Equivalence of Lp Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups

Lp stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is Lp stable for some p > 0. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

Common fixed point theorems for semigroups on metric spaces

This paper consists of two main results. The first one shows that ifSis a left reversible semigroup of selfmaps on a complete metric space(M,d)such that there is a gauge functionφfor whichd(f(x),f(y))≤φ(δ(Of (x,y)))forf∈Sandx,yinM, whereδ(Of (x,y))denotes the diameter of the orbit ofx,yunderf, thenShas a unique common fixed pointξinMand, moreover, for anyfinSandxinM, the sequence of iterates{fn(x)}converges toξ. The second result is a common fixed point theorem for a left reversible uniformly Lipschitzian semigroup of selfmaps on a bounded hyperconvex metric space(M,d).

A Fixed Point Theorem for Semigroups of Proximately Uniformly Lipschitzian Mappings

As a generalization of Kiang and Tan's proximately nonexpansive semigroups, the notion of a proximately uniformly Lipschitzian semigroup is introduced and an existence theorem of common fixed points for such a semigroup is proved in a Banach space whose characteristic of convexity is less than one.

Uniformly Lipschitzian Semigroups in Hilbert Space

Let K be a closed, bounded, convex, nonempty subset of a Hilbert Space . It is shown that if is a left reversible, uniformly k-lipschitzian semigroup of mappings of K into itself, with k < √2, then has a common fixed point in K.