Throughout this paper, a Banach space will always be over the real scalar field. We denote its norm by and its dual space by . The value of at is denoted by and the normalized duality mapping from into is defined by
Let denote the set of all fixed points for a mapping , that is, , and let denote the set of all positive integers. We write (resp. ) to indicate that the sequence weakly (resp. wea) converges to ; as usual will symbolize strong convergence.
In the proof of our main results, we need the following definitions and results. Let denote the unit sphere of a Banach space . is said to have (i) a Gâteaux differentiable norm (we also say that is smooth), if the limit
exists for each ; (ii) a uniformly Gâteaux differentiable norm, if for each in , the limit (2.2) is uniformly attained for ; (iii) a Fréchet differentiable norm, if for each , the limit (2.2) is attained uniformly for ; (iv) a uniformly Fréchet differentiable norm (we also say that is uniformly smooth), if the limit (2.2) is attained uniformly for . A Banach space is said to be (v) strictly convex if (vi) uniformly convex if for all , such that For more details on geometry of Banach spaces, see [21, 22].
If is a nonempty convex subset of a Banach space and is a nonempty subset of , then a mapping is called a retraction if is continuous with . A mapping is called sunny if whenever and . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . We note that if is closed and convex of a Hilbert space , then the metric projection coincides with the sunny nonexpansive retraction from onto . The following lemma is well known which is given in [22, 23].
Lemma 2.1 (see [22, Lemma 5.1.6]).
Let be nonempty convex subset of a smooth Banach space , , the normalized duality mapping of , and a retraction. Then is both sunny and nonexpansive if and only if there holds the inequality:
Hence, there is at most one sunny nonexpansive retraction from onto .
In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.
Let be a convex subset of a smooth Banach space . Let be a subset of and let be the unique sunny nonexpansive retraction from onto . Suppose is a weak contraction with a function on and is a nonexpansive mapping. Then
(i)the composite mapping is a weak contraction on ;
(ii)For each , a mapping is a weak contraction on . Moreover, defined by (2.4) is well definition:
(iii) if and only if is a unique solution of the following variational inequality:
For any , we have
So, is a weakly contractive mapping with a function . For each fixed and , we have
Namely, is a weakly contractive mapping with a function . Thus, Theorem 1.2 guarantees that has a unique fixed point in , that is, satisfying (2.4) is uniquely defined for each . (i) and (ii) are proved.
Subsequently, we show (iii). Indeed, by Theorem 1.2, there exists a unique element such that . Such a fulfils (2.5) by Lemma 2.1. Next we show that the variational inequality (2.5) has a unique solution . In fact, suppose is another solution of (2.5). That is,
Adding up gets
Hence by the property of . This completes the proof.
Let be a sequence of nonexpansive mappings with on a closed convex subset of a Banach space and let be a sequence in with (C1). is said to have Browder's property if for each , a sequence defined by
for , converges strongly. Let be a sequence in with (C1) and (C2). Then is said to have Halpern's property if for each , a sequence defined by
for , converges strongly.
We know that if is a uniformly smooth Banach space or a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, is bounded, is a constant sequence , then has both Browder's and Halpern's property (see [7, 10, 11, 23], resp.).
Lemma 2.3 (see [24, Proposition 4]).
Let have Browder's property. For each , put , where is a sequence in defined by (2.10). Then is a nonexpansive mapping on .
Lemma 2.4 (see [24, Proposition 5]).
Let have Halpern's property. For each , put , where is a sequence in defined by (2.11). Then the following hold: (i) does not depend on the initial point . (ii) is a nonexpansive mapping on .
Let be a smooth Banach space, and have Browder's property. Then is a sunny nonexpansive retract of , and moreover, define a sunny nonexpansive retraction from to .
For each , it is easy to see from (2.10) that
This implies for any and some ,
The smoothness of implies the norm weak continuity of [22, Theorems 4.3.1, 4.3.2], so
By Lemma 2.1, is a sunny nonexpansive retraction from to .
We will use the following facts concerning numerical recursive inequalities (see [25–27]).
Let and be two sequences of nonnegative real numbers, and a sequence of positive numbers satisfying the conditions and . Let the recursive inequality
be given where is a continuous and strict increasing function for all with . Then ( 1) converges to zero, as ; ( 2) there exists a subsequence such that