# New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings

Authors: F. O. Isiogugu; C. Izuchukwu; C. C. Okeke
Journal: Journal of Mathematics
 (2020)

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## Abstract

We introduce a new algorithm (horizontal algorithm) in a real Hilbert space, for approximating a common fixed point of a finite family of mappings, without imposing on the finite family of the control sequencesςnin=1∞i=1N, the condition that ∑i=1Nςni=1, for each n≥1. Furthermore, under appropriate conditions, the horizontal algorithm converges both weakly and strongly to a common fixed point of a finite family of type-one demicontractive mappings. It is also applied to obtain some new algorithms for approximating a common solution of an equilibrium problem and the fixed point problem for a finite family of mappings. Our work is a contribution to ongoing research on iteration schemes for approximating a common solution of fixed point problems of a finite family of mappings and equilibrium problems.


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## Body

## 1. Introduction

LetY be a nonempty set and S:Y⟶Y be a mapping. A point y∈Y is called a fixed point of S if y=Sy. If S:Y⟶2Y is a multivalued mapping, then y is a fixed point of S if y∈Sy. y is called a strict fixed point of S if Sy=y. The set FS=y∈DS:y∈Sy (respectively, FS=y∈DS:y=Sy) is called the set of fixed points of the multivalued (respectively, single-valued) mapping S, while the set FsS=y∈DS:Sy=y is called the set of strict fixed points of S.LetY be a normed space. A subset K of Y is called proximinal if for each y∈Y, there exists k∈K such that(1)y−k=infy−w:w∈K=dy,K.It is known that every convex closed subset of a uniformly convex Banach space is proximinal. We shall denote the family of all nonempty closed and bounded subsets ofY by CBY, the family of all nonempty subsets of Y by 2Y, and the family of all proximinal subsets of Y by PY, for a nonempty set Y.LetD denote the Hausdorff metric induced by the metric d on Y, that is, for every A,B∈CBY,(2)DA,B=maxsupa∈Ada,B,supb∈Bdb,A.LetY be a normed space and S:DS⊆Y⟶2Y be a multivalued mapping on Y. S is called L−Lipschitzian if there exists L≥0 such that, for all x,y∈DS,(3)DSx,Sy≤Lx−y.In (3), if L∈0,1, then S is a contraction, while S is nonexpansive if L=1. S is called quasi-nonexpansive if FS≠∅ and for all p∈FS,(4)DSx,Sp≤x−p.Clearly, every nonexpansive mapping with the nonempty fixed point set is quasi-nonexpansive. The multivalued mappingS is k-strictly pseudo-contractive-type of Isiogugu [1] using the terminology of Browder and Petryshen [2] for single-valued pseudo-contractive mapping and Markin [3] for the monotone operator if there exists k∈0,1 such that given any pair x,y∈DS and u∈Sx, there exists v∈Sy satisfying u−v≤DSx,Sy and(5)D2Sx,Sy≤x−y2+kx−u−y−v2.Ifk=1 in (5), then S is pseudo-contractive-type, while S is nonexpansive-type if k=0. Every multivalued nonexpansive mapping S:DS⊆Y⟶PY is nonexpansive-type. S is of type-one in the sense of Isiogugu et al. [4] if given any pair x,y∈DS, then(6)u−v≤DSx,Sy,for allu∈PSx,v∈PSy,where PSx:=u∈Sx:u−x=dx,Sx. S is called a multivalued demicontractive in the sense of Isiogugu and Osilike [5] using the terminology of Hicks and Kubicek [6] for single-valued demicontractive if FS≠∅ and for all p∈FS and x∈DS, there exists k∈0,1 such that(7)D2Sx,Sp≤x−p2+kd2x,Sx.Ifk=1 in (7), S is hemicontractive in the terminology of Naimpally and Singh [7] for single-valued hemicontractive, while S is quasi-nonexpansive if k=0.Furthermore, every multivaluedk−strictly pseudo-contractive-type in the sense of [1] with the nonempty set of strict fixed points is demicontractive with respect to its set of strict fixed points.A single-valued mappingS:DS⊆H⟶H is called nonspreading in the sense of Kohsaka and Takahashi [8, 9] if(8)Sx−Sy2≤x−y2+2x−Sx,y−Sy,∀x,y∈C.Observe that ifS is nonspreading and FS≠∅, then S is quasi-nonexpansive. S is k-strictly pseudo-nonspreading in the sense of Osilike and Isiogugu [10] if there exists k∈0,1 such that(9)Sx−Sy2≤x−y2+kx−Sx−y−Sy2+2x−Sx,y−Sy,for all x,y∈DS. Clearly, every nonspreading mapping is k-strictly pseudo-nonspreading. If S is k−strictly pseudo-nonspreading and FS≠∅, then S is demicontractive in the sense of [6] (see also [11]).Several algorithms have been introduced by different authors for the approximation of common fixed points of finite family of mappingsSii=1N, where N∈ℕ (the set of nonnegative integers) (see, for example, [12–18] and references therein). One of the motivations for this aspect of research is the well-known convex feasibility problem which is reducible to the problem of finding a point in the intersection of the set of fixed points of a family of nonexpansive mappings (see, for example, [19, 20]). The earliest of such algorithms was the cyclic algorithm introduced by Bauschke [12] using a Halpern-type iterative process considered in [21] for the approximation of a common fixed point of a finite family of nonexpansive self-mappings. He proved the following theorem.Theorem 1 (see [12], Theorem 3.1).
