Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

Abstract

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for a 𝛽-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

1. Introduction

Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻 with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖. Let 𝐹 be a bifunction of 𝐶×𝐶 into ℛ, where ℛ is the set of real numbers, 𝜑∶𝐶→ℛ be a real-valued function. Let Λ be arbitrary index set. The system of mixed equilibrium problem is for finding 𝑥∈𝐶 such that𝐹_𝑘(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,𝑘∈Λ,∀𝑦∈𝐶.(1.1) The set of solutions of (1.1) is denoted by SMEP(𝐹_𝑘), that is, SMEP𝐹_𝑘=𝑥∈𝐶∶=𝐹_𝑘(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,𝑘∈Λ,∀𝑦∈𝐶.(1.2) If Λ is a singleton, then problem (1.1) becomes the following mixed equilibrium problem: finding 𝑥∈𝐶 such that𝐹(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.3) The set of solutions of (1.3) is denoted by MEP(𝐹).

If 𝜑≡0, the problem (1.3) is reduced into the equilibrium problem [1] for finding 𝑥∈𝐶 such that𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.4) The set of solutions of (1.4) is denoted by EP(𝐹). This problem contains fixed-point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the system of mixed equilibrium problem and the equilibrium problem, please consult [2–19].

Recall that, a mapping 𝑆∶𝐶→𝐶 is said to be nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖,(1.5) for all 𝑥,𝑦∈𝐶. If 𝐶 is a bounded closed convex and 𝑆 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑆) is nonempty [20]. Let 𝐴∶𝐶→𝐻 be a mapping, the Hartmann-Stampacchia variational inequality for finding 𝑥∈𝐶 such that⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(1.6) The set of solutions of (1.6) is denoted by VI(𝐶,𝐴). The variational inequality has been extensively studied in the literature [21–28].

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻:𝜃(𝑥)=12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑦⟩,∀𝑥∈𝐹(𝑆),(1.7) where 𝐴 is a linear bounded operator, 𝐹(𝑆) is the fixed point set of a nonexpansive mapping 𝑆, and 𝑦 is a given point in 𝐻 [29].

We denote weak convergence and strong convergence by notations ⇀ and →, respectively. A mapping 𝐴 of 𝐶 into 𝐻 is called monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,(1.8) for all 𝑥,𝑦∈𝐶. A mapping 𝐴 of 𝐶 into 𝐻 is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,(1.9) for all 𝑥,𝑦∈𝐶. It is obvious that any 𝛼-inverse-strongly monotone mappings 𝐴 are monotone and Lipschitz continuous mapping. A linear bounded operator 𝐴 is strongly positive if there exists a constant 𝛾>0 with the property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,(1.10) for all 𝑥∈𝐻. A self-mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,(1.11) for all 𝑥,𝑦∈𝐶. We use Π_𝐶 to denote the collection of all contraction on 𝐶. Note that each 𝑓∈Π_𝐶 has a unique fixed point in 𝐶.

Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2^𝐻 be a set-valued mapping. The variational inclusion problem is to find 𝑥∈𝐻 such that𝜃∈𝐵(𝑥)+𝑀(𝑥),(1.12) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (1.12) is denoted by 𝐼(𝐵,𝑀). The variational inclusion has been extensively studied in the literature, see, for example, [30–32] and the reference therein.

A set-valued mapping 𝑀∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑀(𝑥), and 𝑔∈𝑀(𝑦) impling ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑀 is maximal if its graph 𝐺(𝑀)∶={(𝑓,𝑥)∈𝐻×𝐻∶𝑓∈𝑀(𝑥)} of 𝑀 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑀 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for all (𝑦,𝑔)∈𝐺(𝑀) impling 𝑓∈𝑀(𝑥).

Let 𝐵 be an inverse-strongly monotone mapping of 𝐶 into 𝐻, and let 𝑁_𝐶𝑣 be normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}, and define𝑇𝑣=𝐵𝑣+𝑁^𝐶∅,𝑣,iiff𝑣𝑣∈∉𝐶𝐶.,(1.13) Then, 𝑇 is a maximal monotone and 𝜃∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐵) (see [33]).

