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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 536283, 20 pages
http://dx.doi.org/10.1155/2012/536283
Research Article

An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed and πœ™-Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 22 April 2012; Revised 4 July 2012; Accepted 29 July 2012

Academic Editor: SimeonΒ Reich

Copyright Β© 2012 Kasamsuk Ungchittrakool. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and πœ™-quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (𝐾). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

1. Introduction

It is well known that, in an infinite-dimensional Hilbert space, the normal Mann’s iterative algorithm [1] has only weak convergence, in general, even for nonexpansive mappings [2, 3]. Consequently, in order to obtain strong convergence, Nakajo and Takahashi [4] modified the normal Mann’s iteration algorithm, later well known as a hybrid projection iteration method. Since then, the hybrid method has undergone rapid developments. For the details, the readers are referred to papers [5–9] and the references therein.

Let 𝐸 be a smooth Banach space, and let πΈβˆ— be the dual of 𝐸. The function πœ™βˆΆπΈΓ—πΈβ†’β„ is defined by πœ™(𝑦,π‘₯)=‖𝑦‖2βˆ’2βŸ¨π‘¦,𝐽π‘₯⟩+β€–π‘₯β€–2(1.1) for all π‘₯,π‘¦βˆˆπΈ, which was studied by Alber [10], Kamimura and Takahashi [11], and Reich [12], where 𝐽 is the normalized duality mapping from 𝐸 to 2πΈβˆ— defined by 𝐽(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2ξ€Ύ,(1.2) where βŸ¨β‹…,β‹…βŸ© denotes the duality paring. It is well known that if 𝐸 is smooth, then 𝐽 is single valued and if 𝐸 is strictly convex, then 𝐽 is injective (one to one). Let 𝐢 be a closed convex subset of 𝐸, and let 𝑇 be a mapping from 𝐢 into itself. We denote by 𝐹(𝑇) the set of fixed points of 𝑇. A point 𝑝 in 𝐢 is said to be an asymptotic fixed point of 𝑇 [12] if 𝐢 contains a sequence {π‘₯𝑛}, which converges weakly to 𝑝 such that the strong limπ‘›β†’βˆž(π‘₯π‘›βˆ’π‘‡π‘₯𝑛)=0. The set of asymptotic fixed points of  𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐢 into itself is called nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β©½β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπΆ and relatively nonexpansive [13–16] if 𝐹(𝑇)=𝐹(𝑇) and πœ™(𝑝,𝑇π‘₯)β©½πœ™(𝑝,π‘₯) for all π‘₯∈𝐢 and π‘βˆˆπΉ(𝑇). The asymptotic behavior of relatively nonexpansive mapping was studied in [13–16].

In 2005, Matsushita and Takahashi [16] proposed the hybrid iteration method with generalized projection for relatively nonexpansive mapping 𝑇 in the framework of uniformly smooth and uniformly convex Banach spaces 𝐸 as follows: π‘₯0∈𝐢chosenarbitrarily,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘‡π‘₯𝑛,𝐢𝑛=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,𝑄𝑛=ξ€½π‘§βˆˆπΆβˆΆβŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯0βˆ’π½π‘₯𝑛,π‘₯⟩β‰₯0𝑛+1=Ξ πΆπ‘›βˆ©π‘„π‘›ξ€·π‘₯0ξ€Έ,(1.3) where 𝐽 is the duality mapping on 𝐸 and Π𝐢(β‹…) is the generalized projection from 𝐸 onto a nonempty closed convex subset 𝐢. Based on the guidelines of Matsushita and Takahashi [16], Plubtieng and Ungchittrakool [17, 18] studied and developed (1.3) to the case of two relatively nonexpansive mappings and finite family of relatively nonexpansive mappings, respectively.

On the other hand, for a real Banach space 𝐸 and the dual πΈβˆ—, let 𝐢 be a nonempty closed convex subset of 𝐸. Let Ξ˜βˆΆπΆΓ—πΆβ†’β„ a bifunction, πœ‘βˆΆπΆβ†’β„ a real-valued function, and π΄βˆΆπΆβ†’πΈβˆ— be a nonlinear mapping. The generalized mixed equilibrium problem is to find π‘₯∈𝐢 such that Θ(π‘₯,𝑦)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.4) The solution set of (1.4) is denoted by GMEP(Θ,𝐴,πœ‘), that is, GMEP(Θ,𝐴,πœ‘)={π‘₯∈𝐢∢Θ(π‘₯,𝑦)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ}.(1.5) If 𝐴=0, the problem (1.4) reduces to the mixed equilibrium problem for Θ, denoted by MEP(Θ,πœ‘), which is to find π‘₯∈𝐢 such that Θ(π‘₯,𝑦)+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.6) If Θ=0, the problem (1.4) reduces to the mixed variational inequality of Browder type, denoted by VI(𝐢,𝐴,πœ‘), which is to find π‘₯∈𝐢 such that ⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.7) If 𝐴=0 and πœ‘=0, the problem (1.4) reduces to the equilibrium problem for Θ (for short, EP), denoted by EP(Θ), which is to find π‘₯∈𝐢 such that Θ(π‘₯,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.8) If Θ=0 and 𝐴=0, the problem (1.4) reduces to the minimization problem for πœ‘, denoted by Argmin(πœ‘), which is to find π‘₯∈𝐢 such that πœ‘(π‘₯)β‰€πœ‘(𝑦),βˆ€π‘¦βˆˆπΆ.(1.9)

The above formulation (1.7) was shown in [19] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of (1.8). However, (1.4) is very general, it covers the problems mentioned above as special cases.

In 2007, S. Takahashi and W. Takahashi [20] and Tada and Takahashi [21, 22] proved weak and strong convergence theorems for finding a common element of the set of solution of an equilibrium problem (1.8) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [9] studied a strong convergence theorem by the hybrid method for a family of nonexpansive mappings in Hilbert spaces as follows: π‘₯0∈𝐻, 𝐢1=𝐢 and π‘₯1=𝑃𝐢1π‘₯0, and let 𝑦𝑛=𝛼𝑛π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘›π‘₯𝑛,𝐢𝑛+1=ξ€½π‘§βˆˆπΆπ‘›βˆΆβ€–β€–π‘¦π‘›β€–β€–β‰€β€–β€–π‘₯βˆ’π‘§π‘›β€–β€–ξ€Ύ,π‘₯βˆ’π‘§π‘›+1=𝑃𝐢𝑛+1π‘₯0,π‘›βˆˆβ„•,(1.10) where 0β©½π›Όπ‘›β©½π‘Ž<1 for all π‘›βˆˆβ„• and {𝑇𝑛} is a sequence of nonexpansive mappings of 𝐢 into itself such that β‹‚βˆžπ‘›=1𝐹(𝑇𝑛)β‰ βˆ…. They proved that if {𝑇𝑛} satisfies some appropriate conditions, then {π‘₯𝑛} converges strongly to π‘ƒβ‹‚βˆžπ‘›=1𝐹(𝑇𝑛)π‘₯0.

Motivated by Takahashi et al. [9], Takahashi and Zembayashi [23] (see also [24]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping 𝑆 in the framework of uniformly smooth and uniformly convex Banach space as follows: π‘₯0=π‘₯,𝐢0𝑦=𝐢,𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘†π‘₯𝑛,π‘’π‘›βˆˆπΆsuchthatΞ˜ξ€·π‘’π‘›ξ€Έ+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯,(1.11) where Π𝐢𝑛+1(β‹…) is the generalized projection from 𝐸 onto 𝐢𝑛+1. Under some appropriate assumptions on Θ, {𝛼𝑛}, and {π‘Ÿπ‘›}, they proved that the sequence {π‘₯𝑛} converges strongly to Π𝐹(𝑆)∩EP(Θ)(π‘₯0).

