# Corrigendum to “A Shrinking Projection Algorithm with Errors for Costerro Bounded Mappings”

Authors: Joseph Frank Gordon
Journal: Journal of Mathematics
 (2022)

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## Body

## 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 fixed point problem and its applications to certain types of linear and nonlinear problems, interested readers should refer 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 theShrinking Projection Method with Errors, which was developed by Takahashi et al. (12) and used by Yasunori (13). The 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, the convergence of the iterative sequence in the scheme presented in Yasunori’s paper, that is, when the error termε0=0, is independent of the boundedness of the closed convex subset in a real Hilbert space. With the boundedness being removed, Ezearn and Prempeh further provided a better estimate for the convergence result of the iterative sequence in their algorithm especially in finite dimension and further showed that when the closed convex set is compact, their estimates does not involve the diameter of the subset.In this paper, we show that the strong limit point of the iterative sequencexnn≥1 presented in Section 1.1 always exists in a finite-dimensional space. We also show that, when the space is not finite dimensional, the strong limit point of xnn≥1 is guaranteed when the closed convex subset is compact. Finally, we show that the strong limit point of xnn≥1 also exists when the error term (ε0) is zero regardless of the compactness of the closed convex subset and the dimension of the space.Definition 1.
(normalised duality mapping, see Lunner (18)). LetX be a Banach space with the norm ⋅, and let X∗ be the dual space of X. Denote ⋅,⋅ as the duality product. The normalised duality mapping J from X to X∗ is defined by(1)Jx≔f∈X∗:f∗2=x2=x,f=fx,for all x∈X. The Hahn–Banach theorem guarantees that Jx≠∅ for every x∈X. For our purposes in this paper, our interest mostly lies on the case when Jx is single-valued for all x∈X, which is equivalent to the statement that X is a smooth Banach space.
Throughout this paper,ℜ denotes the real part of a complex number. We also use FT to denote the set of fixed points of the mapping T (that is, FT=x∈C:Tx=x).
The mappings we study in this paper are defined as follows.Definition 2.
(Costerro bounded mappings). LetX be a strictly convex smooth reflexive space and C be a closed convex subset of X. A mapping T:C⟶X is said to be a Costerro bounded mapping if(2)Tx≤x,such that whenever z∈FT, then(3)ℜz,JTx−Jx≥0,∀x,z∈C.
An immediate example of such mappings is the scaling operator given by(4)Tx=ax,where the scaling factor a lies in the closed unit disk.
In order to state our iterative scheme, let us define the following function.Definition 3.
(generalised projection functional, see (16)). LetX be a smooth Banach space, and let X∗ be the dual space of X. The generalised projection functional ϕ⋅,⋅:X×X⟶ℝ is defined by(5)ϕy,x=y2−2ℜy,Jx+x2,for all x,y∈X, where J is the normalised duality mapping from X to X∗. It is obvious from the definition that the generalised projection functional ϕ⋅,⋅ satisfies the following inequality:(6)y−x2≤ϕy,x≤y+x2,for all x,y∈X.
We should note here that the generalised projection functionalϕ⋅,⋅ is continuous.
The next function which is stated in our iterative scheme is established via the theorem as follows.Theorem 1.
(generalised projection, see [18]). LetXbe a uniformly convex smooth Banach space, and letC≠∅be a closed convex subset ofX. Then, for everyx∈X, there exists a uniquey∈Csuch that(7)Λx,C=ϕy,x=infz∈Cϕz,x.The unique pointy satisfying equation (7) is called the generalised projection of x on C. That is, we define the projection operator ΠC:X⟶C by setting(8)ΠCx=y,where y is the only point in C satisfying equation (7).Remark 1.
InTheorem 1, note that if X is a Hilbert space, then ϕy,x=y−x2. Hence, the (generalised) projection ΠC defined in equation (8) coincides with the metric projection onto C in the Hilbert space setting. The converse is not necessarily true in a general Banach space.
The iterative scheme is stated as follows.


### 1.1. Iterative Scheme 1

LetX be a uniformly convex smooth Banach space, and let C≠∅ (not necessarily bounded) be a closed convex subset ofX. LetTkk=1mbe a finite set of Costerro bounded mappings fromCtoXwithF≔∩k=1mFTk≠∅. Letαn,kn≥1andεnn≥1be nonnegative real sequences satisfying the following conditions, for all1≤k≤mandn≥1:(i)
αn,kn≥1⊂0,1(ii)
∑k=1mαn,k=1(iii)
αk≔liminfn⟶∞αn,k>0(iv)
ε0≔limsupn⟶∞εn<∞Then, for any arbitraryu∈X with the assumptions x1∈C1≔C and ϕx1,u<ε12, the sequence xnn≥1 is defined iteratively by the following scheme:(9)ℬn=z∈C:∑k=1mαn,kϕz,Tkxn≤ϕz,xn,(10)Cn+1=ℬn∩Cn,(11)xn+1∈Cn+1,ϕxn+1,u≤Λu,Cn+1+εn+12,for all n≥1.


