ON GENERALIZED LIPSCHITZIAN MAPPING AND EXPANSIVE LIPSCHITZ CONSTANT
Imdad Mohd, Soliman Ahmed H., Barakat Mohamed A. · 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 2 S} on a nonempty convex weakly compact subset C of a Banach space with a weak uniform normal structure X, such that lim infS3t!1 ||T(t)|| = limS3t!1 ||T(t)|| = k < WCS(X)??0, admits a common fixed point where ??0 = inf{?? 1 : ??(1 ??? X(1/??)) (1/2)}, WCS(X) is the weakly convergent sequence coefficient of X, and ||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].