In 2010, Zhou and Gao [25] introduced the definition of a quasi-strict pseudocontraction related to the function πœ™ and proved a hybrid projection algorithm for finding a fixed point of a closed and quasi-strict pseudocontraction in more general framework than uniformly smooth and uniformly convex Banach spaces as follows:π‘₯0∈𝐸,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthat𝐢𝑛+1=ξ‚†π‘§βˆˆπΆπ‘›ξ€·π‘₯βˆΆπœ™π‘›,𝑇π‘₯𝑛≀21βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯π‘›βŸ©ξ‚‡,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ,(1.12) where Π𝐢𝑛+1 is the generalized projection from 𝐸 onto 𝐢𝑛+1. They proved that the sequence {π‘₯𝑛} converges strongly to Π𝐹(𝑇)(π‘₯0).

Motivated and inspired by the above research work, in this paper, we provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Furthermore, by employing the inequality that appeared in Lemma 2.10 together with (1.12) and some facts of Zhou and Gao [25], we create an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and πœ™-quasi-strict pseudocontractions in the framework of reflexive, strictly convex, and smooth Banach spaces with the property (𝐾). The results of this research improve and extend the corresponding results of Zhou and Gao [25] and many others.

2. Preliminaries

In this paper, we denote by 𝐸 and πΈβˆ— a real Banach space and the dual space of 𝐸, respectively. Let 𝐢 be a nonempty closed convex subset of 𝐸. We denote by 𝐽 the normalized duality mapping from 𝐸 to 2πΈβˆ— defined by (1.1). Let 𝑆(𝐸)∢={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–=1} be the unit sphere of 𝐸. Then, a Banach space 𝐸 is said to be strictly convex β€–(π‘₯+𝑦)/2β€–<1 for all π‘₯,π‘¦βˆˆπ‘†(𝐸) and π‘₯≠𝑦. It is also said to be uniformly convex if limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=0 for any two sequences {π‘₯𝑛}and{𝑦𝑛} in 𝑆(𝐸) such that limπ‘›β†’βˆžβ€–(π‘₯𝑛+𝑦𝑛)/2β€–=1. The Banach space 𝐸 is said to be smooth provided lim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.1) exists for each π‘₯,π‘¦βˆˆπ‘†(𝐸). In this case, the norm of 𝐸 is said to be GΓ’teaux differentiable. The norm of 𝐸 is said to be FrΓ©chet differentiable if for each π‘₯βˆˆπ‘†(𝐸), the limit (2.1) is attained uniformly for π‘¦βˆˆπ‘†(𝐸). The norm of 𝐸 is said to be uniformly FrΓ©chet differentiable (and 𝐸 is said to be uniformly smooth) if the limit (2.1) is attained uniformly for π‘₯,π‘¦βˆˆπ‘†(𝐸).

A Banach space 𝐸 is said to have the property (K) (or Kadec-Klee property) if for any sequence {π‘₯𝑛}βŠ‚πΈ such that π‘₯𝑛→π‘₯ weakly and β€–π‘₯𝑛‖→‖π‘₯β€–, then β€–π‘₯π‘›βˆ’π‘₯β€–β†’0. For more information concerning property (𝐾) the reader is referred to [26] and references cited there.

A mapping π‘‡βˆΆπΆβ†’πΆ is said to be closed if for any sequence {π‘₯𝑛}βŠ‚πΆ with π‘₯𝑛→π‘₯ and 𝑇π‘₯𝑛→𝑦, π‘₯=𝑦.

We also know the following properties (see [27–29] for details):(i)if 𝐸 is smooth (β‡”πΈβˆ— is strictly convex), then 𝐽 is single-valued,(ii)if 𝐸 is strictly convex (β‡”πΈβˆ— is smooth), then 𝐽 is one-to-one (i.e., 𝐽(π‘₯)∩𝐽(𝑦)=βˆ… for all π‘₯≠𝑦),(iii)if 𝐸 is reflexive (β‡”πΈβˆ— is reflexive), then 𝐽 is surjective,(iv)if πΈβˆ— is smooth and reflexive, then π½βˆ’1βˆΆπΈβˆ—β†’2𝐸 is single-valued and demicontinuous (i.e., if {π‘₯βˆ—π‘›}βŠ‚πΈβˆ— such that π‘₯βˆ—π‘›β†’π‘₯βˆ—, then π½βˆ’1(π‘₯βˆ—π‘›)β†’π½βˆ’1(π‘₯βˆ—)),(v)𝐸 is uniformly smooth if and only if πΈβˆ— is uniformly convex,(vi)if 𝐸 is uniformly convex, then(a)it is strictly convex,(b)it is reflexive,(c)it satisfies the property (𝐾),(vii)if 𝐸 is a Hilbert space, then 𝐽 is the identity operator.

It is also very well known that if 𝐢 is a nonempty closed convex subset of a Hilbert space 𝐻 and π‘ƒπΆβˆΆπ»β†’πΆ is the metric projection of 𝐻 onto 𝐢, then 𝑃𝐢 is nonexpansive. This fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [10] recently introduced a generalized projection operator Π𝐢 in a Banach space 𝐸 that is an analogue of the metric projection in Hilbert spaces.

Let 𝐢 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, for any π‘₯∈𝐸. The generalized projection Ξ πΆβˆΆπΈβ†’πΆ is a map that assigns to an arbitrary point π‘₯∈𝐸 the minimum point of the functional πœ™(π‘₯,𝑦), that is, Π𝐢π‘₯=π‘₯, where π‘₯ is the solution to the minimization problem πœ™ξ€·ξ€Έπ‘₯,π‘₯=minπ‘¦βˆˆπΆπœ™(𝑦,π‘₯).(2.2) Existence and uniqueness of the operator Π𝐢 follow from the properties of the functional πœ™(π‘₯,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [10, 11, 27, 29, 30]). In Hilbert spaces, πœ™(𝑦,π‘₯)=β€–π‘¦βˆ’π‘₯β€–2 and Π𝐢 coincides with the metric projection 𝑃𝐢.

It is obvious from the definition of function πœ™ that ()β€–π‘¦β€–βˆ’β€–π‘₯β€–2)β‰€πœ™(𝑦,π‘₯)≀(‖𝑦‖+β€–π‘₯β€–2,βˆ€π‘₯,π‘¦βˆˆπΈ,(2.3)πœ™(π‘₯,𝑦)=πœ™(π‘₯,𝑧)+πœ™(𝑧,𝑦)+2⟨π‘₯βˆ’π‘§,π½π‘§βˆ’π½π‘¦βŸ©,βˆ€π‘₯,𝑦,π‘§βˆˆπΈ.(2.4)

Remark 2.1. If 𝐸 is a reflexive strictly convex and smooth Banach space, then, for π‘₯,π‘¦βˆˆπΈ,πœ™(π‘₯,𝑦)=0 if and only if π‘₯=𝑦. It is sufficient to show that if πœ™(π‘₯,𝑦)=0, then π‘₯=𝑦. From (2.3), we have β€–π‘₯β€–=‖𝑦‖. This implies ⟨π‘₯,π½π‘¦βŸ©=β€–π‘₯β€–2=‖𝐽𝑦‖2. From the definitions of 𝐽, we have 𝐽π‘₯=𝐽𝑦. That is, π‘₯=𝑦; one may consult [27, 29] for the details.
𝑇 is said to be a πœ™-quasi-strict pseudocontraction [25, page 230] if there exists a constant π‘˜βˆˆ[0,1) and 𝐹(𝑇)β‰ βˆ… such that πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯)+π‘˜πœ™(π‘₯,𝑇π‘₯) for all π‘₯∈𝐢 and π‘βˆˆπΉ(𝑇). In particular, 𝑇 is said to be πœ™-quasi-nonexpansive if π‘˜=0 and 𝑇 is said to be πœ™-quasi-pseudocontractive if π‘˜=1.
Let 𝐢 be a nonempty closed convex subset of a real Banach space 𝐸 with the dual πΈβˆ—. A mapping 𝐴∢𝐷(𝐴)βŠ‚πΈβ†’πΈβˆ— is said to be monotone if, for each π‘₯,π‘¦βˆˆπ·(𝐴), the following inequality holds: ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β©Ύ0.(2.5)𝐴is said to be π‘Ÿ-inverse strongly monotone if there exists a positive real number π‘Ÿ such that ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β‰₯π‘Ÿβ€–π΄π‘₯βˆ’π΄π‘¦β€–2πΈβˆ—,βˆ€π‘₯,π‘¦βˆˆπ·(𝐴).(2.6)𝐴 is said to be π‘Ÿ-quasi inverse strongly monotone if π΄βˆ’1(0)={π‘§βˆˆπ·(𝐴)βˆΆπ΄π‘§=0}β‰ βˆ… and there exists a positive real number π‘Ÿ such that for each π‘’βˆˆπ΄βˆ’1(0), the following inequality holds: ⟨π‘₯βˆ’π‘’,𝐴π‘₯βŸ©β©Ύπ‘Ÿβ€–π΄π‘₯β€–2πΈβˆ—,βˆ€π‘₯∈𝐷(𝐴).(2.7)
Without loss of generality we can assume that the constant π‘Ÿβˆˆ(0,1/2] since if 𝐴 is Μ‚π‘Ÿ-quasi inverse strongly monotone (or Μ‚π‘Ÿ-inverse strongly monotone), then we can find π‘Ÿβˆˆ(0,1/2] such that Μ‚π‘Ÿβ‰₯π‘Ÿ and then Μ‚π‘Ÿβ€–π΄π‘₯β€–2πΈβˆ—β‰₯π‘Ÿβ€–π΄π‘₯β€–2πΈβˆ— (or Μ‚π‘Ÿβ€–π΄π‘₯βˆ’π΄π‘¦β€–2πΈβˆ—β‰₯π‘Ÿβ€–π΄π‘₯βˆ’π΄π‘¦β€–2πΈβˆ—) for all Μ‚π‘Ÿ>0 (i.e., 𝐴 is π‘Ÿ-quasi inverse strongly monotone).
The following is the example of πœ™-quasi-strict pseudocontractions in the framework of a smooth and strictly convex Banach space 𝐸.

