# A New Study on Halpern and Nonconvex Combination Algorithm for Nonlinear Mappings in Banach Spaces with Applications

Authors: Gaobo Li
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi-ϕ-nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.


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## Body

## 1. Introduction

LetH1 be a Hilbert space, and let C be the nonempty closed convex subset of H1. Let H2 be a real Hilbert space, and let Q be the nonempty closed convex subset of H2. Let A:H1⟶H2 be a linear bounded operator. In 1994, Censor and Elfving [1] introduced the split feasibility problem (SFP) as a generalization of convex feasibility problem as follows:(1)find a pointx∗∈Csuch thatAx∗∈Q.Recntly, the SFP and its variants have been investigated by many authors due to its real applications such as medical imaging, radiation therapy, and treatment planning; see, e.g., [2–5]. For solving SFP (1), it needs to get the inverse A−1 (assuming the existence of A−1) in algorithm of Censor and Elfving [1]. However, few authors continue to study the algorithm of Censor and Elfving since the difficulty of computing A−1, even if it exists. In fact, another algorithm solving SFP (1) is more popular which is called CQ algorithm given by Byrne [6, 7]. The CQ algorithm of Byrne is a gradient projection method in convex minimization. Since the CQ algorithm does need to compute A−1 and only involves the projections PC and PQ, it is easy to implement when PC and PQ have the closed-form expressions. However, the computations of PC and PQ are also difficult if these projections did not have the closed-form expressions which is such that the CQ algorithm of Byrne [6, 7] is not easy to implement in this case. In 2010, Xu [8] investigated the CQ algorithm from the ways of optimization and fixed point, proposed Mann’s algorithm, and relaxed CQ algorithm to solve SFP (1). In the relaxed CQ algorithm, the sets C and Q are level sets of convex functions so that the projections involved in the CQ algorithm are onto half-spaces, which makes the algorithm implementable. Also, in 2010, Moudafi [9] proposed an iterative method to solve a split common fixed point problem for quasi-nonexpansive mappings in which the projection is not involved which is such that the algorithm is easy to implement. In 2014, Kraikaew and Saejung [10] combined the Moudafi method and the Halpern algorithm to propose a new iteration in which the projection is not involved for solving the SFP. In the recent years, many algorithms have been given to solve the SFP in Hilbert spaces; see, for instance, [11–15] and the references therein.However, because of the complexity of properties in Banach space, it is very difficulty to solve SFP (and fixed point problem) in Banach spaces. Until now, only limited works on SFP (and fixed point problem) in Banach spaces have been reported in the literature. For instance, the authors in [16] gave an algorithm to solve SFP in Banach space. In [17], Tang et al. introduced some iterative algorithms to solve a split common fixed point problem for a quasi-strict pseudocontractive mapping and an asymptotically nonexpansive mapping in two Banach spaces and obtained the weak and strong convergence for the proposed algorithms. In [18], Chen et al. proposed a new hybrid projection method for solving split feasibility and fixed point problems involved in Bregman quasi-strictly pseudocontractive mapping in p-uniformly convex and uniformly smooth real Banach spaces. They proved the strong convergence for the proposed algorithm using the Bregman projection method. On the feasible and common fixed point problem, the authors also refer to [19–21].LetE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflective Banach space. Let S:E1⟶E1 be a closed quasi-ϕ-nonexpansive mapping and A:E1⟶E2 be a linear bounded operator. Very recently, Ma et al. [22] proposed a hybrid projection algorithm to solve the following split feasibility problem and fixed point problem:(2)findx∗∈Csuch thatAx∗∈Q,where C=x∈E1:x=Sx and Q⊂E2 is a nonempty closed convex subset. Precisely, their algorithm to solve (2) is as follows:(3)x1∈E1,C1=E1,zn=J−1J1xn+γA∗J2PQ−IAxn,yn=J−1αnJ1zn+1−αnJ1Szn,Cn+1=v∈Cn:ϕv,yn≤ϕv,xn,ϕv,zn≤ϕv,xn,xn+1=ΠCn+1x1,n≥1,where α⊂δ,1 with δ>0, γ∈0,1/A2k2, PQ is the metric projection of E2 onto Q, and ΠCn+1 is the generalized projection of E1 in Cn+1. The authors proved that the sequence generated by (3) strongly converges to a point which solves (2).On the contrary, the most algorithms of approximating the fixed points of quasi-ϕ-nonexpansive mappings in Banach spaces are constructed by the hybrid or shrinking projection methods, see [23–25]. However, in 2018, Hieu and Strodiot [26] introduced a new iterative algorithm for solving pseudomonotone equilibrium problem involving the fixed point problem for quasi-ϕ-nonexpansive mapping in Banach space without using the hybrid or shrinking projection methods. More precisely, their algorithm is(4)yn=argminλnfxn,y+12ϕy,xn:y∈C,zn=argminλnfyn,y+12ϕy,xn:y∈C,xn+1=ΠCJ−1αnJu+1−αnβnJzn+1−βnJSzn,where f:C×C⟶ℝ is a pseudomonotone bifunction and S:C⟶C is a quasi-ϕ-nonexpansive mapping. The authors proved that the sequence generated by (4) strongly converges to a common point that solves the pseudomonotone equilibrium problem on f and is a fixed point of S.In general, there are three kinds of iterations of strong convergence that are used to approximate the fixed point of the nonlinear operator. The iterations are the Halpern iteration, the viscosity iteration, and the hybrid projection iteration. Recently, Hussain et al. [27] proposed a new surprising iteration that strongly converges to a fixed point of a nonexpansive mapping in Hilbert space. More precisely, the iteration is(5)x1∈H,xn+1=αn1−μnxn+1−αnTxn,n≥1,where H is a Hilbert space, T:H⟶H is a nonexpansive mapping, and αn,μn⊂0,1 are the control sequences. The authors proved that xn generated by (5) strongly converges to a fixed point of T under some certain conditions on αn and μn. Later on, Marino et al. [28] extended (5) to strict pseudocontraction.In this paper, motivated by the work of [22, 26, 27], we introduce some algorithms to solve a split common fixed point problem for two families of quasi-ϕ-nonexpansive mappings in Banach spaces and prove the strong convergence for the proposed algorithms. As the application, we solve a split equilibrium problem in Banach space. Finally, we give a numerical example in infinite dimension Banach space to illustrate the main result of this paper. Our results extend the one of Ma et al. [22] from one quasi-nonexpansive mapping to two quasi-nonexpansive mappings and [27] from Hilbert space to Banach space.


