On The Stability of The Quasi-Linear Implicit Equations in Hilbert Spaces

Abstract

We use the generalized theorem of Liapounov to obtain some necessary and sufficient conditions for the stability of the stationary implicit equation $$Ax'(t)=Bx(t) ,quad tgeq 0 ,$$ where $A$ and $B$ are bounded operators in Hilbert spaces. The achieved results can be applied to the stability for the quasi-linear implicit equation $$Ax'(t)=Bx(t)+theta(t,x(t)),quad tgeq 0 .$$