Example 2.2. Let 𝐸 be a smooth and strictly convex Banach space 𝐸 with dual πΈβˆ— and 𝐽 a normalized duality mapping. Let π›Όβˆˆ[0,1], and define the mapping π΄π›ΌβˆΆπΈβ†’πΈ by 𝐴𝛼(π‘₯)=𝛼π‘₯,βˆ€π‘₯∈𝐸.(2.8) It follows that the mapping π½π΄π›ΌβˆΆπΈβ†’πΈβˆ— with the set of zero points is given as ξ€·π½π΄π›Όξ€Έβˆ’1ξ€½(0)=π‘§βˆˆπΈβˆΆ0=𝐽𝐴𝛼=ξ‚»(𝑧)=𝐽(𝛼𝑧)=𝛼𝐽𝑧{0},if0<𝛼≀1,𝐸,if𝛼=0.(2.9) Thus for each π‘’βˆˆ(𝐽𝐴𝛼)βˆ’1(0) and π‘Ÿβˆˆ(0,1/2] if 0<𝛼≀1, we obtain ⟨π‘₯βˆ’π‘’,𝐽𝐴𝛼π‘₯⟩=⟨π‘₯βˆ’0,𝐽𝐴𝛼π‘₯⟩=⟨π‘₯,𝐽(𝛼π‘₯)⟩=⟨π‘₯,𝛼𝐽(π‘₯)⟩=𝛼‖π‘₯β€–2β‰₯𝛼2β€–π‘₯β€–2𝐴=βŸ¨π›Όπ‘₯,𝐽(𝛼π‘₯)⟩=𝛼𝐴π‘₯,𝐽𝛼π‘₯=‖‖𝐴𝛼π‘₯β€–β€–2=‖‖𝐽𝐴𝛼π‘₯β€–β€–2πΈβˆ—β€–β€–β‰₯π‘Ÿπ½π΄π›Όπ‘₯β€–β€–2πΈβˆ—,βˆ€π‘₯∈𝐸.(2.10) On the other hand, if 𝛼=0, we have ⟨π‘₯βˆ’π‘’,𝐽𝐴𝛼π‘₯⟩=0=π‘Ÿβ€–0β€–2πΈβˆ—=π‘Ÿβ€–π½π΄π›Όπ‘₯β€–2πΈβˆ—,forallπ‘₯∈𝐸. Therefore, 𝐽𝐴𝛼 is π‘Ÿ-quasi inverse strongly monotone. So, we can define the following set: 𝐽𝒬=π΄βˆΆπΈβ†’πΈβˆ—βˆ£ξ‚¬ξ‚ξ‚­β€–β€–π½ξ‚β€–β€–π‘₯βˆ’π‘’,𝐽𝐴π‘₯β‰₯π‘Ÿπ΄π‘₯2πΈβˆ—ξ‚€π½ξ‚π΄ξ‚,βˆ€π‘₯∈𝐸,π‘’βˆˆβˆ’1ξ‚Ό,(0)(2.11) where ξ‚π΄βˆΆπΈβ†’πΈ and π‘Ÿβˆˆ(0,1/2] (note that π½π΄π›Όβˆˆπ’¬β‰ βˆ…). In the same way, let ξπ΄βˆΆπΈβ†’πΈ, and define 𝐽=.β„’=π΄βˆˆπ’¬βˆ£π½π‘₯βˆ’π΄π‘₯=𝐽π‘₯βˆ’π½π΄π‘₯,π‘₯,𝐽𝐴π‘₯𝐴π‘₯,𝐽π‘₯,βˆ€π‘₯∈𝐸(2.12)
We claim that π½π΄π›Όβˆˆβ„’. Note that π½π΄π›Όβˆˆπ’¬, and let us consider 𝐽π‘₯βˆ’π΄π›Όπ‘₯ξ€Έ=𝐽(π‘₯βˆ’π›Όπ‘₯)=(1βˆ’π›Ό)𝐽π‘₯=𝐽π‘₯βˆ’π›Όπ½π‘₯=𝐽π‘₯βˆ’π½(𝛼π‘₯)=𝐽π‘₯βˆ’π½π΄π›Όπ‘₯.(2.13) Furthermore, it is also found that ⟨π‘₯,𝐽𝐴𝛼π‘₯⟩=⟨π‘₯,𝐽(𝛼π‘₯)⟩=βŸ¨π›Όπ‘₯,𝐽π‘₯⟩=βŸ¨π΄π›Όπ‘₯,𝐽π‘₯⟩.(2.14) This means that π½π΄π›Όβˆˆβ„’β‰ βˆ…. So, we have the claim.
Now, we pick arbitrarily π½ξπ΄βˆˆβ„’ and claim that (πΌβˆ’π΄)βˆΆπΈβ†’πΈ is a πœ™-quasi-strict pseudocontraction (i.e., βˆƒπ‘˜βˆˆ[0,1) s.t. ξξπœ™(π‘ž,(πΌβˆ’π΄)π‘₯)β©½πœ™(π‘ž,π‘₯)+π‘˜πœ™(π‘₯,(πΌβˆ’π΄)π‘₯) for all π‘₯∈𝐸 and ξπ‘žβˆˆπΉ(πΌβˆ’π΄)).
Since π½ξπ΄βˆˆβ„’, we have ‖‖‖‖𝐴π‘₯2==𝐴=ξ‚¬ξ‚€ξπ΄ξ‚ξ‚­βˆ’ξ‚¬ξ‚€ξπ΄ξ‚ξ‚€ξπ΄ξ‚π‘₯𝐴π‘₯,𝐽𝐴π‘₯π‘₯βˆ’πΌβˆ’π‘₯,𝐽π‘₯βˆ’π½π‘₯βˆ’π½π΄π‘₯π‘₯βˆ’πΌβˆ’π‘₯,𝐽π‘₯π‘₯βˆ’πΌβˆ’π‘₯,π½πΌβˆ’=β€–π‘₯β€–2βˆ’ξπ΄ξ‚ξ‚­βˆ’ξ‚¬ξ‚€ξπ΄ξ‚π‘₯ξ‚­+‖‖𝐴π‘₯β€–β€–ξ‚¬ξ‚€πΌβˆ’π‘₯,𝐽π‘₯π‘₯,π½πΌβˆ’πΌβˆ’2=ξ‚΅β€–π‘₯β€–2𝐴π‘₯ξ‚­+‖‖𝐴π‘₯β€–β€–βˆ’2π‘₯,π½πΌβˆ’πΌβˆ’2ξ‚Ά+𝐴π‘₯ξ‚­βˆ’ξπ΄ξ‚ξ‚­ξ‚€ξ‚€ξπ΄ξ‚π‘₯𝐴π‘₯.π‘₯,π½πΌβˆ’ξ‚¬ξ‚€πΌβˆ’π‘₯,𝐽π‘₯=πœ™π‘₯,πΌβˆ’+⟨π‘₯,𝐽π‘₯βŸ©βˆ’π‘₯,𝐽𝐴π‘₯βˆ’βŸ¨π‘₯,𝐽π‘₯⟩+𝐴π‘₯,𝐽π‘₯=πœ™π‘₯,πΌβˆ’(2.15)
Note that β„’βŠ‚π’¬, and thus (𝐽𝐴)βˆ’1(0)β‰ βˆ…. It is not difficult to check that (𝐽𝐴)βˆ’1(0)=𝐹(πΌβˆ’π΄). Therefore, for each ξπ‘žβˆˆπΉ(πΌβˆ’π΄), we have πœ™ξ‚€ξ‚€ξπ΄ξ‚π‘₯=π‘ž,πΌβˆ’β€–π‘žβ€–2𝐴π‘₯ξ‚­+‖‖𝐴π‘₯β€–β€–βˆ’2π‘ž,π½πΌβˆ’πΌβˆ’2=β€–π‘žβ€–2+‖‖𝐴π‘₯β€–β€–βˆ’2π‘ž,𝐽π‘₯βˆ’π½π΄π‘₯πΌβˆ’2=ξ€·β€–π‘žβ€–2βˆ’2βŸ¨π‘ž,𝐽π‘₯⟩+β€–π‘₯β€–2ξ€Έβˆ’β€–π‘₯β€–2+‖‖𝐴π‘₯β€–β€–+2π‘ž,𝐽𝐴π‘₯πΌβˆ’2=πœ™(π‘ž,π‘₯)βˆ’β€–π‘₯β€–2+‖‖𝐴π‘₯β€–β€–βˆ’2π‘₯βˆ’π‘ž,𝐽𝐴π‘₯+2π‘₯,𝐽𝐴π‘₯πΌβˆ’2β‰€πœ™(π‘ž,π‘₯)βˆ’β€–π‘₯β€–2β€–β€–π½ξβ€–β€–βˆ’2π‘Ÿπ΄π‘₯2+‖‖𝐴π‘₯β€–β€–+2⟨π‘₯,𝐽π‘₯βŸ©βˆ’2π‘₯,𝐽π‘₯βˆ’π½π΄π‘₯πΌβˆ’2‖‖‖‖=πœ™(π‘ž,π‘₯)βˆ’2π‘Ÿπ΄π‘₯2+ξ‚΅β€–π‘₯β€–2𝐴π‘₯ξ‚­+‖‖𝐴π‘₯β€–β€–βˆ’2π‘₯,π½πΌβˆ’πΌβˆ’2𝐴π‘₯𝐴π‘₯𝐴π‘₯𝐴π‘₯,=πœ™(π‘ž,π‘₯)βˆ’2π‘Ÿπœ™π‘₯,πΌβˆ’+πœ™π‘₯,πΌβˆ’=πœ™(π‘ž,π‘₯)+(1βˆ’2π‘Ÿ)πœ™π‘₯,πΌβˆ’=πœ™(π‘ž,π‘₯)+π‘˜πœ™π‘₯,πΌβˆ’(2.16) where π‘˜βˆΆ=(1βˆ’2π‘Ÿ). So, we obtain the desired result.
Next, we claim that the mapping (πΌβˆ’π΄π›Ό) as mentioned previously is closed. Let {π‘₯𝑛}βŠ‚πΈ such that π‘₯𝑛→π‘₯; then we have ξ€·πΌβˆ’π΄π›Όξ€Έπ‘₯𝑛=π‘₯π‘›βˆ’π΄π›Όπ‘₯𝑛=π‘₯π‘›βˆ’π›Όπ‘₯𝑛=(1βˆ’π›Ό)π‘₯π‘›βŸΆ(1βˆ’π›Ό)π‘₯asπ‘›βŸΆβˆž.(2.17) Moreover, (πΌβˆ’π΄π›Ό)π‘₯=π‘₯βˆ’π΄π›Όπ‘₯=π‘₯βˆ’π›Όπ‘₯=(1βˆ’π›Ό)π‘₯. This shows that (πΌβˆ’π΄π›Ό) is a closed mapping.