## 2. Preliminaries

LetE be a Banach space, and let E∗ be the dual space of E. For all x∈E and x∗∈E∗, we denote the value of x∗ at x by x,x∗. The duality mapping J on E is defined by(6)Jx=x∗∈E∗:x,x∗=x2=x∗2,∀x∈E.It is known thatJx is nonempty for all x∈E. A Banach space E is said to be smooth if the limit(7)limn⟶∞x+ty−xtexists for all x,y∈SE=z∈E:z=1. The space E is smooth if and only if the duality mapping J is single-valued.A Banach spaceE is said to be strictly convex if x+y/2<1 for x,y∈E with x=y=1 and x≠y and uniformly convex if for each ϵ∈0,2, there exists δ>0 such that x+y/2≤1−δ for all x,y∈E with x=y=1 and x−y≥ϵ. It is known that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single-valued, one-to-one, and onto. Let E be a smooth Banach space. The function ϕ:E×E⟶ℝ is defined by(8)ϕx,y=x2−2x,Jy+y2,for all x,y∈E. From the definition of ϕ, it is easy to see that, for all x,y,z∈E, the following hold:(9)x−y2≤ϕx,y≤x+y2,ϕx,J−1λJy+1−λJz≤λϕx,y+1−λϕx,z,λ∈0,1.The following is an important property for the functionϕ:(10)ϕx,y=ϕz,y+ϕx,z+2z−x,Jy−Jz,for all x,y,z∈E.Lemma 1 (see [29]).
LetE be a uniformly convex and smooth Banach space, and let xn and yn be two sequences of E. If ϕxn,yn⟶0 and either xn or yn is bounded, then xn−yn⟶0.
For any bounded sequencesxn and yn in a uniformly convex and uniformly smooth Banach space, the following hold:(11)ϕxn,yn⟶0⇔xn−yn⟶0⇔Jxn−Jyn⟶0.
LetΠC:E⟶C be mapping called the generalized projection [30] that assigns to an arbitrary element x∈E the minimum point of the functional ϕx,y; that is, ΠCx=argminy∈Cϕy,x.Lemma 2 (see [30]).
LetE be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then, the following conclusions hold:(a)
ϕx,ΠCy+ϕΠCy,y≤ϕx,y, ∀x∈C, ∀y∈E(b)
Forx∈E, z=ΠCx if and only if z−y,Jx−Jz≥0, ∀y∈C(c)
Forx,y∈E, ϕx,y=0 if and only if x=y
LetE be a strictly convex and reflexive Banach space and C be a nonempty closed and convex subset. The metric projection(12)PCx=argminy∈Cy−x,∀x∈E.Lemma 3 (see [31]).
LetE be a smooth, strictly convex, and reflexive Banach space and C be a nonempty, closed convex subset of E. Let x∈E. Then,(13)z=PCxif and only ifz−y,Jx−z≥0,∀y∈C.
LetE be a strictly convex, smooth, and reflexive Banach space. The duality mapping J∗ from E∗ onto E∗∗=E coincides with the inverse of the duality mapping J from E onto E∗, that is, J∗=J−1. Define a mapping V:E×E∗⟶ℝ [32] by(14)Vx,x∗=x2−2x,x∗+x∗2,∀x,x∗∈E×E∗.Lemma 4 (see [32]).
LetE be a reflexive, smooth, and strictly convex Banach space. Then,(15)Vx,x∗≤Vx,x∗+y∗−2J−1x∗−x,y∗,for all x∈E and x∗,y∗∈E∗. Obviously, Vx,x∗=ϕx,J−1x∗ for all x∈E and x∗∈E.
LetE be a smooth Banach space. A mapping T:E⟶E is said to be closed if for any sequence xn⊂E with xn⟶x and Txn⟶y, then Txn=y. T is said to be quasi-ϕ-nonexpansive mapping if FixT≠∅ and(16)ϕp,Tx≤ϕp,x,for all p∈FixT and x∈E. For a quasi-ϕ-nonexpansive mapping T, FixT is convex. If T is closed, then FixT is closed, see [24].Lemma 5 (see [33]).
Letr>0. A real Banach space E is uniformly convex if and only if there exists a continuous strictly increasing function g:0,∞⟶0,∞ with g0=0 such that(17)tx+1−ty2≤tx2+1−ty2−t1−tgx−y,for all t∈0,1 and x,y∈Br, where Br=x∈E:x≤r.Lemma 6 (see [33]).
Letr>0. Let E be a 2-uniformly smooth Banach space with the best smoothness constants k>0. Then,(18)x+y2≤x2+2y,Jx+2k2y2,for all x,y∈E.Lemma 7 (see [34]).
Letan be a sequence of nonnegative real numbers. Suppose that(19)an+1≤1−γnan+γnδn,∀n∈ℕ,where γn⊂0,1 and δn⊂ℝ satisfy the conditions:(20)limn⟶∞γn=0,∑n=1∞γn=∞,andlim supn⟶∞δn≤0.
Then,limn⟶∞an=0.Lemma 8 (see [35]).
Letan be a sequence of real numbers such that there exists a subsequence ni of n such that ani<ani+1 for all i∈ℕ. Then, there exists a nondecreasing sequence mk⊂ℕ such that mk⟶∞ as k⟶∞, and the following properties are satisfied by all (sufficiently large) numbers k∈ℕ:(21)amk≤amk+1andak≤amk+1.
In fact,mk=maxj≤k:aj<aj+1.Lemma 9 (see [36]).
Suppose thatan and bn are sequences of nonnegative real numbers such that(22)an+1≤an+bn,n≥1.
If∑n=1∞bn<∞, then limn⟶∞an exists.