## 1.1. Iterative Scheme 1

LetX be a uniformly convex smooth Banach space, and let C≠∅ (not necessarily bounded) be a closed convex subset ofX. LetTkk=1mbe a finite set of Costerro bounded mappings fromCtoXwithF≔∩k=1mFTk≠∅. Letαn,kn≥1andεnn≥1be nonnegative real sequences satisfying the following conditions, for all1≤k≤mandn≥1:(i)
αn,kn≥1⊂0,1(ii)
∑k=1mαn,k=1(iii)
αk≔liminfn⟶∞αn,k>0(iv)
ε0≔limsupn⟶∞εn<∞Then, for any arbitraryu∈X with the assumptions x1∈C1≔C and ϕx1,u<ε12, the sequence xnn≥1 is defined iteratively by the following scheme:(9)ℬn=z∈C:∑k=1mαn,kϕz,Tkxn≤ϕz,xn,(10)Cn+1=ℬn∩Cn,(11)xn+1∈Cn+1,ϕxn+1,u≤Λu,Cn+1+εn+12,for all n≥1.


## 2. Preliminaries

The inequalityℜz,JTx−Jx≥0 in Definition 2 can be written equivalently in terms of norms. This is achieved via the elementary lemma by Ezearn in (19). We give the proof here for the sake of completeness.Theorem 2 (see, for instance, (19)).
LetXbe a smooth Banach space, and letx∈X\0and anyy∈X. Then,(12)1xℜy,Jx=limα⟶0x+αy−xα,for all α>0.Lemma 1 (19).
LetXbe a smooth Banach space andx1,x2,…,xm,y∈Xwherem∈ℕ. Then,(13)∑k=1mxk2≤∑k=1mxkxk+αy,for all α∈0,q (where q∈ℝ>0) if and only if(14)ℜy,Jx1+Jx2+…+Jxm≥0.Proof.
We observe that ifα=0, then the lemma is proved trivially and as a result, we assume that α≠0 (without loss of generality, we can equally assume that xk≠0). Now, if ℜ∑k=1my,Jxk≥0, then(15)∑k=1mxk2=ℜ∑k=1mxk,Jxk≤ℜxk+αy,Jxk≤∑k=1mxk+αyxk.
On the other hand, if∑k=1mxk2≤∑k=1mxkxk+αy for every α∈0,q (where q∈ℝ>0), then(16)0≤1α∑k=1mxkxk+αy−xk=∑k=1mxkxk+αy−xkα.
Taking the limit asα⟶0, then by Theorem 2, equation (16) becomes(17)1∥xk∥ℜ∑k=1my,Jxk≥0.
Sincexk≠0, then ℜ∑k=1my,Jxk≥0 and hence proved.Corollary 1.
The inequalityℜz,JTx−Jx≥0is equivalent to(18)Tx2+x2≤TxTx+αz+xx−αz,for all α≥0.Proof.
By considering Lemma1 for the case when m=2, the inequality(19)ℜz,Jx1−Jx2=ℜz,Jx1+J−x2≥0is equivalent to the following condition:(20)∑k=12xk2≤∑k=12xkxk+αy,(21)x12+x22≤x1x1+αy+x2x2+αy.
Now, replacingx1 with Tx, y with z, and x2 with −x, the corollary is proved.
In the following, we give a nontrivial example of Costerro bounded mappings which we refer to as Ezearn nonexpansive mapping. Ezearn, in his thesis (19), had defined certain closely related mappings (named Type III variational nonexpansive mappings).Corollary 2 (Ezearn nonexpansive mapping).
LetCbe a closed convex subset of a strictly convex smooth reflexive spaceX. Then, the following is a nontrivial example of a Costerro bounded mapping:(22)Tx2+Ty2+x2−y2≤TxTx+αTy+xx−αy,for all x,y∈C and all α≥0.Proof.
Forα=0, equation (18) reduces to the following:(23)Tx2+Ty2+x2−y2≤Tx2+x2,(24)Ty2≤y2,(25)Ty≤y,which satisfies the first part of Definition 2. To show the second part of Definition 2, if y∈FT, where FT refers to the fixed point set of T, then equation (18) reduces to the following evaluation:(26)Tx2+y2+x2−y2≤TxTx+αy+xx−αy,(27)Tx2+x2≤TxTx+αy+xx−αy,which by Corollary 1 is equivalent to ℜy,JTx−Jx≥0. Hence proved.Lemma 2 (see, for instance, Ezearn, (19)).
LetCnn≥1be a sequence of nonempty closed convex subsets of a uniformly convex smooth Banach spaceXsuch thatCn+1⊂Cn. Suppose further thatC∞=∩n≥1Cnis nonempty. Then, the sequence of generalized projectionsΠCnxn≥1converges strongly toΠC∞xfor anyx∈X.Proposition 1 (Alber (20), Alber and Reich (21), Kamimura and Takahashi, (22)).
LetXbe a real uniformly convex smooth Banach space andC≠∅be a closed convex subset ofX. Then, the following inequality holds:(28)ϕy,ΠCx+ϕΠCx,x≤ϕy,x,for all y∈C and x∈X.Proposition 2 (continuity in duality pairing).
LetXbe a Banach space, and letX∗be the dual space ofX. Denote⋅,⋅as the duality product. Now, forxnn≥1⊂Xandfnn≥1⊂X∗, suppose either of the following conditions hold:
xnx and fn⟶f, xn⟶x and fn∗f
Then,limn⟶∞xn,fn=x,f.Lemma 3 (weak star-continuity in smooth spaces).
LetXbe a real smooth Banach space. Then,J:X⟶X∗is norm-to-weak star continuous, whereJis the normalized duality mapping.Lemma 4 (Kamimura and Takahashi, strong).
LetXbe a uniformly convex and smooth Banach space, and letxnandynbe two sequences inXsuch that eitherxnorynis bounded. Iflimn⟶∞ϕxn,yn=0, thenlimn⟶∞xn−yn=0.


