Extension of Some Common Fixed Point Theorems to Mean Non-expansive Mappings

Subject Areas : Statistics

Shahram Saeidi ¹ ( Department of Mathematic, University of Kurdistan, Sanandaj, Iran )
Jafar Bakhande ² ( Assistant Professor,Department of Mathematic, University of Kurdistan, Sanandaj, Iran )

Received: 2019-11-10 Accepted : 2020-04-19 Published : 2021-03-01

Keywords: عمل نیم گروه نیم توپولوژیک, میانگینی غیرانبساطی, توابع تقریباً متناوب, نقطه ثابت,

Abstract :

In 1969, DeMarr proved that a commuting family of non-expansive mappings on a nonempty convex compact subset of a Banach space has a fixed point. Takahashi extended DeMarr’s theorem to the case of discrete amenable semi-groups. In recent years, considerable research has been devoted to the theory of fixed points as well as common fixed points. In the case of semi-topological semi-groups (that is, a semi-group with a Hausdorff topology such that the multiplication is separately continuous), Lau and Zhang studied DeMarr’s theorem under more general conditions for example in the case of the amenability of the space of almost periodic functions as well as the space of weakly almost periodic functions. In this paper, we study several fixed point properties of the mean non-expansive semi-topological semi-groups acting on nonempty convex weakly compact subsets of a locally convex space as well as give extensions of the results of Lau and Zhang.

References:

[1] R. Ahmed, S. Altwqi, Convergence theorems for three finite families of multivalued nonexpansive mappings, Journal of the Egyptian Mathematical Society 22 (2014), 459 – 465

 

[2] T.D. Benavides, M.A.J. Pineda, Fixed points of nonexpansive mappings in spaces of continuous functions, Proceeding of the American Mathematical Society 133 (2005), 3037- 3046

 

[3] T.D. Benavides, M.A.J. Pineda, S. Prus, Weak compactness and fixed point property for affine mappings, Journal of Functional Analysis 209 (2004), 1 - 15

 

[4] F.E. Browder, Non-expansive nonlinear operators in Banach spaces, Proceedings of the National Academy of Sciences USA 54 (1965), 1041 - 1044

 

[5] R.K. Bisht, R.P. Pant, A critical remark on Fixed point theorems for occasionally weakly compatible mappings, Journal of the Egyptian Mathematical Society 21 (2013), 273 - 275

 

[6] P.N. Dowling, C.J. Lennard, B. Turett, Weak compactness is equivalent to the fixed point property in c0, Proceeding of the American Mathematical Society 132 (2004), 1659 - 1666.

 

[7] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28, Cambridge Univ. Press, Cambridge, 1990

 

[8] K. Goebel, W.A. Kirk, Classical theory of nonexpansive mappings, in: Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, (2001), 49 - 91

 

[9] J. Kang, Fixed point set of semigroups of non-expansive mappings and amenability, Journal of Mathematical Analysis and Applications 341 (2008), 1445 - 1456

 

[10] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, American Mathematical Monthly 72 (1965), 1004 - 1006

 

[11] A.T.-M. Lau, Amenability of semigroups, in: K.H. Hoffmann, J.D. Lawson, J.S. Pym (Eds.), The Analytic and Topological Theory of Semigroups, de Gruyter, Berlin, (1990), 313 - 334

 

[12] R. DeMarr, Common fixed points for commuting contraction mappings. Pacific Journal of Mathematics 13(1963),1139–1141

 

[13] W. Takahashi, Fixed point theorem for amenable semigroup of nonexpansivemappings, Kodai Mathematical Journal Sem. Rep. 21 (1969), 383–386

 

[14] A.T.-M. Lau, Invariant means on almost periodic functions and fixed pointproperties. Rocky Mountain Journal of Mathematics 3(1973), 69–76

 

[15] A.T.-M. Lau, Y. Zhang, Fixed point properties of semigroups of non-expansive mappings, Journal of Functional Analysis 254 (2008), 2534 - 2554

 

[16] M.M. Day, Amenable semigroups. Illinois Journal of Mathematics 1 (1957), 509–544

 

[17] S. Zhang, About fixed point theory for mean nonexpansive mapping in Banach spaces, Journal of Sichuan University 2 (1975), 67-68

 

[18] C. Wu, L.J. Zhang, Fixed points for mean non-expansive mappings, Acta Mathematica Sinica, English Series 23 (2007), 489-494

 

[19] Y. Yang, Y. Cui, Viscosity

approximation methods for mean non-expansive mappings in Banach spaces, Applied Mathematical Sciences (Ruse) 2 (2008), 627638

 

[20] K. Nakprasit, Mean nonexpansive mappings and Suzuki-generalized nonexpansive mappings, Journal of Nonlinear Analysis and Optimization, 1 (2010), 93-96

 

[21] Z. Zuo, Fixed-Point Theorems for Mean Nonexpansive Mappings in Banach Spaces, Abstract and Applied Analysis, Volume 2014, Article ID 746291, 6 pages

 

[22] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansivemappings, Journal of Mathematical Analysis and Applications 340 (2008), 1088-1095

 

[23] A.H. Soliman, M. A. Barakat, A characterization between fixed point properties of weak nonexpansive semigroups and the existence of a left invariant mean on the space of weakly almost periodic functions, Advanced Fixed Point Theory 7 (2017), 172-182

 

[24] A.T.-M. Lau, Y. Zhang, Fixed point properties for semigroups of nonlinear mappings and amenability, Journal of Functional Analysis 263 (2012), 2949-2977

 

[25] A.T.-M. Lau, Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2010) Art. ID 580956