## 3. Main Results

In this section, letE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflexive Banach space. Define the functions ϕ1 and ϕ2 by(23)ϕ1x,y=x12−2x,J1y1+y12,∀x,y∈E1,ϕ2u,v=u22−2u,J2v2+v22,∀u,v∈E2,where x,J1y1 (resp., u,J1v2) and x1 (resp., u2) denote the value of J1y at x and norm of x (resp., the value of J1v at u and norm of u ) in E1 (resp. E2), respectively. However, for convenience, we use the same symbols ·,·, ·, and ϕ in E1 and E2 without the confusion.LetA:E1⟶E2 be a linear bounded operator with adjoint A∗. Let S:E1⟶E1 and T:E2⟶E2 be the quasi-ϕ-nonexpansive mappings. Consider the following split common fixed point problem:(24)findx∈FixSsuch thatAx∈FixT.Denote the set of solutions of the above split common fixed point problem byΩ. In this section, assume that S and T are closed and I−S and I−T are demiclosed at zeros in E1 and E2. Note that, from the closedness of S and T, it follows that FixS and FixT are closed [24], which implies that Ω is closed. The convexity of Ω is from the convexity of FixS. Assume that Ω is nonempty.Letx∗=ΠΩθ, where θ is the zero element in E1. We will prove that sequence xn generated by the following algorithm converges strongly to x∗.Algorithm 1.
Takex1∈E1, and define a sequence xn by(25)wn=TAxn,Qn=w∈E2:ϕw,wn≤ϕw,Axn,zn=J1−1J1xn−γnA∗J2I−PQnAxn,yn=J1−1βnJ1zn+1−βnJ1Szn,xn+1=J1−1αn1−τnJ1xn+1−αnJ1yn,n≥1,where αn,βn⊂0,1,τn⊂τ,1 with τ∈0,1 and(26)γn=PQn−IAxn22k2A∗J2I−PQnAxn2,ifPQn−IAxn≠0,0,otherwise.Lemma 10.
The sequencexn is well-defined and bounded.Proof.
Sinceϕw,wn≤ϕw,Axn is equivalent to 2w,J2Axn−J2wn≤Axn2−wn2, it follows that Qn is closed and convex for each n≥1. For any p∈Ω, it follows that Ap∈Qn for all n≥1. Hence, each Qn is nonempty closed convex, which implies that PQnAxn is well-defined. Now, we show that PQn−IAxn≠0 implies that A∗J2PQn−IAxn≠0. Assume that A∗J2PQn−IAxn=0. We have Ap−PQnAxn,J2PQn−IAxn≥0 by Lemma 3 and hence(27)0=p−xn,A∗J2PQn−IAxn=Ap−Axn,J2PQn−IAxn=Ap−PQnAxn,J2PQn−IAxn+PQnAxn−Axn,J2PQn−IAxn=Ap−PQnAxn,J2PQn−IAxn+PQn−IAxn2≥PQn−IAxn2.
It is a contradiction. It follows thatPQn−IAxn≠0 implies that A∗J2PQn−IAxn≠0. Hence, zn is well-defined. Furthermore, xn is well-defined.
SinceE1 is a 2-uniformly convex and 2-uniformly smooth real Banach space, E1∗ is 2-uniformly smooth real Banach space, and J1=J1∗−1. From (25) and Lemma 6, we have(28)ϕx∗,zn=x∗2−2x∗,J1xn+γnA∗J2PQn−IAxn+J1xn+γnA∗J2PQn−IAxn2≤x∗2−2x∗,J1xn−2γnx∗,A∗J2PQn−IAxn+xn2+2γnxn,A∗J2PQn−IAxn+2γn2k2A∗J2PQn−IAxn2=ϕx∗,xn−2γnx∗−xn,A∗J2PQn−IAxn+2γn2k2A∗J2PQn−IAxn2.
SinceAx∗∈Qn, Ax∗−PQnAxn,J2PQn−IAxn≥0. Hence, we have(29)2x∗−xn,A∗J2PQn−IAxn=2Ax∗−Axn,J2PQn−IAxn=2Ax∗−PQnAxn,J2PQn−IAxn+2PQn−IAxn2≥2PQn−IAxn2.
Combining (28) with (29), we obtain(30)ϕx∗,zn≤ϕx∗,xn−2γnPQn−IAxn2+2κ2γn2A∗J2PQn−IAxn2=ϕx∗,xn−PQn−IAxn42k2A∗J2PQn−IAxn2≤ϕx∗,xn.
Furthermore, by Lemma5, (25), and (30) we obtain(31)ϕx∗,yn=x∗2−2x∗,βnJ1zn+1−βnJ1Szn+βnJ1zn+1−βnJ1Szn2≤x∗2−2x∗,βnJ1zn+1−βnJ1Szn+βnzn2+1−βnSzn2−βn1−βngJ1zn−J1Szn=βnϕx∗,zn+1−βnϕx∗,Szn−βn1−βngJ1zn−J1Szn≤ϕx∗,xn−PQn−IAxn42k2A∗J2PQn−IAxn2−βn1−βngJ1zn−J1Szn.