## 3. Main Results

We give the proof of the main result of this paper, which is accomplished in Theorem3. The following corollary and lemmas shall aid us in arriving at the conclusion of the main result.Corollary 3.
LetTkk=1mbe a continuous Costerro bounded mapping. If the sequencexnn≥1has a strong limit point, sayx, thenx∈F=∩k=1mFixTk.Proof.
Without loss of generality, we assume that the sequencexnn≥1=x1,x2,x3,… is the subsequence converging to x. Now, for n≥1, since the sets Cn form a decreasing sequence of sets, that is, Cn+1⊂Cn, then from Section 1.1, we have that xn+1∈Cn+1⊂Cn, where xn+1n≥1=x2,x3,x4,…. Hence, we observe that(29)∑k=1mαn,kϕxn+1,Tkxn≤ϕxn+1,xn.
Hence, taking limit asn⟶∞ of the above inequality, we have(30)limn⟶∞∑k=1mαn,kϕxn+1,Tkxn≤limn⟶∞ϕxn+1,xn.
By Proposition2 and Lemma 3, limn⟶∞ϕxn+1,xn⟶0 and as a result, we obtain(31)limn⟶∞∑k=1mαn,kϕxn+1,Tkxn≤0.
Since the generalised functionalϕ⋅,⋅ is nonnegative and the limit infimum of αn,k is nonzero for all k, we have(32)limn⟶∞ϕxn+1,Tkxn=0,for all. k∈1,…,m.
So by Lemma4, we have that(33)limn⟶∞xn+1−Tkxn=0,for all k∈1,…,m and that proves the corollary due to the continuity of the norm functional and the mappings Tk.Lemma 5.
For alln≥1, the setsℬnandCnin Section1.1are closed convex sets.Proof.
BecauseC1≔C is a closed convex set by assumption, it suffices to show that ℬn is a closed convex set for all n. To prove the closure aspect of the lemma, we observe that if zjj≥1⊂ℬn converges to z∈C, then via the continuity of the generalised functional ϕ⋅,⋅, we have the following:(34)∑k=1mαn,kϕz,Tkxn=limj⟶∞∑k=1mαn,kϕzj,Tkxn≤limj⟶∞ϕzj,xn=ϕz,xn,and as a result, z∈ℬn.
Finally, to prove convexity, letu,v∈ℬn and t∈0,1. First, note that whenever z∈ℬn, then we have the following inequality:(35)∑k=1mαn,kϕz,Tkxn≤ϕz,xn,which can be expanded and observed to be equivalent to(36)∑k=1mαn,kTkxn2−xn2≤2ℜ∑k=1mαn,kz,JTkxn−Jxn.
So by making the substitutionz=u and multiplying it by t and adding it to z≔v multiplied by 1−t, we obtain(37)∑k=1mαn,kTkxn2−xn2=t∑k=1mαn,kTkxn2−xn2+1−t∑k=1mαn,kTkxn2−xn2≤2tℜ∑k=1mαn,ku,JTkxn−Jxn+21−tℜ∑k=1mαn,kv,JTkxn−Jxn=2ℜ∑k=1mαn,ktu+1−tv,JTkxn−Jxn.From this, we conclude that(38)∑k=1mαn,kϕtu+1−tv,Tkxn≤ϕtu+1−tv,xn.
Hence,ℬn is convex.
Now, let us define(39)C∞≔∩n≥1Cn.Lemma 6.
The setC∞is a closed convex set containingF. Hence, the sequenceΠCnxn≥1of generalised projections converges strongly toΠC∞xfor any arbitraryxin a uniformly convex smooth Banach spaceX.Proof.
