Uniformly normal structure and uniformly generalized Lipschitzian semigroups
-
1893
Downloads
-
2764
Views
Authors
Ahmed H. Soliman
- Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt.
Mohamed A. Barakat
- Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt.
Abstract
In this work, we introduce some condition on one-parameter semigroup of self-mappings it is called \(k\)-uniformly
generalized Lipschitzian. The condition is weaker than Lipschitzian type conditions. Also, we
show that a \(k\)-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
\(k > 0\). Our results extending the results due to L.C. Ceng, H. K. Xu and J.C. Yao [5]
Share and Cite
ISRP Style
Ahmed H. Soliman, Mohamed A. Barakat, Uniformly normal structure and uniformly generalized Lipschitzian semigroups, Journal of Nonlinear Sciences and Applications, 5 (2012), no. 5, 379--388
AMA Style
Soliman Ahmed H., Barakat Mohamed A., Uniformly normal structure and uniformly generalized Lipschitzian semigroups. J. Nonlinear Sci. Appl. (2012); 5(5):379--388
Chicago/Turabian Style
Soliman, Ahmed H., Barakat, Mohamed A.. "Uniformly normal structure and uniformly generalized Lipschitzian semigroups." Journal of Nonlinear Sciences and Applications, 5, no. 5 (2012): 379--388
Keywords
- Uniformly normal structure
- Uniformly generalized semigroup
- Fixed point
- Characteristic of convexity
- Modulus of convexity.
References
-
[1]
A. G. Aksoy, M. A. Khamsi , Nonstandard methods in fixed point theory, Springer, New York (1990)
-
[2]
R. E. Bruck, On the almost-convergence of iterates of a nonexpansive mappings in Hilbert space and the structure of the weak-limit set, Israel J. Math., 29 (1978), 1-16.
-
[3]
W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math., 86 (1980), 427-436.
-
[4]
E. Casini, E. Maluta, Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Anal., 9 (1985), 103-108.
-
[5]
L. C. Ceng, H. K. Xu, J. C. Yao, Uniformly normal structure and uniformly Lipschitzian semigroups, Nonlinear Anal., doi:10.1016/j.na. 2010.07. 044. (2010)
-
[6]
M. Edelstein, The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc., 78 (1972), 206-208.
-
[7]
K. Goebel, Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compos. Math., 22 (1970), 269-274.
-
[8]
K. Goebel, W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. , 47 (1973), 135-140.
-
[9]
K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, in: pure and Applied Math., A series of Monoghraph and Textbooks, 83, Marcel Dekker, New York (1984)
-
[10]
H. Ishihara, W. Takahashi , Fixed point theorems for unformly Lipschitzian semigroups in Hilbert spaces, J. Math. Anal. Appl. , 127 (1987), 206-210.
-
[11]
M. Kato, L. Maligranda, Y. Takahashi , On James and Jordan-von Neumann constants and normal structure coefficient in Banach spaces, Studia Math. , 144 (2001), 275-293.
-
[12]
E. A. Lifschits, Fixed point theorems for operators in strongly convex spaces, Voronez Gos. Univ. Trudy Math. Fak. , 16 (1975), 23-28.
-
[13]
T. C. Lim, On the normal structure coefficient and the bounded sequence coefficient, Proc. Amer. Math. Soc., 88 (1983), 262-264.
-
[14]
E. Maluta, Uniform normal structure and related coefficients, Paciffic J. Math. , 111 (1984), 357-369.
-
[15]
S. Prus, M. Szczepanik, New coefficients related to uniform normal structure, Nonlinear and Convex Anal., 2 (2001), 203-215.
-
[16]
K. K. Tan, H. K. Xu, Fixed point theorems for Lipschitzian semigroups in Banach spaces, Nonlinear Anal. , 20 (1993), 395-404.
-
[17]
X. Wu, J. C. Yao, L. C. Zeng, Uniformly normal structure and strong convergences theorems for asymptoticlly pseudocontractive mapping, J. Nonlinear convex Anal., 6 (3) (2005), 453-463.
-
[18]
H. K. Xu, Fixed point theorems for uniformly Lipschitzian semigroups in uniformly convex spaces, J. Math. Anal. Appl. , 152 (1990), 391-398.
-
[19]
J. C. Yao, L. C. Zeng , A fixed point theorem for asymptotically regular semigroups in metric spaces with uniform normal structure, J. Nonlinear convex Anal. , 8 (1) (2007), 153-163.
-
[20]
L. C. Zeng, Fixed point theorems for nonlinear semigroups of Lipschitzian mappings in uniformly convex spaces, Chinese Quart. J. Math. , 9 (4) (1994), 64-73.
-
[21]
L. C. Zeng, On the existence of fixed points and nonlinear ergodic retractions for Lipschitzian semigroups without convexity, Nonlinear Anal. , 24 (1995), 1347-1359.
-
[22]
L. C. Zeng, Fixed point theorems for asymptotically regular Lipschitzian semigroups in Banach spaces, Chinese Ann. Math., 16A (6) (1995), 744-751.
-
[23]
L. C. Zeng, Y. L. Yang, On the existence of fixed points for Lipschitzian semigroups in Banach spaces, Chinese Ann. Math. , 22B (3) (2001), 397-404.
-
[24]
L. C. Zeng, Fixed point theorems for asymptotically regular semigroups in Banach spaces, Chinese Ann. Math., 23A (6) (2002), 699-706.
-
[25]
L. C. Zeng, Weak uniform normal structure and fixed points of asymptotically regular semigroups, Acta. Math. Sin. (English Series) , 20 (6) (2004), 977-982.
-
[26]
L. C. Zeng, Uniform normal structure and solutions of Reich's open question, Appl. Math. Mech., 26 (9) (2005), 1204-1211.