It follows from (25), (31), and Lemma 5 that(32)ϕx∗,xn+1=ϕx∗,J1−1αn1−τnJ1xn+1−αnJ1yn=x∗2−2αn1−τnx∗,J1xn−21−αnx∗,J1yn+αn1−τnJ1xn+1−αnJ1yn2≤x∗2−2αn1−τnx∗,J1xn−21−αnx∗,J1yn+αn1−τnJ1xn2+1−αnJ1yn2≤x∗2−2αn1−τnx∗,J1xn−21−αnx∗,J1yn+αn1−τnxn2+1−αnyn2=αn1−τnϕx∗,xn+1−αnϕx∗,yn+αnτnx∗2≤αn1−τnϕx∗,xn+1−αnϕx∗,xn−PQn−IAxn42k2A∗J2PQn−IAxn2−βn1−βngJ1zn−J1Szn+αnτnx∗2=1−αnτnϕx∗,xn+αnτnx∗2−1−αnPQn−IAxn42k2A∗J2PQn−IAxn2+βn1−βngJ1zn−J1Szn≤maxϕx∗,xn,x∗2≤⋯≤maxϕx∗,x1,x∗2,n≥1.
So,ϕx∗,xn is bounded.Lemma 11.
Letxn be the sequence generated by Algorithm 1. Then,(33)ϕx∗,xn+1≤1−αnτnϕx∗,xn+2αnτnx∗−xn+1,J1x∗+1−αnJ1xn−J1yn.Proof.
Lethn=αnJ1xn+1−αnJ1yn. Then, by (31), we have(34)ϕx∗,J1−1hn≤αnϕx∗,xn+1−αnϕx∗,yn≤αnϕx∗,xn+1−αnϕx∗,xn=ϕx∗,xn.
Note that(35)xn+1=J1−11−αnτnhn+αnτn1−αnJ1yn−J1xn.
By (34) and (35) and Lemma 4, we have(36)ϕx∗,xn+1=ϕx∗,J1−11−αnτnhn+αnτn1−αnJ1yn−J1xn=Vx∗,1−αnτnhn+αnτn1−αnJ1yn−J1xn≤Vx∗,1−αnτnhn+αnτn1−αnJ1yn−J1xn+αnτnJ1x∗−1−αnJ1yn−J1xn−2xn+1−x∗,αnτnJ1x∗−1−αnJ1yn−J1xn=Vx∗,1−αnτnhn+αnτnJ1x∗−2xn+1−x∗,αnτnJ1x∗−1−αnJ1yn−J1xn≤1−αnτnϕx∗,J1−1hn+αnτnϕx∗,x∗−2xn+1−x∗,αnτnJ1x∗−1−αnJ1yn−J1xn≤1−αnτnϕx∗,xn+2αnτnx∗−xn+1,J1x∗−1−αnJ1yn−J1xn.Theorem 1.
If the following conditions hold:(37)limn⟶∞αn=0,∑n=1∞αn=∞andliminfn⟶∞βn1−βn>0,then the sequence xn generated by Algorithm 1 converges strongly to the element x∗.Proof.
By (32), we have(38)1−αnPQn−IAxn42k2A∗J2PQn−IAxn2+βn1−βngJ1zn−J1Szn≤1−αnτnϕx∗,xn−ϕx∗,xn+1+αnτnx∗2≤ϕx∗,xn−ϕx∗,xn+1+αnτnx∗2.
Now, we show thatxn−x∗⟶0 by the following two cases:Case 1. Suppose that there exists n0∈ℕ such that ϕx∗,xnn=n0∞ is nonincreasing. In this situation, ϕx∗,xn is convergent. By (37) and (38), we have(39)limn⟶∞PQn−IAxn4A∗J2PQn−IAxn2=limn⟶∞gJ1zn−J1Szn=0,which implies that(40)limn⟶∞J1zn−J1Szn=0.SinceA∗J2PQn−IAxn is bounded, we have(41)limn⟶∞PQn−IAxn=0.By (40), we have(42)J1yn−J1zn=1−βnJ1zn−J1Szn⟶0.Combining (39) with (42), we obtain(43)J1yn−J1xn≤J1yn−J1zn+J1zn−J1xn=J1yn−J1zn+γnA∗J2PQn−IAxn=J1yn−J1zn+PQn−IAxn22κ2A∗J2PQn−IAxn⟶0.On the contrary, from (25) and (43), it follows that(44)J1zn−J1xn+1≤J1zn−J1yn+J1yn−J1xn+1=J1zn−J1yn+αn1−τnJ1xn−J1yn⟶0.SinceE1 is a 2-uniformly convex and 2-uniformly smooth real Banach space, J1 is uniformly norm-to-norm continuous. From (40), (42), and (44), it follows that(45)limn⟶∞zn−Szn=limn⟶∞yn−zn=limn⟶∞zn−xn+1=0.Sincezn is bounded, there exist a subsequence znk of zn converging weakly to p∈E1 such that(46)lim supn⟶∞x∗−zn,J1x∗=limk⟶∞x∗−znk,J1x∗=x∗−p,J1x∗.Now, we show thatp∈Ω. First, by (45) and demi-closeness principle at zero of S, we have p∈FixS. On the contrary, since PQnAxn∈Qn and PQnAxn−Axn⟶0, we have(47)ϕPQnAxn,wn≤ϕPQnAxn,Axn⟶0.By Lemma1, it follows that(48)PQnAxn−wn=PQnAxn−TAxn⟶0.Hence,(49)Axn−TAxn≤Axn−PQnAxn+PQnAxn−TAxn⟶0.SinceA is bounded and linear, by (45), we can conclude that Axnk+1 converges weakly to Ap∈E2. By (49) and demi-closedness principle of T, we obtain that Ap∈FixT. Hence, p∈Ω. Therefore, by (45) and Lemma 3,(50)lim supn⟶∞x∗−xn+1,J1x∗=lim supn⟶∞x∗−zn,J1x∗=x∗−p,J1x∗≤0.Finally, the conclusionxn−x∗⟶0 follows from the hypothesis on αn, (33), (43), (50), and Lemma 4.Case 2. Suppose that there exists a subsequence ni of n such that(51)ϕx∗,xni<ϕx∗,xni+1,for alli∈ℕ.Then, by Lemma5, there exists a nondecreasing sequence mk⊂ℕ such that