An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces
© Hussain et al.; licensee Springer. 2014
Received: 31 August 2014
Accepted: 12 November 2014
Published: 4 December 2014
The Erratum to this article has been published in Fixed Point Theory and Applications 2015 2015:203
We introduce an implicit method for finding an element of the set of common fixed points of a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the common fixed point of a representation of nonexpansive mappings.
exists for each . E is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for all . E is said to be uniformly smooth or is said to have a uniformly Féchet differentiable norm if the limit is attained uniformly for . It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm to weak∗ continuous on each bounded subset of E. A Banach space E is smooth if the duality mapping J of E is single-valued. We know that if E is smooth, then J is norm to weak-star continuous; for more details, see .
Let C be a nonempty closed and convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if for all , and a mapping f is an α-contraction on E if , such that .
for each , where is the adjoint operator of .
Let f be a function of the semigroup S into a reflexive Banach space E such that the weak closure of is weakly compact, and let X be a subspace of containing all the functions with . We know from  that for any , there exists a unique element in E such that for all . We denote such by . Moreover, if μ is a mean on X, then from , .
for all and ;
for every , the mapping is nonexpansive.
We denote by the set of common fixed points of , that is, .
Theorem 2.1 
is a nonexpansive mapping from C into C.
for each .
for each .
If X is -invariant for each and μ is right invariant, then for each .
Remark From Theorem 4.1.6 in , every uniformly convex Banach space is strictly convex and reflexive.
Let D be a subset of B, where B is a subset of a Banach space E, and let P be a retraction of B onto D, that is, for each . Then P is said to be sunny if for each and with , . A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D. We know that if E is smooth and P is a retraction of B onto D, then P is sunny and nonexpansive if and only if for each and , . For more details, see .
Lemma 2.2 
Let S be a semigroup, and let C be a compact convex subset of a real strictly convex and smooth Banach space E. Suppose that is a representation of S as a nonexpansive mapping from C into itself. Let X be a left invariant subspace of such that , and the function is an element of X for each and . If μ is a left invariant mean on X, then and there exists a unique sunny nonexpansive retraction from C onto .
Throughout the rest of this paper, the open ball of radius r centered at 0 is denoted by . Let C be a nonempty closed convex subset of a Banach space E. For and a mapping , we let be the set of ϵ-approximate fixed points of T, i.e., .
3 Main result
In this section, we deal with a strong convergence approximation scheme for finding a common element of the set of common fixed points of a representation of nonexpansive mappings.
strongly converges to Px.
Proof By Proposition 1.7.3 and Theorem 1.9.21 in , any compact subset C of a reflexive Banach space E is weakly compact, and from Proposition 1.9.18 in , any closed convex subset of a weakly compact subset C of a Banach space E is itself weakly compact, and by Proposition 1.9.13 in , any convex subset C of a normed space E is weakly closed if and only if C is closed. Therefore, weak closure of is weakly compact for each .
We divide the proof into five steps.
Step 1. The existence of which satisfies (1).
Therefore, by the Banach contraction principle , there exists a unique point such that .
Step 2. for all .
Since is arbitrary, we get .
Step 3. , where denotes the set of strongly limit points of .
converges strongly to a point z.
Step 5. strongly converges to Px.
That is, . □
Remark 3.2 It would be an interesting problem to prove Theorem 3.1 for continuous representations instead of nonexpansive.
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the referee of the paper for his helpful comments and invaluable suggestions. This research was supported by the Center of Excellence for Mathematics and the Office of Graduate Studies of the Lorestan University and the University of Isfahan.
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