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 G} on a nonempty convex weakly compact subset C of a Banach space with a weak uniform normal structure X, such that _G i t→∞| \|T(t)\| | = _G i t→∞| \|T(t)\| |=k < WCS(X)μ_0 admits a common fixed point where μ_0={μ≥1 : μ(1-δ_X(1/μ))≥(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].