Let 𝑀∶𝐻→2^𝐻 be a set-valued maximal monotone mapping, then the single-valued mapping 𝐽_𝑀,𝜆∶𝐻→𝐻 defined by𝐽_𝑀,𝜆(𝑥)=(𝐼+𝜆𝑀)⁻¹(𝑥),𝑥∈𝐻,(1.14) is called the resolvent operator associated with 𝑀, where 𝜆 is any positive number and 𝐼 is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone, and that a solution of problem (1.12) is a fixed point of the operator 𝐽_𝑀,𝜆(𝐼−𝜆𝐵) for all 𝜆>0, (for more details see [34]).

In 2000, Moudafi [35] introduced the viscosity approximation method for nonexpansive mappings and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥_𝑛} defined by the iterative method below, with the initial guess 𝑥₀∈𝐶 is chosen arbitrarily,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,𝑛≥0,(1.15) where {𝛼_𝑛}⊂(0,1) satisfies certain conditions and converges strongly to a fixed point of 𝑆 (say 𝑥∈𝐶), which is then a unique solution of the following variational inequality:(𝐼−𝑓)𝑥,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆).(1.16)

In 2006, Marino and Xu [29] introduced a general iterative method for nonexpansive mapping. They defined the sequence {𝑥_𝑛} generated by the algorithm 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑥_𝑛,𝑛≥0,(1.17) where {𝛼_𝑛}⊂(0,1), and 𝐴 is a strongly positive linear bounded operator. They proved that if 𝐶=𝐻, and the sequence {𝛼_𝑛} satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.17) converges strongly to a fixed point of 𝑆 (say 𝑥∈𝐻) which is the unique solution of the following variational inequality:(𝐴−𝛾𝑓)𝑥,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆),(1.18) which is the optimality condition for the minimization problemmin_{𝑥∈𝐹(𝑆)∩EP(𝐹)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.19) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities. Let 𝑃_𝐶 be the projection of 𝐻 onto 𝐶. In 2005, Iiduka and Takahashi [36] introduced the following iterative process for 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥0,(1.20) where 𝑢∈𝐶, {𝛼_𝑛}⊂(0,1), and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2𝛽. They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝜆_𝑛}, the sequence {𝑥_𝑛} generated by (1.20) converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say 𝑥∈𝐶) which solve some variational inequality𝑥−𝑢,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆)∩VI(𝐶,A).(1.21)

In 2008, Su et al. [37] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: 𝑥₁,𝑢_𝑛∈𝐻𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝑥⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑢_𝑛,(1.22) for all 𝑛∈ℕ, where {𝛼_𝑛}⊂[0,1) and {𝑟_𝑛}⊂(0,∞) satisfing some appropriate conditions. Furthermore, they proved that {𝑥_𝑛} and {𝑢_𝑛} converge strongly to the same point 𝑧∈𝐹(𝑆)∩VI(𝐶,𝐴)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩VI(𝐶,𝐴)∩EP(𝐹)}𝑓(𝑧).

Let {𝑇_𝑖} be an infinite family of nonexpansive mappings of 𝐻 into itself, and let {𝜆_𝑖} be a real sequence such that 0≤𝜆_𝑖≤1 for every 𝑖∈𝑁. For 𝑛≥1, we defined a mapping 𝑊_𝑛 of 𝐻 into itself as follows:𝑈𝑈_𝑛,𝑛+1∶=𝐼,_𝑛,𝑛𝑈∶⋮=𝜆_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝐼,_𝑛,𝑘𝑈∶⋮=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝐼,_𝑛,2𝑊∶=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝐼,_𝑛∶=𝑈_𝑛,1∶=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼.(1.23)

In 2011, He et al. [38] introduced the following iterative process for {𝑇_𝑛∶𝐶→𝐶} which is a sequence of nonexpansive mappings. Let {𝑧_𝑛} be the sequence defined by𝑧_𝑛+1=𝜖_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝜖_𝑛𝑊_𝑛𝐾^1𝑟_1,𝑛𝐾^2𝑟_2,𝑛⋅⋯⋅𝐾^𝐾𝑟_𝐾,𝑛𝑧_𝑛,∀𝑛∈𝑁.(1.24) The sequence {𝑧_𝑛} defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality, and the generalized equilibrium problem. Recently, Jitpeera and Kumam [39] introduced the following new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a 𝛽-inverse-strongly monotone mapping in a real Hilbert space.