LetK be a nonempty convex closed subset of a real Hilbert space H and S1,S2,…,SN be a finite family of nonexpansive mappings of K into itself with F:=∩i=1NFSi≠∅ with F=FSNSN−1…S1=FS1SN…S2=FSN−1SN−2…S1SN. Given points u,x0∈K, let xn be generated by(10)xn+1=ςnxn+1−ςnSn+1xn+1,n≥0,where Sn≔SnmodN and ςn⊂0,1 satisfies ∑n≥1ςn+N−ςn<∞. Then, xn converges strongly to PFu, where PF:H⟶F is the metric projection.The above algorithm of Bauschke was extended to approximate the family of more general class of strictly pseudo-contractive mappings (see, for example, [22, 23]). Suantai et al. also considered similar algorithms (see, for example, [24]) and references therein.In 2008, Zhang and Guo [25] considered a parallel iteration for approximating the common fixed points of a finite family of strictly pseudo-contractive mapping. They obtained the following theorem.Theorem 2 (see [25], Theorem 4.3).
LetE be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty convex closed subset of E. Let N≥1 be an integer, and for each 1≤i≤N, let Si:K⟶K be a ki-strictly pseudo-contractive mapping for some 0≤ki<1. Let k=minki:1≤i≤N. Assume the common fixed point set ∩i−1NFSi is nonempty. Assume also for each n, λini=1N is a finite sequence of positive numbers such that ∑1=1Nλin=1 for all n and infn≥1λin>0 for all 1≤i≤N. Given x0∈K, let xnn=1∞ be the sequence generated by the algorithm:(11)xn+1=ςnxn+1−ςn∑i=1NλinSixn,n≥0.
Letςnn=1∞ be a real sequence satisfying the conditions(12)∑n=0∞∑i=1Nλin+1−λin<∞;∑n=0∞1−ςnqk−Cq1−ςnq−1=∞.
Then,xn converges weakly to a common fixed point of Sii=1N.Motivated by the parallel algorithm above, many authors have considered in a real Hilbert space, another form of parallel algorithm for a finite familySii=1N of ki-strictly pseudo-contractive mappings defined by(13)xn+1=ςn0xn+∑i=1NςniSixn,n≥1,where ςnin=1∞⊆0,1 for each i and ∑i=0Nςni=1 for each n (see, for example, [13] and references therein).In [14], Iemoto and Takahashi studied the approximation of common fixed points of a nonexpansive self-mapping T and a nonspreading self-mapping S in a real Hilbert space. If T,S:C⟶C are, respectively, nonexpansive and nonspreading mappings, they considered the iterative scheme xnn=1∞ generated from arbitrary x1∈C by(14)xn+1=1−ςnxn+ςnβnSxn+1−βnTxn,n≥1,where ςn and βn are suitable sequences in 0,1. They proved the following main theorem:Theorem 3 (see [14], Theorem 4.1).
LetH be a real Hilbert space. Let C be a nonempty convex and closed subset of H. Let S be a nonspreading mapping of C into itself and T a nonexpansive mapping of C into itself such that FT∩FS≠∅. Define a sequence xnn=1∞ in C as follows:(15)x1∈C,xn+1=1−ςnxn+ςnβnSxn+1−βnTxn,for all n∈ℕ, where ςnn=1∞, βnn=1∞⊂0,1.
Then, the following hold:(i)
Ifliminfn⟶∞ςn1−ςn>0, ∑n=1∞1−βn<∞, then xnn=1∞ converges weakly to v∈FS.(ii)
If∑n=1∞ςn1−ςn=∞ and ∑n=1∞βn<∞, then xnn=1∞ converges weakly to v∈FT.(iii)
Ifliminfn⟶∞ςn1−ςn>0 and liminfn⟶∞βn1−βn>0, then xnn=1∞ converges weakly to v∈FT∩FS.
Motivated by the above result, Osilike and Isiogugu obtained the following result.Theorem 4 (see [10], Theorem 3.1.1).
LetC be a nonempty convex closed subset of a real Hilbert space, and let T:C⟶C be a k-strictly pseudo-nonspreading mapping with a nonempty fixed point set FT. Let β∈k,1, and let ςnn=1∞ be a real sequence in 0,1 such that limn⟶∞ςn=0. Let xnn=1∞ and znn=1∞ be sequences in C generated for arbitrary x1∈C by(16)xn+1=ςnxn+1−ςnβxn+1−βTxn,n≥1,zn=1n∑k=1nxk,n≥1.
Then,znn=1∞ converges weakly to z∈FT, where z=limn⟶∞PFTxn.We observed that all the existing iteration schemes for the approximation of a common fixed point of a finite familyT1,T2,…,TN of mappings for N>2, which are related to the parallel algorithm, require the condition that, for each n, ∑i=1Nςni=1 on the control sequences ςnin=1∞i=1N. However, in real-life applications, if N is very large, it is very difficult or almost impossible to generate a family of such control sequences. Moreover, the computational cost of generating such a family of control sequences is very high and also takes a very long process. On the contrary, the algorithms of Iemoto and Takahashi [14] and Osilike and Isiogugu [10] do not require the imposition ∑i=1Nςni=1 on the control sequences for N=2. Consequently, there is a need to extend the iteration schemes in [10, 14] for N>2.Motivated by the above observations and the algorithms of Iemoto and Takahashi [14] and Osilike and Isiogugu [10], which do not require the imposition ∑i=1Nςni=1 on the control sequences for N=2 and the need to extend the iteration schemes for N>2, the aim of this work is first to study some possible linear combinations of the products of the elements of a family of sequence of real numbers ςnin=1∞i=1N whose sum is unity. Second, to apply the result to construct a new (horizontal) algorithm which does not require the condition ∑i=1Nςni=1 on the finite family of the control sequences ςnin=1∞i=1N. Third, to prove that the new algorithm converges weakly and strongly to an element in the intersection of the set of fixed points of a countable finite family of multivalued type-one demicontractive mappings. We also show that our algorithm is an extension of the algorithm of Osilike and Isiogugu [10] when N=2. Furthermore, the algorithm is applied to establish some new algorithms for the approximation of the common solution of an equilibrium problem and a fixed point problem for a finite family of type demicontractive mappings. The numerical examples and computations of the horizontal algorithm were also presented. The obtained results complement, extend, and improve many results on the iteration schemes for the approximation of common fixed points for a finite family of single-valued and multivalued mappings.


