Abstract

The purpose of this paper is to introduce and analyze the shrinking projection algorithm with errors for a finite set of costerro bounded linear mappings in the setting of uniformly convex smooth Banach spaces. Here, under finite dimensional or compactness restriction or the error term being zero, the strong limit point of the sequence stated in the iterative scheme for these mappings in uniformly convex smooth Banach spaces was studied. This paper extends Ezearn and Prempeh’s result for nonexpansive mappings in real Hilbert spaces.

1. Introduction

Fixed-point theory is a fascinating subject, with a lot of applications in various fields of mathematics and engineering. In a number of situations, one may need to find a common fixed point of a family of mappings. In practice, a modification may be needed to turn the problem into a fixed-point problem (see, for instance, Picard [1] and Lindelöf [2]). For more information on the fixed-point problem and its applications to certain types of linear and nonlinear problems, interested readers should be referred to Tang and Chang [3] (equilibrium problems), Solodov and Svaiter [4] (proximal point algorithm), Takahashi [5, 6] (convex optimization and minimization problems), and Blum and Oettli [7] (variational inequalities).

In practice, finding an exact closed form of a solution to a fixed-point problem is almost a difficult task. For this reason, it has been of particular importance in the development of feasible iterative schemes or methods for approximating fixed points of certain maps, most notably, nonexpansive type of mappings. For instance, Halpern [8], Mann [9], and Ishikawa [10] studied and developed an iterative scheme to approximate the fixed points of nonexpansive mappings in Hilbert spaces under certain conditions. In their scheme, strong convergence is always guaranteed for all closed convex subsets of a Hilbert space. Haugazeau [11] initially proposed the projection method which was later developed by Solodov and Svaiter [4]. A type of projection method which is of relevance and central to this paper is called the Shrinking Projection Method with Errors, which was developed by Takahashi et al. [12] and used by Yasunori [13]. Strong convergence result is always guaranteed for all closed convex subsets of a Hilbert space under certain conditions.

In [14], Ezearn and Prempeh improved the boundedness requirement of Yasunori’s result [13] regarding a shrinking projection algorithm for common fixed points of nonexpansive mappings in a real Hilbert space. In their results, they showed that the boundedness requirement in Yasunori’s results could be removed. That is to say that the convergence of the iterative sequence in the scheme presented in Yasunori’s paper, that is, the error term is independent of the boundedness of the closed convex subset in a real Hilbert space. With the boundedness removed, Ezearn and Prempeh further provided a better estimate for the convergence result of the iterative sequence in their algorithm especially in finite dimensional and further showed that when the closed convex set is compact, their estimates do not involve the diameter of the subset.

In this paper, it is shown that the strong limit point of the iterative sequence presented in Iterative Scheme 1 always exists in a finite-dimensional space. And, it also shown that when the space is not finite dimensional, the strong limit point of is guaranteed when the closed convex subset is compact. Finally, the strong limit point of also exists when the error term () is zero regardless of the compactness of the closed convex subset and the dimension of the space.

Definition 1. (normalised duality mapping, see Lunner [15]). Let be a Banach space with the norm and let be the dual space of . Denote as the duality product. The normalised duality mapping from to is defined byfor all . The Hahn Banach theorem guarantees that for every . For the purposes of this paper, the interest mostly lies on the case when is single valued for all , which is equivalent to the statement that is a smooth Banach space.
Throughout this paper, denotes the real part of a complex number and is used to denote the set of fixed points of the mapping (that is,  =  ).
The mappings studied in this paper are defined in the following.

Definition 2. (costerro bounded linear mappings). Let be a strictly convex smooth reflexive space and a closed convex subset of . A mapping is said to be a costerro bounded linear mapping ifsuch that whenever , thenAn immediate example of such mappings is the scaling operator given bywhere the scaling factor lies in the closed unit disk.
In order to state the iterative scheme, the following function is defined.

Definition 3. (generalised projection functional, see Alber [16]). Let be a smooth Banach space and let be the dual space of . The generalised projection functional is defined byfor all , where is the normalised duality mapping from to . It is obvious from the definition that the generalised projection functional satisfies the following inequality:for all .
Note that the generalised projection functional is continuous.
The next function which is stated in the iterative scheme is established via the following theorem .

Theorem 1. (generalised projection, see Li [17]). Let be a uniformly convex smooth Banach space and let be a closed convex subset of . Then, for every , there exists a unique such thatThe unique point satisfying equation (7) is the called the generalised projection of on . That is, the projection operator is defined by settingwhere is the only point in satisfying equation (7).

Remark 1. In Theorem 1, note that if is a Hilbert space, then . Hence, the (generalised) projection defined in equation (8) coincides with the metric projection onto in the Hilbert space setting. The converse is not necessarily true in a general Banach space.
The iterative scheme is stated as follows.
Iterative Scheme 1. Let be a uniformly convex smooth Banach space and let (not necessarily bounded) be a closed convex subset of . Let be finite set of costerro bounded linear mappings from to with . Let and be nonnegative real sequences satisfying the following conditions:(i)(ii)(iii)(iv)for all and .
Then, for any arbitrary with the assumptions and , the sequence is defined iteratively by the following scheme:for all .

2. Preliminaries

The inequality in Definition 2 can be written equivalently in terms of norms. This is achieved via the elementary lemma by Ezearn in [18]. The proof is given here for the sake of completeness.

Theorem 2. (see, for instance, Ezearn [18]). Let be a smooth Banach space and let and any . Then,for all .