mk⟶∞:(52)ϕx∗,xmk≤ϕx∗,xmk+1andϕx∗,xk≤ϕx∗,xmk+1,∀k≥1.Replacingn with mk in (38), by (52), we have(53)1−αmkPQmk+1−IAxmk42k2A∗J2PQmk+1−IAxmk2+βmk1−βmkgJ1zmk−J1Szmk≤ϕx∗,xmk−ϕx∗,xmk+1+αmkτmkx∗2≤αmkτmkx∗2.Then, by a similar process with proving (43)–(50), we can obtain that(54)limk⟶∞J1xmk−J1ymk=0andlim supn⟶∞x∗−xmk+1,J1x∗≤0.Replacingn with mk in (33), we have(55)ϕx∗,xmk+1≤1−αmkτmkϕx∗,xmk+2αmkτmkx∗−xmk+1,J1x∗+1−αmkJ1ymk−J1xmk,from which we obtain(56)αmkτmkϕx∗,xmk≤ϕx∗,xmk−ϕx∗,xmk+1+2αmkτmkx∗−xmk+1,J1x∗+1−αmkJ1ymk−J1xmk≤2αmkτmkx∗−xmk+1,J1x∗+1−αmkJ1ymk−J1xmk.Sinceαmkτmk>0, by (54) and (56), we have(57)ϕx∗,xmk≤2x∗−xmk+1,J1x∗+1−αmkJ1ymk−J1xmk⟶0.Furthermore, by (54), (55), and (57), it follows that(58)limk⟶∞ϕx∗,xmk+1=0.However,ϕx∗,xk≤xmk+1−x∗ for all k≥1. So, we conclude that ϕx∗,xk⟶0 as k⟶∞ and hence xk−x∗⟶0 as k⟶∞ by Lemma 1. The proof is complete.Remark 1.
IfPQn−IAxn=0 for all n≥1, then γn=0 and zn=xn for all n≥1. In this case, Axn=PQnAxn and ϕAxn,wn=ϕAxn,TAxn≤ϕAxn,Axn=0, which implies that Axn=TAxn for all n≥1. The iterative scheme (25) becomes(59)yn=J1−1βnJ1xn+1−βnJ1Sxn,xn+1=J1−1αn1−τnJ1xn+1−αnJ1yn,n≥1.
By the proof process above, we still can see thatxn converges strongly to x∗=PFixSθ. Since A is linear and bounded, Axn⟶Ax∗, which implies that Axn⇀x∗. Note that Axn=TAxn, for all n≥1, and Axn−TAxn⟶0 as n⟶∞. By the hypothesis that I−T is demi-closedness at zero, we get Ax∗=TAx∗. Hence, x∗∈Ω. Hence, without loss generality, we assume that γn≠0 for all n≥1 in the proof process.Algorithm 2.
Takeu=x1∈E1, and define a sequence xn by(60)wn=TAxn,Qn=w∈E2:ϕw,wn≤ϕw,Axn,zn=J1−1J1xn−γnA∗J2I−PQnAxn,yn=J1−1βnJ1zn+1−βnJ1Szn,xn+1=J1−1αnJ1u+1−αnJ1yn,n≥1,where αn,βn⊂0,1 and(61)γn=PQn−IAxn22k2A∗J2I−PQnAxn2,PQn−IAxn≠0,0,otherwise.Lemma 12.
xn is well-defined and bounded.Proof.
By a similar proof lines of Lemma10, we can show that xn is well-defined. Now, we prove that xn is bounded. By (29)–(31), (60), and Lemma 5, for any x^∈Ω, we have(62)ϕx^,xn+1=ϕx^,J1−1αnJ1u+1−αnJ1yn=x^2−2αnx^,J1u−21−αnx^,J1yn+αnJ1u+1−αnJ1yn2≤x^2−2αnx^,J1u−21−αnx^,J1yn+αnu2+1−αnyn2=αnϕx^,u+1−αnϕx^,yn≤αnϕx^,u+1−αnϕx^,xn−PQn−IAxn42k2A∗J2PQn−IAxn2−βn1−βngJ1zn−J1Szn≤αnϕx^,u+1−αnϕx^,xn≤αnϕx^,u+ϕx^,xn,n≥1.
By the hypothesis onαn and Lemma 9, it follows that the limit of ϕx^,xn exists. Hence, xn is bounded.Theorem 2.
Assume thatS and T are closed. If the interior of Ω is nonempty and αn and βn satisfy the following conditions(63)∑n=1∞αn<∞andlim infn⟶∞βn1−βn>0,then xn generated by Algorithm 2 converges strongly to the element x∗=limn⟶∞ΠΩxn.Proof.
We first show thatxn is a Cauchy sequence and hence converges strongly to some point x∗∈E1. Since the interior of Ω is nonempty, there exist p∈Ω and r>0 such that(64)p+rh∈Ω,whenever h≤1. By (10), we have(65)ϕp,xn=ϕxn+1,xn+ϕp,xn+1+2xn+1−p,J1xn−J1xn+1=ϕxn+1,xn+ϕp,xn+1+2xn+1−p+rh,J1xn−J1xn+1+2rh,J1xn−J1xn+1.
On the contrary, by (10), again we have(66)ϕp+rh,xn=ϕxn+1,xn+ϕp+rh,xn+1+2xn+1−p+rh,J1xn−J1xn+1.
Combining (65) with (66), we obtain(67)2rh,J1xn−J1xn+1=ϕp,xn−ϕxn+1,xn+ϕp,xn+1+2xn+1−p+rh,J1xn−J1xn+1=ϕp,xn−ϕxn+1,xn−ϕp,xn+1−ϕp+rh,xn+ϕxn+1,xn+ϕp+rh,xn+1=12rϕp,xn−ϕp,xn+1+ϕp+rh,xn+1−ϕp+rh,xn.