## 2. Preliminaries

In the sequel, we shall need the following definitions and lemmas.Definition 1 (see, e.g., [26–27]).
LetY be a Banach space and S:DS⊆Y⟶2Y be a multivalued mapping. I−S is weakly demiclosed at zero if for any sequence, xnn=1∞⊆DS such that xn converges weakly to p and a sequence yn with yn∈Sxn for all n∈ℕ such that xn−yn strongly converges to zero. Then, p∈Sp (i.e., 0∈I−Sp).Definition 2.
A Banach spaceY is said to satisfy Opial’s condition [28] if whenever a sequence xn weakly converges to x∈Y, then it is the case that(17)liminfxn−x<liminfxn−y,for all y∈Y, y≠x.Definition 3 (see [29]).
A multivalued mappingS:C⟶PC is said to satisfy condition (1) (see, for example, [29]) if there exists a nondecreasing function f:0,∞⟶0,∞ with f0=0 and fr>0 for all r∈0,∞ such that(18)dx,Sx≥fdx,FS,∀x∈C.Definition 4 (see [4]).
LetY be a normed space and S:DS⊆Y⟶2Y be a multivalued map. S is of type-one if given any pair x,y∈DS, then(19)u−v≤DSx,Sy,for  allu∈PSx,v∈PSy.Lemma 1 (see [30]).
Letan and γn be sequences of nonnegative real numbers satisfying the following relation:(20)an+1≤an+γn,∀n∈N.
If∑γn<∞, then limn⟶∞an exists.


