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 , where is completely continuous and 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 denote a closed convex nonempty subset of the Banach space of bounded continuous functions . Consider the Cauchy problem
Let be uniformly Lipschitz in for ), . Let the operator defined by
be continuous on and the image of the set be compact. Let the operator defined by
be contracting on with the constant .
Then if for each a unique solution to , , is in , then a solution to the Cauchy problem above is also in .