Sincep+rh∈Ω, from (62) and (67), it follows that(68)2rh,J1xn−J1xn+1≤ϕp,xn−ϕp,xn+1+αnϕp+rh,u−ϕp+rh,xn≤ϕp,xn−ϕp,xn+1+αnϕp+rh,u.
Sinceh with h≤1 is arbitrary, we have(69)J1xn−J1xn+1≤12rϕp,xn−ϕp,xn+1+αnϕp+rh,u.
So, for allm>n,(70)J1xn−J1xm=J1xn−J1xn+1+J1xn+1−⋯−J1xm−1+J1xm−1+J1xm≤∑i=nm−1J1xi−J1xi+1≤12r∑i=nm−1ϕp,xi−ϕp,xi+1+αiϕp+rh,u=12r∑i=nm−1ϕp,xi−ϕp,xi+1+ϕp+rh,u2r∑i=nm−1αi=ϕp,xn−ϕp,xm+ϕp+rh,u2r∑i=nm−1αi.
Since the limit ofϕp,xn exists and ∑n=1∞αn<∞, from (70), we see(71)limm,n⟶∞J1xn−J1xm=0,which implies that J1xn is a Cauchy sequence in E1∗. Hence, J1xn converges strongly to some point in E1∗. Since E1∗ has a Fréchet differentiable norm, then J1−1 is continuous on E1∗. Hence, xn converges strongly to some point x∗ in E1.
For anyx^∈Ω, by (62), we have(72)1−αnPQn−IAxn42k2A∗J2PQn−IAxn2+βn1−βngJ1zn−J1Szn≤αnϕx^,u+1−αnϕx^,xn−ϕx^,xn+1≤αnϕx^,u+ϕx^,xn−ϕx^,xn+1.
Since the limit ofϕx^,xn exists, by the hypothesis on αn and βn, it follows that(73)limn⟶∞PQn−IAxn4A∗J2PQn−IAxn2=limn⟶∞gJ1zn−J1Szn=0,which implies that(74)limn⟶∞PQn−IAxn=0andlimn⟶∞J1zn−J1Szn=0,and hence(75)zn−Szn⟶0.
On the contrary, by (60) and (73), we have(76)J1zn−J1xn=γnA∗J2I−PQnAxn=I−PQnAxn2A∗J2I−PQnAxn⟶0.
It follows that(77)zn−xn⟶0.
Hence,zn converges strongly to x∗∈E1. Since S is closed, by (75), we get x∗=Sx∗.
Now, we show thatAx∗=TAx∗. From (49), it follows that Axn−TAxn⟶0. Since A is linear bounded, Axn⟶Ax∗. From the closedness of T, we get Ax∗=TAx∗. Therefore, x∗∈Ω. Finally, we show that x∗=limn⟶∞ΠΩxn. In fact, since x∗∈Ω, by Lemma 2, we have(78)ϕx∗,ΠΩxn≤ϕx∗,xn⟶0.
It follows thatx∗=limn⟶∞ΠΩxn. The proof is complete.
LetQ be a nonempty closed convex subset of E2. In Algorithms 1 and 2, if putting T=I and Q1=Q, we have wn=Axn and Qn=Q for all n≥1. Then, we have the following results.Corollary 1.
LetE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflexive Banach space with a nonempty closed convex subset Q⊂E2. Let A:E1⟶E2 be a linear bounded operator with adjoint A∗. Let S:E1⟶E1 and Q⊂E2 be a nonempty subset. Assume that I−S is demi-closedness at zero and Γ≠∅, where Γ=x∈E1:x∈FixS,Ax∈Q. Let x1∈E1 and define a sequence xn by(79)zn=J1−1J1xn−γnA∗J2I−PQAxn,yn=J1−1βnJ1zn+1−βnJ1Szn,xn+1=J1−1αn1−τnJ1xn+1−αnJ1yn,n≥1,where αn,βn⊂0,1,τn⊂τ,1 with τ∈0,1 and(80)γn=PQn−IAxn22k2A∗J2I−PQnAxn2,PQn−IAxn≠0,0,otherwise.
If the following conditions hold,(81)limn⟶∞αn=0,∑n=1∞αn=∞andlim infn⟶∞βn1−βn>0,then the sequence xn generated by (60) converges strongly to the element x∗=ΠΓθ, where θ is the zero element in E1.Corollary 2.
LetE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflexive Banach space with a nonempty closed convex subset Q⊂E2. Let A:E1⟶E2 be a linear bounded operator with adjoint A∗. Let S:E1⟶E1 and Q⊂E2 be a nonempty subset. Assume that S is closed and the interior of Γ is nonempty, where Γ=x∈E1:x∈FixS,Ax∈Q. Let u=x1∈E1 and define a sequence xn by(82)zn=J1−1J1xn−γnA∗J2I−PQAxn,yn=J1−1βnJ1zn+1−βnJ1Szn,xn+1=J1−1αnJ1u+1−αnJ1yn,n≥1,where αn,βn⊂0,1 and γn=PQn−IAxn2/2k2A∗J2I−PQnAxn2,PQn−IAxn≠0,0,otherwise.
If the following conditions hold(83)limn⟶∞αn=0,∑n=1∞αn<∞andlim infn⟶∞βn1−βn>0,then the sequence xn generated by (82) converges strongly to some element x∗=limn⟶∞ΠΓxn.