## 3. Main Results

LetK be a nonempty convex and closed subset of a real Hilbert space H. Suppose that Sii=1N, N≥2 is a countable finite family of mappings Si:K⟶K, and we consider the horizontal iteration process generated from arbitrary x1 for the finite family of mappings Sii=1N, using a finite family of the control sequences ςnin=1∞i=1N as follows:ForN = 2,(21)xn+1=ςn1xn+1−ςn1ςn2S1xn+1−ςn2S2xn.ForN = 3,(22)xn+1=ςn1xn+1−ςn1ςn2S1xn+1−ςn2ςn3S2xn+1−ςn3S3xn.For arbitrary but finiteN≥2,(23)xn+1=ςn1xn+1−ςn1ςn2S1xn+1−ςn2ςn3S2xn+1−ςn3…ςNSN−1xn+1−ςNSNxn…=ςn1xn+∑i=2Nςni∏j=1i−11−ςnjSi−1xn+∏j=1N1−ςnjSNxn,n≥1.We now present the following results which are very useful in establishing our convergence theorems.Proposition 1.
Letςii=1N⊆ℝ be a countable subset of the set of real numbers ℝ, where N≥2 is an arbitrary integer. Then, the following holds:(24)ς1+∑i=2Nςi∏j=1i−11−ςj+∏j=1N1−ςj=1.Proof.
ForN=2,(25)ς1+∑i=22ςi∏j=1i−11−ςj+∏j=121−ςj=ς1+ς21−ς1+1−ς11−ς2=ς1+1−ς1ς2+1−ς2=ς1+1−ς1=1.
We assume it is true forN and prove for N+1.(26)ς1+∑i=2N+1ςi∏j=1i−11−ςj+∏j=1N+11−ςj=ς1+∑i=2Nςi∏j=1i−11−ςj+ςN+1∏j=1N1−ςj+∏j=1N+11−ςj=ς1+∑i=2Nςi∏j=1i−11−ςj+∏j=1N1−ςjςN+1+1−ςN+1=ς1+∑i=2Nςi∏j=1i−11−ςj+∏j=1N1−ςj=1.Remark 1.
Proposition1 holds if ςii=1N is replaced with ςii=0N.Proposition 2.
Letςii=kN⊆ℝ be a countable subset of the set of real numbers ℝ, where k is a fixed nonnegative integer and N∈ℕ is any integer with k+1≤N. Then, the following holds:(27)ςk+∑i=k+1Nςi∏j=ki−11−ςj+∏j=kN1−ςj=1.Proof.
Fork=0, N=1, and k=1,N=2, the proofs follow from Remark 1 and Proposition 1, respectively.
We assume it is true fork and N. Now, for k and N+1,(28)ςk+∑i=k+1N+1ςi∏j=ki−11−ςj+∏j=kN+11−ςj=ςk+∑i=k+1Nςi∏j=ki−11−ςj+ςN+1∏j=kN1−ςj+∏j=kN+11−ςj=ςk+∑i=k+1Nςi∏j=ki−11−ςj+∏j=kN1−ςjςN+1+1−ςN+1=ςk+∑i=k+1Nςi∏j=ki−11−ςj+∏j=kN1−ςj=ςk+∑i=kNςi∏j=ki−11−ςj+∏j=kN1−ςj=1.Proposition 3.
Lett,u, and v be arbitrary elements of a real Hilbert space H. Let k be a fixed nonnegative integer and N∈ℕ be such that k+1≤N. Let vii=kN−1⊆H and ςii=kN⊆0,1 be a countable finite subset of H and ℝ, respectively. Define(29)y=ςkt+∑i=k+1Nςi∏j=ki−11−ςjvi−1+∏j=kN1−ςjv.Then,(30)y−u2=ςkt−u2+∑i=k+1Nςi∏j=ki−11−ςjvi−1−u2+∏j=kN1−ςjv−u2−ςk∑i=k+1Nςi∏j=ki−11−ςjt−vi−12+∏j=kN1−ςjt−v2−1−ςk∑i=k+1N−1ςi∏j=ki1−ςjvi−1−ςi+1vi+wi+12+ςN∏j=kN1−ςjv−vN−12,where wk=∑i=k+1Nςi∏j=ki−11−ςjvi−1+∏j=kN1−ςjv, k=1,2,…,N−1, and wN=1−ςNv.Proof.
Using the well-known identity,(31)tx+1−ty2=tx2+1−ty2−t1−tx−y2,which holds for all x,y∈H and for all t∈0,1, we prove by (i) direct computation and (ii) induction.
