with Hölder continuous derivatives in Banach spaces, including unbounded operators in Hilbert spaces.

Findings

Lyapunov stability under certain integral conditions

Boundedness and convergence of solutions established

Extension of stability results to unbounded operators in Hilbert spaces

Abstract

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fr\'echet derivative of $F(t,u)$ is H\"{o}lder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\|F'_u(t,w) - F'_u(t,v)\|\le \alpha(t)\|v - w\|^{q}$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v} = F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if $\sup_{t\ge 0}\int_0^t \alpha(\xi)\|U(t,\xi)\|\, d\xi<\infty$ and $\sup_{t\ge 0}\|U(t)\|<\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi)$ is the solution to the equation $\frac{d}{dt}{U}(t,\xi) = F'_u(t,0)U(t,\xi)$, $t\ge \xi$, $U(\xi,\xi)=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for $\lim_{t\to\infty}u(t) = 0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.