Remark 2.3. A relatively nonexpansive mapping is a πœ™-quasi-strict pseudocontraction, but the converse may be not true.

Example 2.4. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. Let π΄βŠ‚πΈΓ—πΈβˆ— be a maximal monotone mapping such that π΄βˆ’1(0) is nonempty. Then, π½π‘Ÿ=(𝐽+π‘Ÿπ΄)βˆ’1𝐽 is a closed and πœ™-quasi-strict pseudocontraction from 𝐸 onto 𝐷(𝐴) with constant π‘˜=0 and 𝐹(π½π‘Ÿ)=π΄βˆ’1(0).

Example 2.5. Let Π𝐢 be the generalized projection from a smooth, strictly convex, and reflexive Banach space 𝐸 onto a nonempty closed convex subset 𝐢 of 𝐸. Then, Π𝐢 is a closed and πœ™-quasi-strict pseudocontraction from 𝐸 onto 𝐢 with constant π‘˜=0 and 𝐹(Π𝐢)=𝐢.
For solving the equilibrium problem for a bifunction Ξ˜βˆΆπΆΓ—πΆβ†’β„, let us assume that Θ satisfies the following conditions:(A1)Θ(π‘₯,π‘₯)=0 for all π‘₯∈𝐢,(A2)Θ is monotone, that is, Θ(π‘₯,𝑦)+Θ(𝑦,π‘₯)≀0 for all π‘₯,π‘¦βˆˆπΆ,(A3)for each π‘₯,𝑦,π‘§βˆˆπΆ, lim𝑑↓0Θ(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)β‰€Ξ˜(π‘₯,𝑦),(2.18) (A4)for each π‘₯∈𝐢, π‘¦β†¦Ξ˜(π‘₯,𝑦) is convex and lower semicontinuous.

Lemma 2.6 (Blum and Oettli [19]). Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, and let Θ be a bifunction of 𝐢×𝐢 into ℝ satisfying (𝐴1)–(𝐴4). Letβ€‰β€‰π‘Ÿ>0  and π‘₯∈𝐸. Then, there exists π‘§βˆˆπΆ such that1Θ(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.19)

Lemma 2.7 (Takahashi and Zembayashi [24]). Let 𝐢 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, and let Θ be a bifunction from 𝐢×𝐢 to ℝ satisfying (𝐴1)–(𝐴4). For π‘Ÿ>0 and π‘₯∈𝐸, define a mapping 𝑇rβˆΆπΈβ†’πΆ as follows: π‘‡π‘Ÿξ‚†1π‘₯=π‘§βˆˆπΆβˆΆΞ˜(𝑧,𝑦)+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.20) for all π‘₯∈𝐢. Then, the following hold:(i)π‘‡π‘Ÿ is single valued,(ii)π‘‡π‘Ÿ is firmly nonexpansive-type mapping, that is, for any π‘₯,π‘¦βˆˆπ», βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π½π‘‡π‘Ÿπ‘₯βˆ’π½π‘‡π‘Ÿπ‘¦βŸ©β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©,(2.21)(iii)𝐹(π‘‡π‘Ÿ)=EP(Θ),(iv)EP(Θ) is closed and convex

Lemma 2.8 (Takahashi and Zembayashi [24]). Let 𝐢 be a closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸, let Θ be a bifunction from 𝐢×𝐢 to ℝ satisfying (𝐴1)–(𝐴4), and let π‘Ÿ>0. Then, for π‘₯∈𝐸 and π‘žβˆˆπΉ(π‘‡π‘Ÿ), πœ™ξ€·π‘,π‘‡π‘Ÿπ‘₯𝑇+πœ™π‘Ÿξ€Έπ‘₯,π‘₯β‰€πœ™(𝑝,π‘₯).(2.22)