Observe that, fork≤N−1, wk=1−ςkςk+1vk+wk+1. Consequently, by the direct computation, we have(32)y−u2=ςkt+∑i=k+1Nςi∏j=ki−11−ςjvi−1+∏j=kN1−ςjv−u2=ςkt+wk−u2=ςkt+1−ςkςk+1vk+wk+1−u2=ςkt−u2+1−ςkςk+1vk+wk+1−u2−ςk1−ςkt−ςk+1vk+wk+12=ςkt−u2+1−ςkςk+1vk−u2+1−ςk+1ςk+2vk+1+wk+2−u2−ςk+11−ςk+1vk−ςk+2vk+1+wk+22−ςk1−ςkςk+1t−vk2+1−ςk+1t−ςk+2vk+1+wk+22−ςk+11−ςk+1vk−ςk+2vk+1+wk+22=ςkt−u2+1−ςkςk+1vk−u2+1−ςk1−ςk+1ςk+2vk+1+wk+2−u2−1−ςkςk+11−ςk+1vk−ςk+2vk+1+wk+22−ςk1−ςkςk+1t−vk2−ςk1−ςk1−ςk+1t−ςk+2vk+1+wk+22+ςk1−ςkςk+11−ςk+1vk−ςk+2vk+1+wk+22=ςkt−u2+1−ςkςk+1vk−u2−ςk1−ςkςk+1t−vk2+1−ςk1−ςk+1ςk+2vk+1+wk+2−u2−ςk1−ςk1−ςk+1t−ςk+2vk+1+wk+22−ςk+11−ςk1−ςk+11−ςkvk−ςk+2vk+1+wk+22=ςkt−u2+1−ςkςk+1vk−u2−ςk1−ςkςk+1t−vk2−ςk+11−ςk1−ςk+11−ςkvk−ςk+2vk+1+wk+22+1−ςk1−ςk+1ςk+2vk+1+wk+2−u2−ςk1−ςk1−ςk+1t−ςk+2vk+1+wk+22=ςkt−u2+1−ςkςk+1vk−u2−ςk1−ςkςk+1t−vk2−ςk+11−ςk1−ςk+11−ςkvk−ςk+2vk+1+wk+22+1−ςk1−ςk+1ςk+2vk+1+1−ςk+2ςk+3vk+2+wk+3−u2−ςk1−ςk1−ςk+1ςk+2vk+1+1−ςk+2ςk+3vk+2+wk+3−t2=ςkt−u2+1−ςn1ςk+1vk−u2−ςn11−ςn1ςk+1t−vk2+1−ςk1−ςk+1ςk+2vk+1−u2+1−ςk1−ςk+11−ςk+2ςk+3vk+2+wk+3−u2−1−ςk1−ςk+1ςk+21−ςk+2vk+1−ςk+3vk+2+wk+32−ςk1−ςk1−ςk+1ςk+2vk+1−t2−ςk1−ςk1−ςk+11−ςk+2ςk+3vk+2+wk+3−t2+ςk1−ςk1−ςk+1ςk+21−ςk+2vk+1−ςk+3vk+2+wk+32−ςk+11−ςk1−ςk+11−ςkvk−ςk+2vk+1+wk+22=ςkt−u2+1−ςkςk+1vk−u2−ςk1−ςkςk+1t−vk2+1−ςk1−ςk+1ςk+2vk+1−u2+1−ςk1−ςk+11−ςk+2ςk+3vk+2+wk+3−u2−ςk1−ςk1−ςk+1ςk+2vk+1−t2−ςk1−ςk1−ςk+11−ςk+2ςk+3vk+2+wk+3−t2−ςk+11−ςk1−ςk+11−ςkvk−ςk+2vk+1+wk+22−1−ςk21−ςk+1ςk+21−ςk+2vk+1−ςk+3vk+2+wk+32,=ςkt−u2+∑i=k+1k+2ςi∏j=ki−11−ςjvi−1−u2−ςk∑i=k+1k+2ςi∏j=ki−11−ςjt−vi−12−1−ςk∑i=k+1k+2ςi∏j=ki1−ςjvi−1−ςi+1vi+wi+12+∏j=kk+21−ςjςk+3vk+2+wk+3−u2−ςk∏j=kk+21−ςjt−ςk+3vk+2+wk+32=⋮=ςkt−u2+∑i=k+1Nςi∏j=ki−11−ςjvi−1−u2+∏j=kN1−ςjv−u2−ςk∑i=k+1Nςi∏j=ki−11−ςjt−vi−12+∏j=kN1−ςjt−v2−1−ςk∑i=k+1N−1ςi∏j=ki1−ςjvi−1−ςi+1vi+wi+12+ςN∏j=kN1−ςjv−vN−12.
Therefore, it holds fork,N from direct computation.
Since induction holds for a fixedk and each N from direct computation, then it is true for k,N=1,2,3. Thus, to prove by induction, we then assume that it is true for k,N and prove for k and N+1. From(33)y=ςkt+∑i=k+1N+1ςi∏j=ki−11−ςjvi−1+∏j=kN+11−ςjv,we have that(34)y−u2=ςkt+∑i=k+1N+1ςi∏j=ki−11−ςjvi−1+∏j=kN+11−ςjv−u2=ςkt+∑i=k+1Nςi∏j=ki−11−ςjvi−1+ςN+1∏j=kN1−ςjjvN+∏j=kN+11−ςjv−u2=ςkt+∑i=k+1Nςi∏j=ki−11−ςjvi−1+∏j=kN1−ςjςN+1vN+1−ςN+1v−u2=ςkt+∑i=k+1Nςi∏j=ki−11−ςjvi−1+∏j=kN1−ςjv∗−u2=ςkt−u2+∑i=k+1Nςi∏j=ki−11−ςjvi−1−u2+∏j=kN1−ςjv∗−u2−ςk∑i=k+1Nςi∏j=ki−k1−ςjt−vi−12+∏j=kN1−ςjt−v∗2−1−ςk∑i=k+1N−1ςi∏j=ki1−ςjvi−1−ςi+1vi+wi+12+ςN∏j=kN1−ςjv∗−vN−12.
Observe that(35)t−v∗2=ςN+1vN+1−ςN+1v−t2=ςN+1t−vN2+1−ςN+1t−v2−ςN+11−ςN+1vN−v2.
Also,(36)v∗−vN−12=ςN+1vN+1−ςN+1v−vN−12=vN−1−ςN+1vN+1−ςN+1v2=vN−1−ςN+1vN+wN+12.
Furthermore,(37)v∗−u2=ςN+1vN+1−ςN+1v−u2=ςN+1u−vN2+1−ςN+1v−u2−ςN+11−ςN+1vN−v2.