In this paper, we modify the iterative methods (1.17), (1.22), and (1.24) by purposing the following new general viscosity iterative method: 𝑥₀,𝑢_𝑛∈𝐶,𝑢𝑥_𝑛=𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,(1.25) for all 𝑛∈ℕ, where {𝛼_𝑛}⊂(0,1), {𝑟_𝑛}⊂(0,2𝜎), and 𝜆∈(0,2𝛽) satisfy some appropriate conditions. The purpose of this paper shows that under some control conditions the sequence {𝑥_𝑛} converges strongly to a common element of the set of common fixed points of nonexpansive mappings, the solution of the system of mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space. Moreover, we apply our results to the class of strictly pseudocontractive mappings. Finally, we give a numerical example which supports our main theorem in the last part. Our results improve and extend the corresponding results of Marino and Xu [29], Su et al. [37], He et al. [38], and some authors.

2. Preliminaries

Let 𝐻 be a real Hilbert space and 𝐶 be a nonempty closed and convex subset of 𝐻. Recall that the (nearest point) projection 𝑃_𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥∈𝐻 and the unique point in 𝑃_𝐶𝑥∈𝐶 satisfies the property ‖‖𝑥−𝑃_𝐶𝑥‖‖=min_𝑦∈𝐶‖𝑥−𝑦‖,(2.1) which is equivalent to the following inequality⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦⟩≥0,∀𝑦∈𝐶.(2.2) The following characterizes the projection 𝑃_𝐶. We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1. The function 𝑢∈𝐶 is a solution of the variational inequality if and only if 𝑢∈𝐶 satisfies the relation 𝑢=𝑃_𝐶(𝑢−𝜆𝐵𝑢) for all 𝜆>0.

Lemma 2.2. For a given 𝑧∈𝐻, 𝑢∈𝐶, 𝑢=𝑃_𝐶𝑧⇔⟨𝑢−𝑧,𝑣−𝑢⟩≥0,∀𝑣∈𝐶.
It is well known that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶 and satisfies ‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²≤⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩,∀𝑥,𝑦∈𝐻.(2.3) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and for all 𝑥∈𝐻,𝑦∈𝐶, ⟨𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0.(2.4)

Lemma 2.3 (see [40]). Let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐵∶𝐻→𝐻 be a monotone and Lipshitz continuous mapping. Then the mapping 𝐿=𝑀+𝐵∶𝐻→2^𝐻 is a maximal monotone mapping.

Lemma 2.4 (see [41]). Each Hilbert space 𝐻 satisfies Opial's condition, that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖, hold for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [42]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛,∀𝑛≥0,(2.5) where {𝛾_𝑛}⊂(0,1) and {𝛿_𝑛} is a sequence in ℛ such that(i)∑^∞𝑛=1𝛾_𝑛=∞, (ii)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0 or ∑^∞𝑛=1|𝛿_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.6 (see [43]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛⟶𝑥,𝑥_𝑛−𝑇𝑥_𝑛⟶0,(2.6) implying 𝑥=𝑇𝑥.

For solving the mixed equilibrium problem, let us assume that the bifunction 𝐹∶𝐶×𝐶→ℛ and the nonlinear mapping 𝜑∶𝐶→ℛ satisfy the following conditions: (A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶; (A3)for each fixed 𝑦∈𝐶, 𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous; (A4)for each fixed 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous; (B1)for each 𝑥∈𝐶 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊆𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥,𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥−𝜑(𝑧)+1𝑟⟨𝑦^𝑥−𝑧,𝑧−𝑥⟩<0,(2.7)(B2)𝐶 is a bounded set.

