ON GENERALIZED LIPSCHITZIAN MAPPING AND EXPANSIVE LIPSCHITZ CONSTANT - Imdad, M.; Soliman, A. H.; Barakat, M. A. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2013)
In this paper, we will introduce a new uniformly generalized Lipschitzian type conditions for a one-parameter semigroups of self-mappings and, we show that a uniformly generalized Lipschitzian semigroup of nonlinear self-mappings of a nonempty closed convex subset $C$ of real Banach space $X$ admits a common fixed point if the semigroup has a bounded orbit and if $k$ is appropriately larger than one. Finally, we prove that a semigroup of self mappings $T = \{T(t) : t \in G\}$ on a nonempty convex weakly compact subset $\overline{C}$ of a Banach space with a weak uniform normal structure $X$, such that $\lim \inf_{G i t\rightarrow\infty}| \|T(t)\| | = \lim_{G i t\rightarrow\infty}| \|T(t)\| |=k < WCS(X)\mu_0$ admits a common fixed point where $\mu_0=\inf\{\mu\geq1 : \mu(1-\delta_X(1/\mu))\geq(1/2)\}$, and $WCS(X)$ is the weakly convergent sequence coefficient of $X$ while $| \|T(t)\| |$ is the exact Lipschitz constant of $T(t)$. Our result represents an extension of the result of L.C. Ceng, H. K. Xu and J.C. Yao [5] and L. C. Zeng [29].