It then follows from (34–37) that(38)y−u2=ςkt−u2+∑i=k+1N+1ςi∏j=ki−11−ςjvi−1−u2+∏j=kN+11−ςjv−u2−ςk∑i=k+1N+1ςi∏j=ki−i1−ςjt−vi−12+∏j=kN+11−ςjt−v2−1−ςk∑i=k+1Nςi∏j=ki1−ςjvi−1−ςi+1vi+wi+12+ςN+1∏j=kN+11−ςjv−vN2.
We now apply Propositions2 and 3 to prove the following weak and strong convergence theorems for type-one demicontractive mappings.Theorem 5.
LetK be a nonempty convex and closed subset of a real Hilbert space H. Suppose that Sii=1N, N≥2 is a countable finite family of type-one demicontractive mappings Si:K⟶PK from K into the family of all proximinal subsets of K with contractive coefficients λi∈0,1 for each i. Suppose that ∩i=1NFSi≠∅ and for each i, I−Si is weakly demiclosed at zero; then, the sequence of the horizontal algorithm defined by(39)xn+1=ςn,1xn+∑i=2Nςn,i∏j=1i−11−ςn,jyn,i−1+∏j=1N1−ςn,jyn,N,n≥1,converges weakly to q∈∩i=1NFSi, where yn,i∈Sixn for each i and ςn,in=1∞i=1N is a countable finite family of real sequences in [0, 1] satisfying the following:(i)
ςn,1>λ>maxλii=1N; ςn,i<ς<1, for each i.(ii)
liminfn⟶∞ςn,i∏j=1i−11−ςn,jςn,1−λi−1>0, i=2,3,…,N.(iii)
liminfn⟶∞∏j=1N1−ςn,jςn,1−λN>0.
Also, if, in addition,Si is L-Lipschitzian and satisfies condition (1) for each i, then xn converges strongly to q∈∩i=1NFSi.Proof.
Settingxn+1=y, xn=t, p=u, k=1, and yn,N∈SNxn=v in Proposition 3, we obtain(40)xn+1−p2≤ςn,1xn−p2+∑i=2Nςn,i∏j=1i−11−ςn,jyn,i−1−p2+∏j=1N1−ςn,jyn,N−p2−ςn,1∑i=2Nςn,i∏j=1i−11−ςn,jxn−yn,i−12+∏j=1N1−ςn,jxn−yn,N2.
Applying type-one demicontractive condition on eachSi, we obtain(41)xn+1−p2≤ςn,1xn−p2+∑i=2Nςn,i∏j=1i−11−ςn,jxn−p2+λi−1xn−yn,i−12+∏j=1N1−ςn,jxn−p2+λNxn−yn,N2−ςn,1∑i=2Nςn,i∏j=1i−11−ςn,jxn−yn,i−12+∏j=1N1−ςn,jxn−yn,N2=ςn,1+∑i=2Nςn,i∏j=1i−11−ςn,j+∏j=1N1−ςn,jxn−p2−ςn,1−λi−1∑i=2Nςn,i∏j=1i−11−ςn,jxn−yn,i−12−ςn,1−λN∏j=1N1−ςn,jxn−yn,N2.
Consequently, if we setk = 1 in Proposition 2, we obtain(42)xn+1−p2≤xn−p2−∑i=2Nςn,i∏j=1i−11−ςn,jςn,1−λi−1xn−yn,i−12+∏j=1N1−ςn,jςn,1−λNxn−yn,N2.
Furthermore, condition (i) on the control sequences implies thatlimn⟶∞xn−p exists; hence, xn is bounded. Similarly, conditions (ii) and (iii) imply that limn⟶∞xn−yn,i0,i=1,2,…,N. Finally, the demiclosedness property of each I−Si, boundedness of xn, uniqueness of the limit of a weakly convergent sequence, and Opial property of a real Hilbert space guarantee the weak convergence of xn to q∈∩i=1NFSi. Also, since Si is L-Lipschitzian and satisfies condition (1) for each i, it then follows from standard argument that xn converges strongly to q∈∩i=1NFSi.Remark 2.
IfN=2 and we set ςn,1=ςn and ςn,2=β, for all n, Itheidentitymapping=S1, and S=S2, we obtain(43)xn+1=ςn,1xn+∑i=22ςn,i∏j=1i−11−ςn,jSn,i−1+∏j=121−ςn,jSn,2=ςnxn+1−ςnβnIxn+1−ςn1−βnSxn=ςnxn+1−ςnβxn+1−βSxn,which was considered by Osilike and Isiogugu [10].