Lemma 2.7 (see [44]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→ℛ be a bifunction mapping satisfying (A1)–(A4), and let 𝜑∶𝐶→ℛ be a convex and lower semicontinuous function such that 𝐶∩dom𝜑≠∅. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, then there exists 𝑢∈𝐶 such that 𝐹(𝑢,𝑦)+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝑢−𝑥⟩≥0.(2.8) Define a mapping 𝐾_𝑟∶𝐻→𝐶 as follows: 𝐾_𝑟(𝑥)=𝑢∈𝐶∶𝐹(𝑢,𝑦)+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝑢−𝑥⟩≥0,∀𝑦∈𝐶,(2.9) for all 𝑥∈𝐻. Then, the following hold: (i)𝐾_𝑟 is single-valued; (ii)𝐾_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻, ‖𝐾_𝑟𝑥−𝐾_𝑟𝑦‖²≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝑥−𝑦⟩;(iii)𝐹(𝐾_𝑟)=MEP(𝐹); (iv)MEP(𝐹) is closed and convex.

Lemma 2.8 (see [29]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.9 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {𝑇_𝑖}_𝑖∈𝑁 be an infinite family of nonexpansive mappings of C into itself such that ∩_𝑖∈𝑁𝐹(𝑇_𝑖)≠∅, and let {𝜆_𝑖} be a real sequence such that 0≤𝜆_𝑖≤b<1 for every 𝑖∈𝑁. Then F(W)=∩_𝑖∈𝑁𝐹(𝑇_𝑖)≠∅.

Lemma 2.10 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {𝑇_𝑖} be an infinite family of nonexpansive mappings of C into itself, and let {𝜆_𝑖} be a real sequence such that 0≤𝜆_𝑖≤b<1 for every 𝑖∈𝑁. Then, for every 𝑥∈𝐶 and 𝑘∈𝑁, the limit lim_𝑛→∞𝑈_𝑛,𝑘 exist.
In view of the previous lemma, we define 𝑊𝑥∶=lim_𝑛→∞𝑈_𝑛,1𝑥=lim_𝑛→∞𝑊_𝑛𝑥.(2.10)