By induction, we observe that the setsCn are all closed convex subsets by the help of Lemma 5 and the definition of Cn+1 in Section 1.1 Moreover, by inclusion, these sets Cn form a decreasing sequence of sets. That is, Cn+1⊂Cn for all n≥1. So C∞ is either empty or nonempty. We claim that C∞⊃F by induction. By the assumption in Section 1.1, it is observed that F⊂C=C1 and x1 is given. Now, let us suppose F⊂Cm for all m≤n and choose arbitrary z∈F. Then, we have the following evaluation:(40)∑k=1mαn,kϕz,Tkxn=∑k=1mαn,kz2+Tkxn2−2ℜz,JTkxn=∑k=1mαn,kz2+∑k=1mαn,kTkxn2−2ℜ∑k=1mαn,kz,JTkxn=∑k=1mαn,kz2+∑k=1mαn,kTkxn2−2ℜ∑k=1mαn,kz,JTkxn≤∑k=1mαn,kz2+∑k=1mαn,kxn2−2ℜ∑k=1mαn,kz,Jxn=z2+xn2−2ℜz,Jxn=ϕz,xn,where we have used the fact that the mappings are Costerro bounded mappings in the third step. Hence, we have shown that z∈Cn+1. From Lemma 2, we conclude that ΠCnxn≥1 converges strongly to ΠC∞x.Lemma 7.
The sequencexnn≥1satisfies the following inequality:(41)ϕxn,ΠCnu≤εn2.Proof.
SinceΛu,Cn≔infx∈Cnϕx,u=ϕΠCnu,u, then for every εn>0, we can find our xn∈Cn such that(42)ϕxn,u≤ΛCn,u+εn2=ϕΠCnu,u+εn2,which implies that(43)ϕxn,u−ϕΠCnu,u≤εn2.
However, Proposition1 implies that(44)ϕxn,ΠCu≤ϕxn,u−ϕΠCu,u,and so in addition to equation (43), we have the following inequality:(45)ϕxn,ΠCnu≤εn2.
This completes the proof.
The main result of this paper is given by the following theorem.Theorem 3 (main result).
LetXbe a uniformly convex smooth Banach space and suppose any of the following cases hold:(1)
The spaceX is finite dimensional(2)
The convex setC is compact(3)
ε0=0
Then,ωxnn≥1≠∅ and ωxnn≥1⊆∩k=1mFTk, where ω denotes the (strong) limit set of the iterative sequence xnn≥1.Proof.
First we observe thatxnn≥1 is a bounded sequence. As a matter of fact, by Lemma 7, we have(46)ϕxn,ΠCnu=xn2−2ℜxn,JΠCnu+ΠCnu2≤εn2,which simplifies to(47)xn2+ΠCnu2−2xnΠCnu=xn−ΠCnu2≤εn2.
Hence, we havexn≤ΠCnu+εn. So by Lemma 6 and the conditions in Section 1.1, we have that xnn≥1 is a bounded sequence.
We now consider the following cases stated in Theorem3.Case 1.
Given thatxnn≥1 is bounded and X is finite dimensional, then by the Bolzano–Weierstrass Theorem, xnn≥1 has a limit point, say z, and by Corollary 3, z∈F, and as a result, a subsequence of xnn≥1 converges strongly to z∈F.Case 2.
Given thatC is compact, since in metric spaces, compactness implies sequential compactness, then xnn≥1 being a bounded sequence has a limit point, say z, and by Corollary 3, z∈F, and as a result, a subsequence of xnn≥1 converges strongly to z∈F.Case 3.
Given thatε0=0, then by Lemma 7, we have(48)limn⟶∞xn−ΠCnu=0.
By continuity of the norm function and Lemma6, we have that xnn≥1⟶ΠC∞u and as result, ΠC∞u∈F since it is the only limit point of the sequence xnn≥1.Remark 2.
We also note that for infinite dimensions, we can also say that the sequencexnn≥1has a weak limit point since a uniformly convex smooth Banach is a reflexive space.



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