ON GENERALIZED LIPSCHITZIAN MAPPING AND EXPANSIVE LIPSCHITZ CONSTANT

Authors

  • M. Imdad Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
  • A. Soliman Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
  • M. Barakat Department of Mathematics, Faculty of Science, King Khaled University, Abha 9004, Saudi Arabia

Abstract

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 $\lim \inf_{G\ni t\rightarrow\infty}| \|T(t)\| | = \lim_{G\ni t\rightarrow\infty}| \|T(t)\| |=k < WCS(X)\mu_0$ 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].

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Published

2013-05-22

How to Cite

Imdad, M., Soliman, A., & Barakat, M. (2013). ON GENERALIZED LIPSCHITZIAN MAPPING AND EXPANSIVE LIPSCHITZ CONSTANT. Journal of Nonlinear Analysis and Optimization: Theory & Applications, 5(1), 11-25. Retrieved from http://www.math.sci.nu.ac.th/ojs302/index.php/jnao/article/view/199

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Section

Research Article