## 4. Application

LetE1 and E2 be two Banach spaces and f1:E1×E1⟶ℝ and f2:E2×E2⟶ℝ be the bifunctions. Let A:E1⟶E2 be a linear bounded operator. In this section, we consider a split equilibrium problem: find a point x∗∈E1 such that(84)x∗∈EPf1andAx∗∈EPf2,where EPf1=x∈E1:f1x,y≥0,∀y∈E1 and EPf2=u∈E2:f2u,v≥0,∀v∈E2. We denote the set of solution of problem (84) by Λ. That is, Λ=x∈EPf1:Ax∈EPf2.The split equilibrium problem has been studied by many authors in Hilbert space, see [37–41]. However, few results on the split equilibrium problem in Banach space is reported by far.Lemma 13 (see [24]).
LetE be a strictly convex, reflexive, and uniform smooth Banach space and f:E×E⟶ℝ be a bifunction satisfying the following conditions:(A1)
fx,x=0 for all x∈E.(A2)
f is monotone, i.e., fx,y+fy,x≤0 for all x,y∈E.(A3)
For allx,y,z∈E,(85)lim supt⟶0+ftz+1−tx,y≤fx,y.(A4)
For allx∈E, fx,· is convex and lower semicontinuous.Forr>0 and x∈E, define a mapping Tr:E⟶E as follows:(86)Trfx=z∈E:fz,y+1ry−z,Jz−Jx≥0for ally∈E,for allx∈E. Then, the following hold:(1)
Trf is single-valued(2)
FixTrf=EPf(3)
EPf is closed and convex(4)
ϕq,Trfx+ϕTrfx,x≤ϕq,x for all x∈E and q∈EPf, which shows that Trf is a quasi-ϕ-nonexpansive mappingNow, we show that the mappingI−Trf is demi-closedness at zero on a bounded subset of E.Lemma 14.
LetE be a strictly convex, reflexive, and uniform smooth Banach space and f:E×E⟶ℝ be a bifunction satisfying conditions (A1)–(A4). Let r>0 and define the mapping Trf as (86). Assume that EPf≠∅. Then, I−Trf is demi-closedness at zero on a bounded set. That is, if xn⊂E is bounded and weakly converges to x∈E and xn−Trfxn⟶0 as n⟶∞, then x=Trfx.Proof.
etxn⊂E be bounded and converges weakly to x∈E and xn−Trfxn⟶0 as n⟶∞. For each x∗∈EPf=FixTrf, since Trf is quasi-ϕ-nonexpansive, we have(87)ϕx∗,Trfxn≤ϕx∗,xn,n≥1,which implies that Trfxn is bounded. On the contrary, since J is uniformly norm-to-norm continuous on bounded sets, it follows that(88)limn⟶∞JTrfxn−Jxn=0.
By (A2), we have(89)1ry−Trfxn,JTrfxn−Jxn≥−fTrfxn,y≥fy,Trfxn,∀y∈E.
Lettingn⟶0 in (89), by (A4) and (88), we obtain(90)fy,x≤0,∀y∈E.
For0<t≤1 and y∈E, let yt=ty+1−tx. Note that (90) implies that fyt,x≤0. By (A1), we have(91)0=fyt,yt≤tfyt,y+1−tfyt,x≤tfyt,y.
Dividing byt, we obtain(92)fyt,y≥0,∀y∈E.
Lett⟶0+, by (A3), we have(93)fx,y≥0,∀y∈E.
It follows thatx∈EPf. That is, x=Trfx by Lemma 13. This completes the proof.
Based on the results in Section3, we give the following conclusion directly.Theorem 3.
LetE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflexive Banach space. Let A:E1⟶E2 be a linear bounded operator with adjoint A∗. Let f1:E1×E1⟶ℝ and f2:E2×E2⟶ℝ be the bifunctions satisfying conditions (A1)–(A4). Assume that Λ≠∅, where Λ=x∈E1:x∈EPf1,Ax∈EPf2. Let r>0. Take x1∈E1 and put Q1=E2. Define a sequence xn by(94)wn=Trf2Axn,Qn=w∈Qn:ϕw,wn≤ϕw,Axn,zn=J−1J1xn+γnA∗J2PQn−IAxn,yn=J−11−βnJ1zn+1−βnJ1Trf1zn,xn+1=J1−1αn1−τnJ1xn+1−αnJ1yn,n≥1,where αn,βn⊂0,1,τn⊂τ,1 with τ∈0,1 and(95)γn=PQn−IAxn22k2A∗J2I−PQnAxn2,PQn−IAxn≠0,0,else.
If the following conditions hold(96)limn⟶∞αn=0,∑n=1∞αn=∞andlim infn⟶∞βn1−βn>0,then the sequence xn generated by (94) converges strongly to the element x∗=ΠΛθ, where θ is the zero element in E1.Theorem 4.
LetE1 be a 2-uniformly convex and 2-uniformly smooth real Banach space with the best smoothness constant k>0 and E2 be a uniformly smooth, strictly convex, and reflexive Banach space. Let A:E1⟶E2 be a linear bounded operator with adjoint A∗. Let f1:E1×E1⟶ℝ and f2:E2×E2⟶ℝ be the bifunctions satisfying conditions (A1)–(A4). Assume that the interior of Λ is nonempty, where Λ=x∈E1:x∈EPf1,Ax∈EPf2. Let r>0. Take u,x1∈E1 and put Q1=E2. Define a sequence xn by(97)wn=Trf2Axn,Qn=w∈Qn:ϕw,wn≤ϕw,Axn,zn=J−1J1xn+γnA∗J2PQn−IAxn,yn=J−11−βnJ1zn+1−βnJ1Trf1zn,xn+1=J1−1αnJ1u+1−αnJ1yn,n≥1,where αn,βn⊂0,1 and(98)γn=PQn−IAxn22k2A∗J2I−PQnAxn2,PQn−IAxn≠0,0,else.
If the following conditions hold(99)limn⟶∞αn=0,∑n=1∞αn<∞andlim infn⟶∞βn1−βn>0,then the sequence xn generated by (97) converges strongly to the element x∗=limn⟶∞ΠΛxn.


