# A Shrinking Projection Algorithm with Errors for Costerro Bounded Linear Mappings

Authors: Joseph Frank Gordon
Journal: Journal of Mathematics
 (2020)

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## 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.


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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 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 ε0=0 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 sequencexnn≥1 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 xnn≥1 is guaranteed when the closed convex subset is compact. Finally, 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 [15]). Let X 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 the purposes of this paper, the 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 and FT is used to denote the set of fixed points of the mapping T (that is, FT  =  x∈C:Tx=x).
The mappings studied in this paper are defined in the following.Definition 2.
(costerro bounded linear mappings). LetX be a strictly convex smooth reflexive space and C a closed convex subset of X. A mapping T:C⟶X is said to be a costerro bounded linear 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 the iterative scheme, the following function is defined.Definition 3.
(generalised projection functional, see Alber [16]). Let X 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.
Note that the generalised projection functionalϕ⋅,⋅ 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 X be a uniformly convex smooth Banach space and let C≠∅ be a closed convex subset of X. Then, for every x∈X, there exists a unique y∈C such that(7)Λx,C=ϕy,x=infz∈Cϕz,x.
The unique pointy satisfying equation (7) is the called the generalised projection of x on C. That is, the projection operator ΠC:X⟶C is defined by setting(8)ΠCx=y,where y is the only point in C satisfying equation (7).Remark 1.
In Theorem1, 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.
Iterative Scheme 1. LetX be a uniformly convex smooth Banach space and let C≠∅ (not necessarily bounded) be a closed convex subset of X. Let Tkk=1m be finite set of costerro bounded linear mappings from C to X with F≔∩k=1mFTk≠∅. Let αn,kn≥1and εnn≥1 be nonnegative real sequences satisfying the following conditions:(i)
αn,kn≥1⊂0,1(ii)
∑k=1mαn,k=1(iii)
αk≔liminfn⟶∞αn,k>0(iv)
ε0≔limsupn⟶∞εn<∞
for all1≤k≤m and n≥1.
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,Cn+1=ℬn∩Cn,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 [18]. The proof is given here for the sake of completeness.Theorem 2.
(see, for instance, Ezearn [18]). Let X be a smooth Banach space and let x∈X\0 and any y∈X. Then,(10)1xℜy,Jx=limα⟶0x+αy−xα,for all α>0.Lemma 1.
(see Ezearn [18]). Let X be a smooth Banach space and x1,x2,…,xm,y∈X where m∈ℕ. Then,(11)∑k=1mxk2≤∑k=1mxkxk+αy,for all α∈0,q (where q∈ℝ>0) if and only if(12)ℜy,Jx1+Jx2+⋯+Jxm≥0.Proof.
Ifα=0, then the lemma is proved trivially, and as a result, it is assumed that α≠0 (without loss of generality, it is equally assumed that xk≠0). Now, if ℜ∑k=1my,Jxk≥0, then(13)∑k=1mxk2=ℜ∑k=1mxk,Jxk≤ℜ∑k=1mxk+αy,Jxk≤∑k=1mxk+αyxk.
On the contrary, if∑k=1mxk2≤∑k=1mxkxk+αy for every α∈0,q (where q∈ℝ>0), then(14)0≤1α∑k=1mxkxk+αy−xk=∑k=1mxkxk+αy−xkα.
Taking the limit asα⟶0, then by Theorem 2, equation (14) becomes(15)1xkℜ∑k=1my,Jxk≥0.
Sincexk≠0, then ℜ∑k=1my,Jxk≥0 and hence proved.Corollary 1.
The inequalityℜz,JTx−Jx≥0 is equivalent to(16)Tx2+x2≤TxTx+αz+xx−αz,for all α≥0.Proof.
By considering Lemma1 for the case when m=2, the inequality(17)ℜz,Jx1−Jx2=ℜz,Jx1+J−x2≥0,is equivalent to the following condition:(18)∑k=12xk2≤∑k=12xkxk+αy,x12+x22≤x1x1+αy+x2x2+αy.
Now, replacingx1 with Tx, y with z, and x2 with −x, 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). LetC be a closed convex subset of a strictly convex smooth reflexive space X. Then, the following is a nontrivial example of a costerro bounded linear mapping:(19)Tx2+Ty2+x2−y2≤TxTx+αTy+xx−αy,for all x,y∈C and all α≥0.Proof.
Forα=0, equation (19) reduces to the following:(20)Tx2+Ty2+x2−y2≤Tx2+x2,Ty2≤y2,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 (19) reduces to the following evaluation:(21)Tx2+y2+x2−y2≤TxTx+αy+xx−αy,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 [18]). Let Cnn≥1 be a sequence of nonempty closed convex subsets of a uniformly convex smooth Banach space X such that Cn+1⊂Cn. Suppose that further that C∞=∩n≥1Cn is nonempty. Then, the sequence of generalized projections ΠCnxn≥1 converges strongly to ΠC∞x for any x∈X.Proposition 1.
(seeAlber [19], Alber and Reich [20], and Kamimura and Takahashi [21]). Let X be a real uniformly convex smooth Banach space and C≠∅ be a closed convex subset of X. Then, the following inequality holds:(22)ϕy,ΠCx+ϕΠCx,x≤ϕy,x,for all y∈C and x∈X.Proposition 2.
(continuity in duality pairing). LetX be a Banach space and let X∗ be the dual space of X. Denote ⋅,⋅ as the duality product. Now, for xnn≥1⊂X and fnn≥1⊂X∗, suppose either of the following conditions hold:(i)
xn⇀x and fn⟶f(ii)
xn⇀x and fn⇀∗f
Then,limn⟶∞xn,fn=x,f.Lemma 3.
(weak star-continuity in smooth spaces). LetX be a real smooth Banach space. Then, J:X⟶X∗ is norm-to-weak star continuous, where J is the normalized duality mapping.Lemma 4.
(see Kamimura and Takahashi [21]). Let X be a uniformly convex and smooth Banach space and let xn and yn be two sequences in X such that either xn or yn is bounded. If limn⟶∞ϕxn,yn=0, then limn⟶∞xn−yn=0.