Lemma 2.9 (Zhang [31]). Let 𝐢 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let π΄βˆΆπΆβ†’πΈβˆ— be a continuous and monotone mapping, πœ‘βˆΆπΆβ†’β„ a lower semi-continuous and convex function, and Θ a bifunction from 𝐢×𝐢 to ℝ satisfying (𝐴1)–(𝐴4). For π‘Ÿ>0 and π‘₯∈𝐸, there exists π‘’βˆˆπΆ such that 1Θ(𝑒,𝑦)+βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘(𝑒)+π‘ŸβŸ¨π‘¦βˆ’π‘’,π½π‘’βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.23) Define a mapping πΎπ‘ŸβˆΆπΆβ†’πΆ as follows: πΎπ‘Ÿξ‚†1(π‘₯)=π‘’βˆˆπΆβˆΆΞ˜(𝑒,𝑦)+βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘(𝑒)+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘’,π½π‘’βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.24) for all π‘₯∈𝐢. Then, the following conclusions hold:(1)πΎπ‘Ÿ is single valued,(2)πΎπ‘Ÿ is firmly nonexpansive type, that is, for all π‘₯,π‘¦βˆˆπΈ, βŸ¨πΎπ‘Ÿπ‘₯βˆ’πΎπ‘Ÿπ‘¦,π½πΎπ‘Ÿπ‘₯βˆ’π½πΎπ‘Ÿπ‘¦βŸ©β‰€βŸ¨πΎπ‘Ÿπ‘₯βˆ’πΎπ‘Ÿπ‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©,(2.25)(3)𝐹(πΎπ‘Ÿ)=GMEP(Θ,𝐴,πœ‘), (4)GMEP(Θ,𝐴,πœ‘) is closed and convex,(5)πœ™(𝑝,πΎπ‘Ÿπ‘₯)+πœ™(πΎπ‘Ÿπ‘₯,π‘₯)β‰€πœ™(𝑝,π‘₯)βˆ€π‘βˆˆπΉ(πΎπ‘Ÿ),π‘₯∈𝐸.

The following lemmas are crucial for the proofs of the main results in this paper.

Lemma 2.10. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. Assume that 𝐢 is a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a πœ™-quasi-strict pseudocontraction, and let πΎπ‘ŸβˆΆπΆβ†’πΆ be as in Lemma 2.9 such that Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)β‰ βˆ…. Then πœ™ξ€·π‘₯,πΎπ‘Ÿξ€Έξ€·πΎπ‘‡π‘₯+πœ™π‘Ÿξ€Έβ‰€2𝑇π‘₯,𝑇π‘₯1βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽𝑇π‘₯βˆ’π½πΎπ‘Ÿπ‘‡π‘₯⟩(2.26) for all π‘₯∈𝐢 and π‘βˆˆΞ©.

Proof. Let π‘₯∈𝐢 and π‘βˆˆΞ©. By the πœ™-quasi-strict pseudocontractility of 𝑇 and (2.4) we have 2πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯)+π‘˜πœ™(π‘₯,𝑇π‘₯)β‡”πœ™(𝑝,π‘₯)+πœ™(π‘₯,𝑇π‘₯)+2βŸ¨π‘βˆ’π‘₯,𝐽π‘₯βˆ’π½π‘‡π‘₯βŸ©β‰€πœ™(𝑝,π‘₯)+π‘˜πœ™(π‘₯,𝑇π‘₯)β‡”πœ™(π‘₯,𝑇π‘₯)≀1βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩.(2.27) It follows from (2.4), Lemma 2.9 (5), and (2.27) that πœ™ξ€·(𝑝,π‘₯)+πœ™π‘₯,πΎπ‘Ÿξ€Έπ‘‡π‘₯+2βŸ¨π‘βˆ’π‘₯,𝐽π‘₯βˆ’π½πΎπ‘Ÿξ€·π‘‡π‘₯⟩=πœ™π‘,πΎπ‘Ÿξ€Έξ€·πΎπ‘‡π‘₯β‰€πœ™(𝑝,𝑇π‘₯)βˆ’πœ™π‘Ÿξ€Έξ€·πΎπ‘‡π‘₯,𝑇π‘₯β‰€πœ™(𝑝,π‘₯)+π‘˜πœ™(π‘₯,𝑇π‘₯)βˆ’πœ™π‘Ÿξ€Έπ‘˜π‘‡π‘₯,𝑇π‘₯β‰€πœ™(𝑝,π‘₯)+2𝐾1βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯βŸ©βˆ’πœ™π‘Ÿξ€Έ,𝑇π‘₯,𝑇π‘₯(2.28) and then πœ™ξ€·π‘₯,πΎπ‘Ÿξ€Έξ€·πΎπ‘‡π‘₯+πœ™π‘Ÿξ€Έπ‘˜π‘‡π‘₯,𝑇π‘₯≀21βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½πΎπ‘Ÿπ‘˜π‘‡π‘₯⟩=21βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽𝑇π‘₯βˆ’π½πΎπ‘Ÿ=ξ‚€2π‘˜π‘‡π‘₯βŸ©ξ‚1βˆ’π‘˜+2⟨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽𝑇π‘₯βˆ’π½πΎπ‘Ÿ=2𝑇π‘₯⟩1βˆ’π‘˜βŸ¨π‘₯βˆ’π‘,𝐽π‘₯βˆ’π½π‘‡π‘₯⟩+2⟨π‘₯βˆ’π‘,𝐽𝑇π‘₯βˆ’π½πΎπ‘Ÿπ‘‡π‘₯⟩.(2.29)

Lemma 2.11 (Alber and Reich [10]). Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝐸, π‘₯0∈𝐢, and π‘₯∈𝐸. Then, π‘₯0=Π𝐢π‘₯ if and only if ⟨π‘₯0βˆ’π‘¦,𝐽π‘₯βˆ’π½π‘₯0⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.30)

Lemma 2.12 (Alber [30]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐢 be a nonempty closed convex subset of 𝐸, and let π‘₯∈𝐸. Then πœ™ξ€·π‘¦,Π𝐢π‘₯ξ€Έξ€·Ξ +πœ™πΆξ€Έπ‘₯,π‘₯β‰€πœ™(𝑦,π‘₯),βˆ€π‘¦βˆˆπΆ.(2.31)

3. Main Result

Theorem 3.1. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that 𝐸 and πΈβˆ— have the property (𝐾). Assume that 𝐢 is a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a closed and πœ™-quasi-strict pseudocontraction, Θ a bifunction from 𝐢×𝐢 to ℝ satisfying (𝐴1)–(𝐴4), πœ‘βˆΆπΆβ†’β„ a lower semi-continuous and convex function, and π΄βˆΆπΆβ†’πΈβˆ— a continuous and monotone mapping such that Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)β‰ βˆ…. Define a sequence {π‘₯𝑛} in 𝐢 by the following algorithm: π‘₯0∈𝐸,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthatΞ˜ξ€·π‘’π‘›ξ€Έ,𝑦+βŸ¨π΄π‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ+1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘‡π‘₯π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ‚†π‘§βˆˆπΆπ‘›ξ€·π‘₯βˆΆπœ™π‘›,𝑒𝑛𝑒+πœ™π‘›,𝑇π‘₯𝑛≀21βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯π‘›βŸ©+2⟨π‘₯π‘›βˆ’π‘§,𝐽𝑇π‘₯π‘›βˆ’π½π‘’π‘›βŸ©ξ‚‡,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ,(3.1) where π‘˜βˆˆ[0,1) and π‘Ÿπ‘›>0 for all π‘›βˆˆβ„• with liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Then, {π‘₯𝑛} converges strongly to Ξ Ξ©(π‘₯0).