## 4. Applications

We now present the application of Theorem5 in the construction of algorithms for approximating a common solution of an equilibrium problem and fixed point problem.For solving the equilibrium problems for a bifunctionF:C×C⟶ℝ, let us assume that F satisfies the following conditions:(A1):Fg,g=0 for all g∈C(A2):F is monotone, that is, Fg,h+Fh,g≤0, for all g,h∈C(A3): for eachg,h,z∈C, limt↓0Ftz+1−tg,h≤Fg,h(A4): for eachg∈C, h↦Fg,h is convex and lower semicontinuousLemma 2 (see [31]).
LetC be a nonempty convex closed subset of a real Hilbert space H and F:C×C⟶ℝ, a bifunction satisfying (A1)–(A4). Let r>0 and g∈H. Then, there exists z∈C such that(44)Fz,h+1rh−z,z−g≥0,∀h∈C.Lemma 3 (see [32]).
LetC be a nonempty convex closed subset of a real Hilbert space H. Assume that F:C×C⟶ℝ satisfies (A1)–(A4). Let r>0 and g∈H. Define Tr:H⟶2C by(45)Trg=z∈C:Fz,h+1rh−z,z−g≥0,∀h∈C.
Then, the following hold:(1)
Tr is single valued.(2)
Tr is firmly nonexpansive, that is, for any g,h∈H, Trg−Trh2≤Trg−Trh,g−h.(3)
FTr=EPF.(4)
EPF is convex and closed.Lemma 4 (see [33]).
LetC be a nonempty convex closed subset of a real Hilbert space H and F:C×C⟶ℝ, a bifunction satisfying (A1)–(A4). Let r>0 and g∈H. Then, for all g∈H and p∈FTr,(46)p−Trg2+Trg−g2≤p−g2.Lemma 5.
LetH be a real Hilbert space, and let C be a nonempty convex closed subset of H. Let PC be the convex projection onto C. Then, convex projection is characterized by the following relations:(i)
g∗=PCg⇔g−g∗,h−g∗≤0, for all h∈C.(ii)
g−PCg2≤g−h2−h−PCg2.(iii)
g−PCh2≤g−h2−PCh−h2.Motivated by Algorithm 19 of Isiogugu et al. [34], we obtain the following result using a selection of Algorithm 4.2 above in the sense of [34].Theorem 6.
LetC be a nonempty convex closed subset of a real Hilbert space H, f:C×C⟶ℝ, a bifunction satisfying (A1)–(A4) and Tii=1N be such that Ti:C⟶PC is type-one λi-strictly pseudo-contractive-type mappings, and I−Ti is weakly demiclosed at zero for each i=1,2,…,N. Suppose that F=∩i=1NFsTi∩EPf≠∅. Let xn be a sequence generated from arbitrary x0∈C as follows:Algorithm 1.
(47)
y
n
=
ς
n
,
1
x
n
+
∑
i
=
2
N
ς
n
,
i
∏
j
=
1
i
−
1
1
−
ς
n
,
j
y
n
,
i
−
1
+
∏
j
=
1
N
1
−
ς
n
,
j
y
n
,
N
,
u
n
∈
K
such that
F
u
n
,
y
+
1
r
n
y
−
u
n
,
u
n
−
y
n
≥
0
,
∀
y
∈
K
,
x
n
+
1
=
1
2
u
n
+
x
n
,
whereyn,i∈Tixn for each i and ςn,in=1∞i=1N is a finite family of real sequences in [0, 1] for each i satisfying
(i)
ς
n
,
1
>
λ
>
max
λ
i
i
=
1
N; ςn,i<ς<1, for each i.
(ii)
lim
inf
n
⟶
∞
ς
n
,
i
∏
j
=
1
i
−
1
1
−
ς
n
,
j
ς
n
,
1
−
λ
i
−
1
>
0, i=2,3,…,N.
(iii)
lim
inf
n
⟶
∞
∏
j
=
1
N
1
−
ς
n
,
j
ς
n
,
1
−
λ
N
>
0.
Also, if, in addition,Ti satisfies condition (1) for each i,
(iv)
r
n
⊂
a
,
∞ for some a>0.
Then,xn converges strongly to p∈F.Proof.
(48)
x
n
+
1
−
p
2
=
1
2
x
n
+
u
n
−
p
2
=
1
2
x
n
−
p
2
+
1
2
u
n
−
p
2
−
1
4
x
n
−
u
n
2
≤
1
2
x
n
−
p
2
+
1
2
y
n
−
p
2
−
1
4
x
n
−
u
n
2
=
1
2
x
n
−
p
2
−
1
4
x
n
−
u
n
2
+
1
2
ς
n
,
1
x
n
+
∑
i
=
2
N
ς
n
,
i
∏
j
=
1
i
−
1
1
−
ς
n
,
j
y
n
,
i
−
1
+
∏
j
=
1
N
1
−
ς
n
,
j
y
n
,
N
−
p
2
≤
1
2
x
n
−
p
2
−
1
4
x
n
−
u
n
2
+
1
2
x
n
−
p
2
−
∑
i
=
2
N
ς
n
,
i
∏
j
=
1
i
−
1
1
−
ς
n
,
j
ς
n
,
1
−
λ
i
−
1
x
n
−
y
n
,
i
−
1
2
+
∏
j
=
1
N
1
−
ς
n
,
j
ς
n
,
1
−
λ
N
x
n
−
y
n
,
N
2
=
−
1
4
x
n
−
u
n
2
+
x
n
−
p
2
−
1
2
∑
i
=
2
N
ς
n
,
i
∏
j
=
1
i
−
1
1
−
ς
n
,
j
ς
n
,
1
−
λ
i
−
1
x
n
−
y
n
,
i
−
1
2
+
∏
j
=
1
N
1
−
ς
n
,
j
ς
n
,
1
−
λ
N
x
n
−
y
n
,
N
2
.
It then follows thatlimn⟶∞xn−p exists; hence, xn is bounded. Also,(49)∑n=1∞12ςn,i∏j=1i−11−ςn,jςn,1−λi−1xn−yn,i−12≤x0−p2<∞,i=1,2,…,N−1.∑n=1∞12∏j=1N1−ςn,jςn,1−λNxn−yn,N2≤x0−p2<∞.
Thus, from (i), (ii), and (iii), we have thatlimn⟶∞xn−yn,i=0, for all i=1,2,…,N. Furthermore, limn⟶∞xn−un=0. Consequently, limn⟶∞xn+1−xn2=limn⟶∞1/2xn−un2=0 which implies that xn is a Cauchy sequence in K. Also, since K is convex and closed, xn converges strongly to some q∈K. From the Opial condition of H and the demiclosedness property of Ti, we have that q∈Tiq, for all i−1,2,…,N.
The remaining part of the proof is similar to the method of [34], Theorem 20. Therefore, it is omitted.