Lemma 1. (see Ezearn [18]). Let be a smooth Banach space and where . Then,for all (where ) if and only if

Proof. If , then the lemma is proved trivially, and as a result, it is assumed that (without loss of generality, it is equally assumed that ). Now, if , thenOn the contrary, if for every (where ), thenTaking the limit as , then by Theorem 2, equation (14) becomesSince , then and hence proved.

Corollary 1. The inequality is equivalent tofor all .

Proof. By considering Lemma 1 for the case when , the inequalityis equivalent to the following condition:Now, replacing with , with , and with , the corollary is proved.
Below, a nontrivial example of costerro bounded linear mappings is given which is referred to as Ezearn nonexpansive mapping. Ezearn, in his thesis [18], had defined certain closely related mappings (named type III variational nonexpansive mappings).

Corollary 2. (Ezearn nonexpansive mapping). Let be a closed convex subset of a strictly convex smooth reflexive space . Then, the following is a nontrivial example of a costerro bounded linear mapping:for all and all .

Proof. For , equation (19) reduces to the following:which satisfies the first part of Definition 2. To show the second part of Definition 2, if , where refers to the fixed point set of , then equation (19) reduces to the following evaluation:which by Corollary 1 is equivalent to . Hence proved.

Lemma 2. (see, for instance, Ezearn [18]). Let be a sequence of nonempty closed convex subsets of a uniformly convex smooth Banach space such that . Suppose that further that is nonempty. Then, the sequence of generalized projections converges strongly to for any .

Proposition 1. (seeAlber [19], Alber and Reich [20], and Kamimura and Takahashi [21]). Let be a real uniformly convex smooth Banach space and be a closed convex subset of . Then, the following inequality holds:for all and .

Proposition 2. (continuity in duality pairing). Let be a Banach space and let be the dual space of . Denote as the duality product. Now, for and , suppose either of the following conditions hold:(i) and (ii) and Then, .

Lemma 3. (weak star-continuity in smooth spaces). Let be a real smooth Banach space. Then, is norm-to-weak star continuous, where is the normalized duality mapping.

Lemma 4. (see Kamimura and Takahashi [21]). Let be a uniformly convex and smooth Banach space and let and be two sequences in such that either or is bounded. If , then .

3. Main Results

The proof of the main result of this paper is given in this section, which is accomplished in Theorem 3. The following corollary and lemmas shall aid in arriving at the conclusion of the main result.

Corollary 3. If the sequence has a strong limit point, say , then .

Proof. Without loss of generality, it is assumed that the sequence is the subsequence converging to . Now, for , since the sets form a decreasing sequence of sets, that is, , then from Iterative Scheme 1, , where . Hence, it is observed thatHence, taking limit as of the above inequality, the following is obtained:By Proposition 2 and Lemma 3, and as a result, the following is obtained:Since the generalised functional is nonnegative and the limit infimum of is nonzero for all , the following is obtained:for all .
So by Lemma 4,for all and that proves the corollary due to the continuity of the norm functional and the mappings .

Lemma 5. For all , the sets and in Iterative Scheme 1 are closed convex sets.

Proof. Because is a closed convex set by assumption, it suffices to show that is a closed convex set for all . To prove the closure aspect of the lemma, if converges to , then via the continuity of the generalised functional , the following is obtained:and as a result, .
Finally, to prove convexity, let and . First, note that whenever , then the inequality is obtained:which can be expanded and observed to be equivalent toSo by making the substitution and multiplied by and adding it to multiplied by , the following is obtained:from which it is concluded thatHence, is convex.
Now, define

Lemma 6. The set is a closed convex set containing . Hence, the sequence of generalised projections converges strongly to for any arbitrary in a uniformly convex smooth Banach space .

Proof. By induction, it is observed that the sets are all closed convex subsets by the help of Lemma 5 and the definition of in Iterative Scheme 1. Moreover, by inclusion, these sets form a decreasing sequence of sets. That is, for all . So, is either empty or nonempty. is claimed by induction.
By the assumption in Iterative Scheme 1, it is observed that and is given. Now, supposing for all and choosing arbitrary . Then, the following evaluation is obtained:where the fact that the mappings are costerro bounded linear mappings in the third step is used. Hence, it is shown that . From Lemma 2, it is concluded that converges strongly to .

Lemma 7. The sequence satisfies the inequality

Proof. Since , then for every , is found such thatwhich implies thatHowever, Proposition 1 implies thatand so in addition to equation (37), the following inequality is obtained:which completes the proof.

The main result of this paper is given by the following theorem.

Theorem 3. (main result). Let be a uniformly convex smooth Banach space and suppose any of the following cases hold:(i)The space is finite dimensional.(ii)The convex set is compact.(iii).Then, and , where denotes the (strong) limit set of the iterative sequence .

Proof. First observe that is a bounded sequence. As a matter of fact, by Lemma 7,which simplifies toHence, . So by Lemma 6 and the conditions in Iterative Scheme 1, it is obtained is a bounded sequence.
Consider the following cases stated in Theorem 3. Case (i): given that is bounded and is finite dimensional, then by the Bolzano–Weierstrass theorem, has a limit point, say , and by Corollary 3, and as a result, a subsequence of converges strongly to . Case (ii): given that is compact, since in metric spaces, compactness implies sequential compactness, then being a bounded sequence has a limit point, say , and by Corollary 3, and as a result, a subsequence of converges strongly to . Case (iii): given that , then by Lemma 7,By continuity of the norm function and Lemma 6, it is obtained that and as result since it is the only limit point of the sequence .