## 5. Numerical Examples

In this section, we give the following examples to illustrate the effectiveness of Algorithms1 and 2. The program is performed by Matlab R2016b running on a PC Desktop with Core(TM) i5CPU M550 3.20 GHz with 4 GB Ram.We first show the convergence of Algorithm1 by the following example which has been used by Ma et al. [22]. In [22], the authors compare the computed results using their algorithm (25) with algorithm (100) in Kraikaew and Saejung [10] by the example. Here, we also compare the convergence of our Algorithm 1 with algorithm (25) in [22] and algorithm (100) in [10].Example 1.
LetE1=ℝ, E2=ℝ2, Q=0,∞×−∞,0, Sx=x/4, for all x∈E1, Tx=PQx for all x∈E2, where PQ is the metric projection from E2 onto Q, and A:E1⟶E2 be a mapping defined by Ax=x/2,x/3 for all x∈E1. Then, A∗u,v=u/2+v/3, for all u,v∈E2. It is easy to see that Ω=x∈E1:x∈FixS,Ax∈FixT=0.Algorithm 3.
Letxn be the sequence generated by (25) in this paper with αn=1/2n and βn=τn=6/7. Then, scheme (25) can be simplified as(100)x1∈E1,wn=PQxn2,xn3,Qn=w∈E2:wn−w≤xn2,xn3−w,Axn=xn2,xn3,zn=xn+γnA∗PQn−IAxn,yn=67zn+128zn,xn+1=114nxn+2n−12nyn,n≥1,where(101)γn=PQn−IAxn22A∗I−PQnAxn2,ifPQn−IAxn≠0,0,otherwise.Algorithm 4.
Letxn be the sequence generated by algorithm (100) in [10] with αn=1/2n and γ=1. Then, scheme (100) in [10] can be simplified as(102)x1∈E1,xn+1=12nx1+2n−18nxn+A∗T−IAxn,n≥1.Algorithm 5.
Letxn be the sequence generated by algorithm (25) in [22] with αn=1/2n and γ=1. Then, scheme (25) in [22] can be simplified as(103)x1∈E1,Axn=xn2,xn3,zn=xn+A∗T−IAxn,yn=2n−12nzn+18nzn,Cn+1=vn:Cn:yn−v≤xn−v,zn−v≤xn−v,xn+1=PCn+1x1,n≥1.
We perform schemes (100)–(103) with the different initial points. Figures 1–4 show that the sequence xn generated by (100)–(103) converge to 0.Figure 1
Convergence for Algorithms3–5 with different initial points x1=3.Figure 2
Convergence for Algorithms3–5 with different initial points x1=7.Figure 3
Convergence for Algorithms3–5 with different initial points x1=−3.Figure 4
Convergence for Algorithms3–5 with different initial points x1=−7.Remark 2.
(a) Although Theorem1 in [22] requires that αn in Algorithm 5, i.e., algorithm (25) in [22], takes values in δ,1 with δ∈0,1; here, for comparing the convergence rate of three schemes, we put the same αn=1/2n. This does not affect the effectiveness of Algorithm 5 since the program stops in finite iterations. (b) Figures 1–4 above show that the convergence rate of Algorithm 3 is faster than that of Algorithms 4 and 5.
Next, we illustrate Theorem2 by the following example.Example 2.
LetE1=ℝ2 and E2=ℝ. Define the mappings S:E1⟶E1 by Sx=x1/2,x2 for all x=x1,x2∈E1, and T:E2⟶E2 by Tx=x/2 if x≤1 and Tx=1 if x>1.
LetA:E1⟶E2 be a mapping defined by Ax=x1 for all x=x1,x2∈E1. Then, A is linear and bounded and A∗y=y,0 for all y∈E2. It is easy to see that Ω=0,x2:x2∈ℝ. All the conditions on S,T,andΩ are satisfied for Theorem 2.
By Algorithm2, we generate a sequence xn with αn=1/n2 and βn=1/21−en/2 for all n≥1. Theorem 2 shows that xn will converge to the point PΩxn. We will stop the program when xn−PΩxn<10−4. The computed results of the sequence xn are given in Tables 1 and 2. Figures 5 and 6 show the convergence of the sequence xn.Table 1
Convergence for Algorithm2 with initial point x1=3,6.
Iteration stepsxn1−2,−52−1.48705,−5.000003−1.13985,−4.999994−0.89031,−4.999995−0.70347,−4.99999⋮⋮100−0.00085,−4.99999261−0.00012,−4.99998262−0.00012,−4.99998263−0.00011,−4.99998264−0.00011,−4.99998265−0.00011,−4.99998⋮⋮286−0.00009,−4.99998Table 2
Convergence for Algorithm2 with initial point x1=3,6.
Iteration stepsxn13,622.23056,0.9999931.70977,1.5555541.33545,1.83331⋮⋮1000.00128,1.999851010.00125,2.000241020.00122,2.000223000.00014,1.999973010.00013,2.000023020.00013,2.000033030.00013,1.99997⋮⋮3490.00010,2.000023500.00010,2.000003510.00009,2.00003Figure 5
Convergence for Algorithm2 with different initial points x1=−2,−2.Figure 6
Convergence for Algorithm2 with different initial points x1=3,6.


## 6. Conclusion

For finding a solution of the split common fixed problem of quasi-ϕ-nonexpansive mappings in Banach space, we introduced a Halpern algorithm and a nonconvex combination algorithm where the norm of the linear bounded operator does not need to be known in advance. The convergence of the algorithms was investigated and some numerical examples were given to illustrate the convergence of the algorithms.



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