Instability of differential equations with respect to the first approximation (Q1065995)

The objective of this paper is to construct an example of differential equation \(\dot x=Ax+F(x)\) (1) in an infinite Banach space, where a nonempty subset of the operator spectrum \(\sigma\) (A) has a positive real part, a nonlinear function satisfies the condition \(\lim_{x\to o^+}\| F(x)\| \cdot \| x\|^{-1}=0,\) and the trivial solution of equation (1) is Lyapunov stable.