The authors present an application of the fixed-point theory in stability. They suggest a generalization of Krasnosel’skii’s theorem on fixed-points of operators of the form $A+B$, where $A$ is completely continuous and $B$ is contracting, and use their result to prove new theorems on the exponential stability of solutions to Cauchy problems. General theorems are applied to perturbed Liénard equations.
One of the main results is as follows:
Let $M$ denote a closed convex nonempty subset of the Banach space $U$ of bounded continuous functions $\varphi: [0,\infty) \to\bbfR^d$. Consider the Cauchy problem $$x'=b(t,x) +a(t,x), \quad x(0)=x_0\in \bbfR^d,\ t \in [0,\infty).$$ Let $b(t,x)$ be uniformly Lipschitz in $x$ for $t\in[0, \infty $), $x\in\bbfR^d$. Let the operator $A$ defined by $$y\mapsto \int^t_0 a\bigl( s,y(s) \bigr)ds$$ be continuous on $M$ and the image $A(M)$ of the set $M$ be compact. Let the operator $B$ defined by $$y\mapsto \int^t_0 b\bigl(s,y(s) \bigr) ds$$ be contracting on $U$ with the constant $\alpha<1$.
Then if for each $y\in M$ a unique solution $x$ to $x'=b(t,x)+ a(t,y)$, $x(0)= x_0$, is in $M$, then a solution to the Cauchy problem above is also in $M$.