Proof. The proof is divided into seven steps.
Step  1. Show that Ξ© is closed and convex.
From step 1 of Zhou and Gao [25], 𝐹(𝑇) is closed and convex and by Lemma 2.9(4) GMEP(Θ,𝐴,πœ‘) is closed and convex. So, Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘) is closed and convex.
Step  2. Show that 𝐢𝑛 is closed and convex for all 𝑛β‰₯1.
For 𝑛=1, 𝐢1=𝐢 is closed and convex. Assume that πΆπ‘˜ is closed and convex for some π‘˜βˆˆβ„•. For π‘§βˆˆπΆπ‘˜+1, one obtains that πœ™ξ€·π‘₯π‘˜,π‘’π‘˜ξ€Έξ€·π‘’+πœ™π‘˜,𝑇π‘₯π‘˜ξ€Έβ‰€21βˆ’π‘˜βŸ¨π‘₯π‘˜βˆ’π‘§,𝐽π‘₯π‘˜βˆ’π½π‘‡π‘₯π‘˜βŸ©+2⟨π‘₯π‘˜βˆ’π‘§,𝐽𝑇π‘₯π‘˜βˆ’π½π‘’π‘˜βŸ©.(3.2) It is not hard to see that the continuity and linearity of βŸ¨β‹…,𝐽π‘₯π‘˜βˆ’π½π‘‡π‘₯π‘˜βŸ© and βŸ¨β‹…,𝐽𝑇π‘₯π‘˜βˆ’π½π‘’π‘˜βŸ© allow πΆπ‘˜+1 to be closed and convex. Then, for all 𝑛β‰₯1, 𝐢𝑛 is closed and convex.
Step  3. Show that Ξ©βŠ‚πΆπ‘› for all 𝑛β‰₯1.
It is obvious that Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)βŠ‚πΆ=𝐢1. Suppose that Ξ©βŠ‚πΆπ‘˜ for some π‘˜βˆˆβ„•. For any π‘βˆˆΞ©, we have π‘βˆˆπΆπ‘˜. Notice that π‘₯π‘˜=Ξ πΆπ‘˜(π‘₯0), and then by Lemma 2.9 we obtain π‘’π‘˜=πΎπ‘Ÿπ‘˜π‘‡π‘₯π‘˜. So, it follows from Lemma 2.10 that πœ™ξ€·π‘₯π‘˜,π‘’π‘˜ξ€Έξ€·π‘’+πœ™π‘˜,𝑇π‘₯π‘˜ξ€Έβ‰€21βˆ’π‘˜βŸ¨π‘₯π‘˜βˆ’π‘,𝐽π‘₯π‘˜βˆ’π½π‘‡π‘₯π‘˜βŸ©+2⟨π‘₯π‘˜βˆ’π‘,𝐽𝑇π‘₯π‘˜βˆ’π½π‘’π‘˜βŸ©.(3.3) This means that π‘βˆˆπΆπ‘˜+1. By mathematical induction, Ξ©βŠ‚πΆπ‘› for all 𝑛β‰₯1. Therefore, β‹‚Ξ©βŠ‚βˆžπ‘›=1𝐢𝑛=βˆΆπ·β‰ βˆ….
Step 4. Show that limπ‘›β†’βˆžπœ™(π‘₯𝑛,π‘₯0) exists.
By π‘₯𝑛=Π𝐢𝑛π‘₯0 and Lemma 2.12, we have πœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·Ξ =πœ™πΆπ‘›π‘₯0,π‘₯0ξ€Έξ€·β‰€πœ™π‘€,π‘₯0ξ€Έξ€·βˆ’πœ™π‘€,π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘€,π‘₯0ξ€Έ,(3.4) for each π‘€βˆˆΞ©βŠ‚πΆπ‘› and for all 𝑛β‰₯1. Therefore, the sequence {πœ™(π‘₯𝑛,π‘₯0)} is bounded.
On the other hand, noticing that π‘₯𝑛=Π𝐢𝑛π‘₯0 and π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0βˆˆπΆπ‘›+1βŠ‚πΆπ‘›, πœ™(π‘₯𝑛,π‘₯0)=minπ‘§βˆˆπΆπ‘›πœ™(𝑧,π‘₯0)β‰€πœ™(π‘₯𝑛+1,π‘₯0) for all 𝑛β‰₯1. Therefore, πœ™(π‘₯𝑛,π‘₯0) is nondecreasing. It follows that the limit of πœ™(π‘₯𝑛,π‘₯0) exists.
Step  5. Show that π‘₯π‘›β†’π‘ž as π‘›β†’βˆž, where π‘ž=Π𝐷π‘₯0.
Since π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0βˆˆπΆπ‘›+1βŠ‚πΆπ‘› for any positive integer 𝑛 and by Lemma 2.12, we have πœ™ξ€·π‘₯𝑛+1,π‘₯𝑛π‘₯=πœ™π‘›+1,Π𝐢𝑛π‘₯0ξ€Έξ€·π‘₯β‰€πœ™π‘›+1,π‘₯0ξ€Έξ€·Ξ βˆ’πœ™πΆπ‘›π‘₯0,π‘₯0ξ€Έξ€·π‘₯=πœ™π‘›+1,π‘₯0ξ€Έξ€·π‘₯βˆ’πœ™π‘›,π‘₯0ξ€Έ.(3.5) Letting π‘›β†’βˆž in (3.5), one has πœ™(π‘₯𝑛+1,π‘₯𝑛)β†’0 as π‘›β†’βˆž. Without loss of generality, we can assume that π‘₯π‘›β†’π‘ž weakly as π‘›β†’βˆž (passing to a subsequence if necessary). It is easy to show that π‘žβˆˆπΆπ‘› for all 𝑛β‰₯1. Hence β‹‚π‘žβˆˆβˆžπ‘›=1𝐢𝑛=𝐷. Noticing that πœ™(π‘₯𝑛,π‘₯0)β‰€πœ™(π‘₯𝑛+1,π‘₯0)β‰€πœ™(π‘ž,π‘₯0), we have ξ€·π‘ž,π‘₯0≀liminfπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛,π‘₯0≀limsupπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛,π‘₯0ξ€Έξ€·β‰€πœ™π‘ž,π‘₯0ξ€Έ,(3.6) which implies that πœ™(π‘₯𝑛,π‘₯0)β†’πœ™(π‘ž,π‘₯0) as π‘›β†’βˆž. Hence β€–π‘₯π‘›β€–β†’β€–π‘žβ€–. By the property (𝐾) of 𝐸, we have π‘₯π‘›β†’π‘ž. From Lemma 2.11, we have ⟨π‘₯π‘›βˆ’π‘¦,𝐽π‘₯0βˆ’π½π‘₯π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπ·.(3.7) Hence βŸ¨π‘žβˆ’π‘¦,𝐽π‘₯0βˆ’π½π‘žβŸ©β‰₯0,βˆ€π‘¦βˆˆπ·,(3.8) which implies that π‘ž=Π𝐷π‘₯0.
Step  6. Show that π‘žβˆˆΞ©.
We prove first that {𝑇π‘₯𝑛} and {𝑒𝑛}={πΎπ‘Ÿπ‘›π‘‡π‘₯𝑛} are bounded. Indeed, taking π‘βˆˆΞ©=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)βŠ‚πΆπ‘›+1, we have ‖𝑝‖2β€–β€–βˆ’2‖𝑝‖𝑇π‘₯𝑛‖‖+‖‖𝑇π‘₯𝑛‖‖2=ξ€·β€–β€–β€–π‘β€–βˆ’π‘‡π‘₯𝑛‖‖2ξ€·β‰€πœ™π‘,𝑇π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯𝑛π‘₯+π‘˜πœ™π‘›,𝑇π‘₯π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯𝑛+2π‘˜1βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯π‘›βŸ©ξ€·β‰€πœ™π‘,π‘₯𝑛+2π‘˜β€–β€–π‘₯1βˆ’π‘˜π‘›β€–β€–β€–β€–π‘₯βˆ’π‘π‘›β€–β€–+2π‘˜β€–β€–π‘₯1βˆ’π‘˜π‘›β€–β€–β€–β€–βˆ’π‘π‘‡π‘₯𝑛‖‖.(3.9) Then ‖‖𝑇π‘₯𝑛‖‖2β‰€ξ‚€πœ™ξ€·π‘,π‘₯π‘›ξ€Έβˆ’β€–π‘β€–2+2π‘˜β€–β€–π‘₯1βˆ’π‘˜π‘›β€–β€–β€–β€–π‘₯βˆ’π‘π‘›β€–β€–ξ‚+ξ‚€2π‘˜β€–β€–π‘₯1βˆ’π‘˜π‘›β€–β€–ξ‚β€–β€–βˆ’π‘+2‖𝑝‖𝑇π‘₯𝑛‖‖‖‖≀𝑀+𝐾𝑇π‘₯𝑛‖‖1=𝑀+2ξ€·β€–β€–2𝐾𝑇π‘₯𝑛‖‖1≀𝑀+2𝐾2+‖‖𝑇π‘₯𝑛‖‖21=𝑀+2𝐾2+12‖‖𝑇π‘₯𝑛‖‖2,(3.10) where π‘€βˆΆ=sup{πœ™(𝑝,π‘₯𝑛)βˆ’β€–π‘β€–2+(2π‘˜/(1βˆ’π‘˜))β€–π‘₯π‘›βˆ’π‘β€–β€–π‘₯π‘›β€–βˆΆπ‘›βˆˆβ„•} and 𝐾∢=sup{(2π‘˜/(1βˆ’π‘˜))β€–π‘₯π‘›βˆ’π‘β€–+2β€–π‘β€–βˆΆπ‘›βˆˆβ„•}. Thus ‖‖𝑇π‘₯𝑛‖‖2≀2𝑀+𝐾2(3.11) for all π‘›βˆˆβ„•. Therefore {𝑇π‘₯𝑛} is bounded. Note that πœ™(𝑝,πΎπ‘Ÿπ‘›π‘‡π‘₯𝑛)β‰€πœ™(𝑝,𝑇π‘₯𝑛) for all π‘›βˆˆβ„•. Therefore {𝑒𝑛}={πΎπ‘Ÿπ‘›π‘‡π‘₯𝑛} is also bounded. From π‘₯𝑛+1βˆˆπΆπ‘›+1, one has πœ™ξ€·π‘₯𝑛,𝑒𝑛𝑒+πœ™π‘›,𝑇π‘₯𝑛≀2π‘₯1βˆ’π‘˜π‘›βˆ’π‘₯𝑛+1,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯𝑛π‘₯+2π‘›βˆ’π‘₯𝑛+1,𝐽𝑇π‘₯π‘›βˆ’π½π‘’π‘›ξ¬.(3.12) By Step  5, we obtain that π‘₯𝑛+1βˆ’π‘₯𝑛→0. Taking limit on both sides of (3.12), we obtain πœ™(π‘₯𝑛,𝑒𝑛)+πœ™(𝑒𝑛,𝑇π‘₯𝑛)β†’0 as π‘›β†’βˆž. Noting that 0≀(β€–π‘₯π‘›β€–βˆ’β€–π‘’π‘›β€–)2β‰€πœ™(π‘₯𝑛,𝑒𝑛), 0≀(β€–π‘’π‘›β€–βˆ’β€–π‘‡π‘₯𝑛‖)2β‰€πœ™(𝑒𝑛,𝑇π‘₯𝑛), and β€–π‘₯π‘›β€–β†’β€–π‘žβ€–, it is implied that β€–π‘žβ€–=limπ‘›β†’βˆžβ€–β€–π‘₯𝑛‖‖=limπ‘›β†’βˆžβ€–β€–π‘’π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π‘‡π‘₯𝑛‖‖,(3.13) and consequently β€–π½π‘žβ€–=limπ‘›β†’βˆžβ€–β€–π½π‘₯𝑛‖‖=limπ‘›β†’βˆžβ€–β€–π½π‘’π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π½π‘‡π‘₯𝑛‖‖.(3.14) This implies that {𝐽(𝑒𝑛)} and {𝐽(𝑇π‘₯𝑛)} are bounded. Since 𝐸 is reflexive, πΈβˆ— is also reflexive. So we can assume that π½ξ€·π‘’π‘›ξ€ΈβŸΆπ‘“βˆˆπΈβˆ—(3.15) weakly. On the other hand, in view of the reflexivity of 𝐸, one has 𝐽(𝐸)=πΈβˆ—, which means that, for π‘“βˆˆπΈβˆ—, there exists π‘₯π‘“βˆˆπΈ, such that 𝐽π‘₯𝑓=𝑓. It follows that πœ™ξ€·π‘₯𝑛,𝑒𝑛=β€–β€–π‘₯𝑛‖‖2π‘₯βˆ’2𝑛𝑒,𝐽𝑛+‖‖𝑒𝑛‖‖2=β€–β€–π‘₯𝑛‖‖2π‘₯βˆ’2𝑛𝑒,𝐽𝑛+‖‖𝐽𝑒𝑛‖‖2.(3.16) Taking liminfπ‘›β†’βˆž on both sides of the equality above, we have 0β‰₯β€–π‘žβ€–2βˆ’2βŸ¨π‘ž,π‘“βŸ©+‖𝑓‖2=β€–π‘žβ€–2ξ«π‘’βˆ’20,𝐽π‘₯𝑓+‖‖𝐽π‘₯𝑓‖‖2ξ€·=πœ™π‘ž,π‘₯𝑓.(3.17) Therefore πœ™(π‘ž,π‘₯𝑓)=0 and consequently π‘ž=π‘₯𝑓, which implies that 𝑓=π½π‘ž. Hence π½ξ€·π‘’π‘›ξ€ΈβŸΆπ½π‘žβˆˆπΈβˆ—(3.18) weakly. Since ‖𝐽(𝑒𝑛)β€–β†’β€–π½π‘žβ€– and πΈβˆ— has the property (𝐾), we have β€–β€–π½ξ€·π‘’π‘›ξ€Έβ€–β€–βˆ’π½π‘žβŸΆ0.(3.19) Noting that π½βˆ’1βˆΆπΈβˆ—β†’πΈ is demicontinuous, we have π‘’π‘›βŸΆπ‘žβˆˆπΈ(3.20) weakly. Since β€–π‘’π‘›β€–β†’β€–π‘žβ€– and 𝐸 has the property (𝐾), we obtain that π‘’π‘›β†’π‘ž as π‘›β†’βˆž. Similarly, it is not difficult to show that 𝑇π‘₯π‘›β†’π‘ž as π‘›β†’βˆž. From π‘₯π‘›β†’π‘ž and the closeness property of 𝑇, we have π‘‡π‘ž=π‘ž.
Next, we want to show that π‘žβˆˆGMEP(Θ,𝐴,πœ‘). Define πΊβˆΆπΆΓ—πΆβ†’β„ by 𝐺(π‘₯,𝑦)=Θ(π‘₯,𝑦)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯) for all π‘₯,π‘¦βˆˆπΆ. It is not hard to verify that 𝐺 satisfies conditions (𝐴1)–(𝐴4). It follows from 𝑒𝑛=πΎπ‘Ÿπ‘›π‘‡π‘₯𝑛 and (𝐴2) that 1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘₯π‘›ξ€·βŸ©β‰₯𝐺𝑦,𝑒𝑛,βˆ€π‘¦βˆˆπΆ.(3.21) By using (𝐴4) and liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, we obtain 0β‰₯𝐺(𝑦,π‘ž) for all π‘¦βˆˆπΆ. For π‘‘βˆˆ(0,1] and π‘¦βˆˆπΆ, let 𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)π‘ž. So, from (𝐴1) and (𝐴4), we have 𝑦0=𝐺𝑑,𝑦𝑑𝑦=𝐺𝑑𝑦,𝑑𝑦+(1βˆ’π‘‘)π‘žβ‰€π‘‘πΊπ‘‘ξ€Έ+𝑦,𝑦(1βˆ’π‘‘)𝐺𝑑𝑦,π‘žβ‰€π‘‘πΊπ‘‘ξ€Έ.,𝑦(3.22) Dividing by 𝑑, we have 𝐺𝑦𝑑,𝑦β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.23) From (𝐴3) we have 0≀lim𝑑→0𝐺(𝑦𝑑,𝑦)=lim𝑑→0𝐺(𝑑𝑦+(1βˆ’π‘‘)π‘ž,𝑦)≀𝐺(π‘ž,𝑦) for all π‘¦βˆˆπΆ, and hence π‘žβˆˆGMEP(Θ,𝐴,πœ‘). So, π‘žβˆˆπΉ(𝑇)∩GMEP(Θ,𝐴,πœ‘)=Ξ©.
Step  7. Show that π‘ž=Ξ Ξ©π‘₯0.
It follows from Steps 5 and 6 that πœ™ξ€·π‘ž,π‘₯0ξ€Έξ€·Ξ β‰€πœ™Ξ©π‘₯0,π‘₯0ξ€Έξ€·β‰€πœ™π‘ž,π‘₯0ξ€Έ,(3.24) which implies that πœ™(Ξ Ξ©π‘₯0,π‘₯0)=πœ™(π‘ž,π‘₯0). Hence, π‘ž=Ξ Ξ©π‘₯0. Then {π‘₯𝑛} converges strongly to π‘ž=Ξ Ξ©π‘₯0. This completes the proof.