## 3. Main Results

The proof of the main result of this paper is given in this section, which is accomplished in Theorem3. The following corollary and lemmas shall aid in arriving at the conclusion of the main result.Corollary 3.
If the sequencexnn≥1 has a strong limit point, say x, then x∈F=∩k=1mFTk.Proof.
Without loss of generality, it is assumed 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 Iterative Scheme 1, xn+1∈Cn+1⊂Cn, where xn+1n≥1=x2,x3,x4,…. Hence, it is observed that(23)∑k=1mαn,kϕxn+1,Tkxn≤ϕxn+1,xn.
Hence, taking limit asn⟶∞ of the above inequality, the following is obtained:(24)limn⟶∞∑k=1mαn,kϕxn+1,Tkxn≤limn⟶∞ϕxn+1,xn.
By Proposition2 and Lemma 3, limn⟶∞ϕxn+1,xn⟶0 and as a result, the following is obtained:(25)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, the following is obtained:(26)limn⟶∞ϕxn+1,Tkxn=0,for all k∈1,…,m.
So by Lemma4,(27)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 sets ℬn and Cn in Iterative Scheme 1 are 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, if zjj≥1⊂ℬn converges to z∈C, then via the continuity of the generalised functional ϕ⋅,⋅, the following is obtained:(28)∑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 the inequality is obtained:(29)∑k=1mαn,kϕz,Tkxn≤ϕz,xn,which can be expanded and observed to be equivalent to(30)∑k=1mαn,kTkxn2−xn2≤2ℜ∑k=1mαn,kz,JTkxn−Jxn.
So by making the substitutionz=u and multiplied by t and adding it to z≔v multiplied by 1−t, the following is obtained:(31)∑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 which it is concluded that(32)∑k=1mαn,kϕtu+1−tv,Tkxn≤ϕtu+1−tv,xn.
Hence,ℬn is convex.
Now, define(33)C∞≔∩n≥1Cn.Lemma 6.
The setC∞ is a closed convex set containing F. Hence, the sequence ΠCnxn≥1 of generalised projections converges strongly to ΠC∞x for any arbitrary x in a uniformly convex smooth Banach space X.Proof.
By induction, it is observed that the setsCn are all closed convex subsets by the help of Lemma 5 and the definition of Cn+1 in Iterative Scheme 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. C∞⊃F is claimed by induction.
By the assumption in Iterative Scheme 1, it is observed thatF⊂C=C1 and x1 is given. Now, supposing F⊂Cm for all m≤n and choosing arbitrary z∈F. Then, the following evaluation is obtained:(34)∑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,kxn2−2ℜ∑k=1mαn,kz,Jxn=z2+xn2−2ℜz,Jxn=ϕz,xn,where the fact that the mappings are costerro bounded linear mappings in the third step is used. Hence, it is shown that z∈Cn+1. From Lemma 2, it is concluded that ΠCnxn≥1 converges strongly to ΠC∞x.Lemma 7.
The sequencexnn≥1 satisfies the inequality(35)xn,ΠCnu≤εn2.Proof.
SinceΛu,Cn≔infx∈Cnϕx,u=ϕΠCnu,u, then for every εn>0, xn∈Cn is found such that(36)ϕxn,u≤ΛCn,u+εn2=ϕΠCnu,u+εn2,which implies that(37)ϕxn,u−ϕΠCnu,u≤εn2.
However, Proposition1 implies that(38)ϕxn,ΠCu≤ϕxn,u−ϕΠCu,u,and so in addition to equation (37), the following inequality is obtained:(39)ϕxn,ΠCnu≤εn2,which completes the proof.The main result of this paper is given by the following theorem.Theorem 3.
(main result). LetX be a uniformly convex smooth Banach space and suppose any of the following cases hold:(i)
The spaceX is finite dimensional.(ii)
The convex setC is compact.(iii)
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 observe thatxnn≥1 is a bounded sequence. As a matter of fact, by Lemma 7,(40)ϕxn,ΠCnu=xn2−2ℜxn,JΠCnu+ΠCnu2≤εn2,which simplifies to(41)xn2+ΠCnu2−2xnΠCnu=xn−ΠCnu2≤εn2.
Hence,xn≤ΠCnu+εn. So by Lemma 6 and the conditions in Iterative Scheme 1, it is obtainedxnn≥1 is a bounded sequence.
Consider the following cases stated in Theorem3.Case (i): 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 (ii): 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 (iii): given thatε0=0, then by Lemma 7,(42)limn⟶∞xn−ΠCnu=0.
By continuity of the norm function and Lemma6, it is obtained 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≥1 has a weak limit point since a uniformly convex smooth Banach is a reflexive space.



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