Paper's Title:

Strong Convergence Theorem for a Common Fixed Point of an Infinite Family of J-nonexpansive Maps with Applications

Author(s):

Charlse Ejike Chidume, Otubo Emmanuel Ezzaka and Chinedu Godwin Ezea

African University of Science and Technology,
Abuja,
Nigeria.
E-mail: cchidume@aust.edu.ng

Ebonyi State University,
Abakaliki,
Nigeria.
E-mail: mrzzaka@yahoo.com

 Nnamdi Azikiwe University,
Awka,
Nigeria.
E-mail: chinedu.ezea@gmail.com

Abstract:

Let E be a uniformly convex and uniformly smooth real Banach space with dual space E^*. Let {T_i}^∞_i=1 be a family of J-nonexpansive maps, where, for each i,~T_i maps E to 2^E*. A new class of maps, J-nonexpansive maps from E to E^*, an analogue of nonexpansive self maps of E, is introduced. Assuming that the set of common J-fixed points of {T_i}^∞_i=1 is nonempty, an iterative scheme is constructed and proved to converge strongly to a point x^* in ∩^∞_n=1F_JT_i. This result is then applied, in the case that E is a real Hilbert space to obtain a strong convergence theorem for approximation of a common fixed point for an infinite family of nonexpansive maps, assuming existences. The theorem obtained is compared with some important results in the literature. Finally, the technique of proof is also of independent interest.