If 𝑇 is closed πœ™-quasi-nonexpansive, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.2. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that 𝐸 and πΈβˆ— have the property (𝐾). Assume that 𝐢 is a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a πœ™-closed quasi-nonexpansive mapping, Θ a bifunction from 𝐢×𝐢 to ℝ satisfying (𝐴1)–(𝐴4), πœ‘βˆΆπΆβ†’β„ a lower semi-continuous and convex function, and π΄βˆΆπΆβ†’πΈβˆ— a continuous and monotone mapping such that Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)β‰ βˆ…. Define a sequence {π‘₯𝑛} in 𝐢 by the following algorithm: π‘₯0∈𝐸,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthatΞ˜ξ€·π‘’π‘›ξ€Έ,𝑦+βŸ¨π΄π‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ+1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘‡π‘₯π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·π‘₯βˆΆπœ™π‘›,𝑒𝑛𝑒+πœ™π‘›,𝑇π‘₯𝑛≀2⟨π‘₯π‘›βˆ’π‘§,𝐽π‘₯π‘›βˆ’π½π‘’π‘›βŸ©ξ€Ύ,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ,(3.25) where π‘Ÿπ‘›>0 for all π‘›βˆˆβ„• with liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Then {π‘₯𝑛} converges strongly to Ξ Ξ©(π‘₯0).