## 5. Examples

We present the numerical computation of the iteration scheme of Theorem5.LetH=ℝm,.,≤Rm with the usual norm “.” on Rm and partial order “≤” on ℝ, C=x¯=x1,x2,…,xt,…,xm∈ℝm:x1=x2=…=xt=…=xm. Observe that C,.,≤ is a convex closed linear total ordered subset of Rn with a¯≤b¯ if and only if at≤bt for all t=1,2,3,…,m. Denote the order interval a¯≤x¯≤b¯ by a¯,b¯, and let Sii=1∞ be a countable infinite family of mappings and Si:C⟶CBC define for each i and x¯∈C by(50)Six¯=−4i2i+1x¯,−3i2i+1x¯,x¯≥0¯,−3i2i+1x¯,−4i2i+1x¯,x¯<0¯.Clearly, for eachi,(I)FSi=0¯.(II)PSix¯=−3i/2i+1x¯.(51)DSix¯,Siy¯=4i2i+1x¯−y¯,x¯,y¯≥0¯,4i2i+1x¯−y¯,x¯,y¯<0¯,3i2i+1x¯−4i2i+1y¯≥3i2i+1x¯−3i2i+1y¯,x¯≥0¯,y¯<0¯.(III)u¯−v¯=3i/2i+1x¯−y¯≤DSix¯,Siy¯, for all u¯∈PSix¯,v¯∈PSiy¯.(IV)∩i=1NFSi=0¯.(V)d2x¯,Six¯=x¯−−3i/2i+1x¯2=x¯+3i/2i+1x¯2=5i+1/2i+12x¯2.(VI)dx¯,FSi=dx¯,0¯=x¯.(VII)H2Six¯,S0¯=4i/2i+1x¯2=x¯2+4i/2i+12−1x¯2=x¯2+12i2−4i−1/2i+12x¯2.It then follows from (V) and (VII) that(VIII)H2Six¯,S0¯=x¯−0¯2+12i2−4i−1/25i2+10i+1d2x¯,Six¯.Also, from (V) and (VI), we obtain that(IX)dx¯,Six¯≥fdx¯,FSi, where f:0,∞⟶0,∞ is defined by fr=r.In summary, for eachi, we have from (III), (VIII), and (IX) that Si is type-one demicontractive mapping with contraction coefficient λi=12i2−4i−1/25i2+10i+1 and satisfies condition (1).Observe thatsupλii=1∞=12/25=limi⟶∞λi. Therefore, if we set −3i/2i+1x¯n=yn,i∈Six¯n and define ςn,in=1∞i=1N⊆20,1 by(52)ςn,i=38ni2+3750ni2+1,then(i)
ςn,1>37/50>12/25=supλii=1∞; ςn,i<38/50<1.(ii)
liminfn⟶∞ςn,i∏j=1i−11−ςn,jςn,1−λi−1=limn⟶∞ςn,i∏j=1i−11−ςn,jςn,1−λi−1=38/501−38/50i−138/50−λi−1>0, 2≤i<N−1.(iii)
liminfn⟶∞∏j=1N1−ςn,jςn,1−λiN=limn⟶∞∏j=1N1−ςn,jςn,1−λiN=1−38/50N38/50−λiN>0.Table1 and Figure 1 show the sequences for N = 5 and N = 10. The values are rounded up to 9 decimal places.Table 1
Sequences of coordinates
x
¯
t
,
n
n
=
1
∞
o
f
x
¯
n
n
=
1
∞
=
x
1
,
n
,
x
2
,
n
,
…
,
x
t
,
n
,
…
x
m
,
n
n
=
1
∞
for each
t
=
1,2
,
…
,
m.
N = 5, xt,1 = 10
N = 5, xt,1 = −10
N = 10, xt,1 = −10
N = 10, xt,1 = 10
n
x
t,n
N
x
t,n
n
x
t,n
n
x
t,n
1
−10
1
10
1
10
1
−10
2
−4.863391744
2
4.863391744
2
4.86333914
2
−4.86333914
3
−2.426008398
3
2.426008398
3
2.42595782
3
−2.42595782
4
−1.220221064
4
1.220221064
4
1.22018365
4
−1.22018365
5
−0.615820431
5
0.615820431
5
0.61579556
5
−0.61579556
6
−0.311310767
6
0.311310767
6
0.31129518
6
−0.31129518
7
−0.157521707
7
0.157521707
7
0.1575123
7
−0.1575123
8
−0.079750993
8
0.079750993
8
0.07974546
8
−0.07974546
9
−0.040392014
9
0.040392014
9
0.04038883
9
−0.04038883
10
−0.020462938
10
0.020462938
10
0.02046113
10
−0.02046113
11
−0.01036864
11
0.01036864
11
0.01036763
11
−0.01036763
12
−0.005254554
12
0.005254554
12
0.00525399
12
−0.00525399
13
−0.002663152
13
0.002663152
13
0.00266284
13
−0.00266284
14
−0.00134987
14
0.00134987
14
0.0013497
14
−0.0013497
15
−0.000684252
15
0.000684252
15
0.00068416
15
−0.00068416
16
−0.000346866
16
0.000346866
16
0.00034682
16
−0.00034682
17
−0.000175843
17
0.000175843
17
0.00017582
17
−0.00017582
18
−0.000089144
18
0.000089144
18
0.00008913
18
−0.00008913
19
−0.000045193
19
0.000045193
19
0.00004519
19
−0.00004519
20
−0.000022912
20
0.000022912
20
0.00002291
20
−0.00002291
21
−0.000011615
21
0.000011615
21
0.00001161
21
−0.00001161
22
−0.000005887
22
0.000005887
22
0.00000588
22
−0.00000588
23
−0.000002983
23
0.000002983
23
0.00000298
23
−0.00000298
24
−0.000001509
24
0.000001509
24
0.00000151
24
−0.00000151
25
−0.000000762
25
0.000000762
25
0.00000076
25
−0.00000076
26
−0.000000384
26
0.000000384
26
0.00000038
26
−0.00000038
27
−0.000000191
27
0.000000191
27
0.00000019
27
−0.00000019
28
−0.000000095
28
0.000000095
28
0.00000009
28
−0.00000009
29
−0.000000046
29
0.000000046
29
0.00000004
29
−0.00000004
30
−0.00000002
30
0.00000002
30
0.00000002
30
−0.00000002
31
−0.000000007
31
0.000000007
31
0.00000001
31
−0.00000001
32
0
32
0Figure 1
Errors vs. iteration numbers (n): case 1a (a); case 1b (b); case 2a (c); case 2b (d).
(a)
(b)
(c)
(d)


## 6. Conclusion

A horizontal iteration scheme for the approximation of a common fixed point of a finite family of mappings is introduced in a real Hilbert space. This algorithm does not require the imposition of sum = 1 on the control sequences. Its applicability in developing other algorithms is demonstrated in Algorithm1. Furthermore, its computability is also exhibited in our numerical computations presented in Section 5.



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*Source: 3287968-2020-05-01.xml*