If 𝐸=𝐻 is a Hilbert space, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.3. Let 𝐻 be a Hilbert space. Assume that 𝐢 is a nonempty closed convex subset of 𝐻. Let π‘‡βˆΆπΆβ†’πΆ be a closed and πœ™-quasi-strict pseudocontraction, Θ a bifunction from 𝐢×𝐢 to ℝ satisfying (A 1)–(A 4), πœ‘βˆΆπΆβ†’β„ a lower semi-continuous and convex function, and π΄βˆΆπΆβ†’π» a continuous and monotone mapping such that Ω∢=𝐹(𝑇)∩GMEP(Θ,𝐴,πœ‘)β‰ βˆ…. Define a sequence {π‘₯𝑛} in 𝐢 by the following algorithm: π‘₯0∈𝐻,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthat,Ξ˜ξ€·π‘’π‘›ξ€Έ,𝑦+βŸ¨π΄π‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ+1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π‘’π‘›βˆ’π‘‡π‘₯π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ‚†π‘§βˆˆπΆπ‘›βˆΆβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–2+β€–β€–π‘’π‘›βˆ’π‘‡π‘₯𝑛‖‖2≀21βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘§,π‘₯π‘›βˆ’π‘‡π‘₯π‘›βŸ©+2⟨π‘₯π‘›βˆ’π‘§,𝑇π‘₯π‘›βˆ’π‘’π‘›βŸ©ξ‚‡,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ,(3.26) where π‘˜βˆˆ[0,1) and π‘Ÿπ‘›>0 for all π‘›βˆˆβ„• with liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Then {π‘₯𝑛} converges strongly to Ξ Ξ©(π‘₯0).

If 𝐴=0 and πœ‘=0, then we have the following corollary.

Corollary 3.4. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space such that 𝐸 and πΈβˆ— have the property (𝐾). Assume that 𝐢 is a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a closed and πœ™-quasi-strict pseudocontraction and Θ a bifunction from 𝐢×𝐢 to ℝ satisfying (A 1)–(A 4) such that Ξ›βˆΆ=𝐹(𝑇)∩EP(Θ)β‰ βˆ…. Define a sequence {π‘₯𝑛} in 𝐢 by the following algorithm: π‘₯0∈𝐸,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthatΞ˜ξ€·π‘’π‘›ξ€Έ+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘‡π‘₯π‘›πΆβŸ©β‰₯0βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ‚†π‘§βˆˆπΆπ‘›ξ€·π‘₯βˆΆπœ™π‘›,𝑒𝑛𝑒+πœ™π‘›,𝑇π‘₯𝑛≀21βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯π‘›βŸ©+2⟨π‘₯π‘›βˆ’π‘§,𝐽𝑇π‘₯π‘›βˆ’π½π‘’π‘›βŸ©ξ‚‡,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ,(3.27) where π‘˜βˆˆ[0,1) and π‘Ÿπ‘›>0 for all π‘›βˆˆβ„• with liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Then {π‘₯𝑛} converges strongly to Ξ Ξ©(π‘₯0).

Corollary 3.5 ([25, Theorem 3.1]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space such that 𝐸 and πΈβˆ— have the property (𝐾). Assume that 𝐢 is a nonempty closed convex subset of 𝐸. Let π‘‡βˆΆπΆβ†’πΆ be a πœ™-closed quasi-strict pseudocontraction. Define a sequence {π‘₯𝑛} in 𝐢 by the following algorithm: π‘₯0∈𝐸,chosenarbitrarily,𝐢1π‘₯=𝐢,1=Π𝐢1ξ€·π‘₯0ξ€Έ,π‘’π‘›βˆˆπΆsuchthat𝐢𝑛+1=ξ‚†π‘§βˆˆπΆπ‘›ξ€·π‘₯βˆΆπœ™π‘›,𝑇π‘₯𝑛≀21βˆ’π‘˜βŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯π‘›βˆ’π½π‘‡π‘₯π‘›βŸ©ξ‚‡,π‘₯𝑛+1=Π𝐢𝑛+1ξ€·π‘₯0ξ€Έ(3.28) where π‘˜βˆˆ[0,1). Then {π‘₯𝑛} converges strongly to Ξ Ξ©(π‘₯0).

Proof. Put Θ=0, 𝐴=0, πœ‘=0, and π‘Ÿπ‘›=1 for all 𝑛β‰₯1 in Theorem 3.1. Then, πΎπ‘Ÿπ‘›=Π𝐢 for all 𝑛⩾1. So, 𝑒𝑛=Π𝐢𝑇π‘₯𝑛 for all 𝑛⩾1 (note that π‘₯1=Π𝐢π‘₯0). Since π‘₯𝑛=Π𝐢𝑛π‘₯0βˆˆπΆπ‘›βŠ‚πΆ and then 𝑇π‘₯π‘›βˆˆπΆ for all 𝑛⩾1, we have 𝑒𝑛=𝑇π‘₯𝑛 for all 𝑛⩾1. Thus πœ™(π‘₯𝑛,𝑒𝑛)+πœ™(𝑒𝑛,𝑇π‘₯𝑛)=πœ™(π‘₯𝑛,𝑇π‘₯𝑛) and 𝐽𝑇π‘₯π‘›βˆ’π½π‘’π‘›=0 for all 𝑛β‰₯1. For this reason, (1.12) is a special case of (3.1). Applying Theorem 3.1, we have the desired result.

Remark 3.6. It is well known that every uniformly convex and uniformly smooth Banach space satisfies all assumptions of Banach space in Theorem 3.1. On the other hand, in general, Musielak-Orlicz space [26] need not be uniformly convex or uniformly smooth, however, any strictly convex, reflexive and smooth Musielak-Orlicz space [26] satisfies all Banach-space assumptions of Theorem 3.1. It can be written as the following diagram:

536283.fig.22

For this reason, Theorem 3.1 can be viewed as a more general one and can be applied widely in both the fixed point problems and the equilibrium problems.

Acknowledgments

The author is supported by Thailand Research Fund under Grant no. MRG5380249 and the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand. The author would like to thank Simeon Reich and an anonymous referee for their valuable comments and suggestions, which were helpful in improving the paper.

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