3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the common fixed points, the common solution of a system of mixed equilibrium problems and variational inclusion of inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty close and convex subset of 𝐻, and let 𝐵 be a 𝛽-inverse-strongly monotone mapping. Let 𝜑∶𝐶→𝑅 be a convex and lower semicontinuous function, 𝑓∶𝐶→𝐶 a contraction mapping with coefficient 𝛼(0<𝛼<1), and 𝑀∶𝐻→2^𝐻 a maximal monotone mapping. Let 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼 and 𝜆∈(0,2𝛽). Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐻 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐼(𝐵,𝑀)≠∅.(3.1) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily and 𝑢𝑥_𝑛=𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,(3.2) for all 𝑛=1,2,3,…, where K^𝐹𝑖_{𝑟𝑖,𝑛}(𝑥)=𝑢_𝑛∈𝐶∶𝐹_𝑖𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑖,𝑛⟨𝑦−𝑢_𝑛𝑖,𝑢=_𝑛−1,𝑥2_𝑛,⟩3,≥…0,,𝑁∀,𝑦∈𝐶,(3.3) and the following conditions are satisfied(C1):{𝜖_𝑛}⊂(0,1),lim_𝑛→0𝜖_𝑛=0,∑^∞𝑛=1𝜖_𝑛=∞,∑^∞𝑛=1|𝜖_𝑛+1−𝜖_𝑛|<∞; (C2):{𝑟_𝑛}⊂[𝑐,𝑑] with 𝑐,𝑑∈(0,2𝜎) and ∑^∞𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃,(3.4) which is the optimality condition for the minimization problem min_𝑞∈𝜃12⟨𝐴𝑞,𝑞⟩−ℎ(𝑞),(3.5) where h is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. For condition (C1), we may assume without loss of generality, and 𝜖_𝑛∈(0,‖𝐴‖⁻¹) for all 𝑛. By Lemma 2.8, we have ‖𝐼−𝜖_𝑛𝐴‖≤1−𝜖_𝑛𝛾. Next, we will assume that ‖𝐼−𝐴‖≤‖1−𝛾‖.
Next, we will divide the proof into six steps.
Step 1. First, we will show that {𝑥_𝑛} and {𝑢_𝑛} are bounded. Since 𝐵 is 𝛽-inverse-strongly monotone mappings, we have ‖(𝐼−𝜆𝐵)𝑥−(𝐼−𝜆𝐵)𝑦‖²=‖𝐼𝑥−𝜆𝐵𝑥−𝐼𝑦+𝜆𝐵𝑦‖²=‖𝑥−𝑦−𝜆𝐵𝑥+𝜆𝐵𝑦‖²=‖(𝑥−𝑦)−𝜆(𝐵𝑥+𝐵𝑦)‖²≤‖𝑥−𝑦‖²−2𝜆⟨𝑥−𝑦⟩⟨𝐵𝑥+𝐵𝑦⟩+𝜆²‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²−2𝜆𝛽‖𝐵𝑥+𝐵𝑦‖²+𝜆²‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²+𝜆(𝜆−2𝛽)‖𝐵𝑥+𝐵𝑦‖²,(3.6) if 0<𝜆<2𝛽, then 𝐼−𝜆𝐵 is nonexpansive.
Put 𝑦_𝑛∶=𝐽_𝑀,𝜆(𝑢_𝑛−𝜆𝐵𝑢_𝑛),𝑛≥0. Since 𝐽_𝑀,𝜆 and 𝐼−𝜆𝐵 are nonexpansive mapping, it follows that ‖‖𝑦_𝑛−𝑞‖‖=≤‖‖‖‖𝐽𝑢_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞)_𝑛‖‖≤‖‖𝑢−𝜆𝐵𝑢_𝑛−(𝑞−𝜆𝐵𝑞)_𝑛‖‖−𝑞‖‖.(3.7) By Lemma 2.7, we have 𝜏𝑢_𝑛=𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,for𝑛≥0_𝑘𝑛=𝐾^𝐹𝑘_{𝑟𝑘,𝑛}⋅𝐾^𝐹𝑘−1_{𝑟𝑘−1,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛,for𝑘∈{0,1,2,…,𝑁},(3.8) and 𝜏^0𝑛=𝐼forall𝑛∈𝑁,𝑞=𝜏^𝐹𝑘_{𝑟𝑘,𝑛}𝑞,𝑢_𝑛=𝜏^𝑁𝑟_𝑘,𝑁𝑥_𝑛 Then, we have ‖‖𝑢_𝑛−𝑞‖‖²=‖‖𝜏^𝑁𝑟_𝑘,𝑛𝑥_𝑛−𝜏^𝐹𝑘_{𝑟𝑘,𝑛}𝑞‖‖²=‖‖𝑥_𝑛−𝑞‖‖².(3.9) Hence, we get ‖‖𝑦_𝑛−𝑞‖‖≤‖‖𝑥_𝑛−𝑞‖‖.(3.10) From (3.2), we deduce that ‖‖𝑥_𝑛+1−𝑞‖‖=≤‖‖‖‖𝑃𝜖_𝐶𝑛𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑃_𝐶𝑞‖‖≤𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞‖‖≤𝜖𝛾𝜖‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖+1−𝜖_𝑛𝛾‖‖𝑦_𝑛−𝑞‖‖+1−‖‖𝑥𝜖_𝑛𝑛−𝑞‖‖+𝜖_𝑛‖𝛾𝑓(𝑔)−𝐴𝑞‖=1−𝛾‖‖𝑥_𝑛−𝑞‖‖=1−𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑞‖‖−𝜖_𝑛‖𝛾𝑓(𝑞)−𝐴𝑞‖𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑞‖‖+𝛾−𝛾𝜖𝜖_𝑛‖𝛾𝑓(𝑞)−𝐴𝑞‖≤m⋮ax‖‖𝑥_𝑛𝛾−𝛾𝜖−𝑞‖‖,‖𝛾𝑓(𝑞)−𝐴𝑞‖𝛾−𝛾𝜖.(3.11) It follows by induction that ‖‖𝑥_𝑛−𝑞‖‖≤max‖‖𝑥₀−𝑞‖‖,‖𝛾𝑓(𝑞)−𝐴𝑞‖𝛾−𝛾𝜖,𝑛≥0.(3.12) Therefore {𝑥_𝑛} is bounded, so are {𝑦_𝑛},{𝐵𝑢_𝑛},{𝑓(𝑥_𝑛)}, and {𝐴𝑊_𝑛𝑦_𝑛}.Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑦_𝑛+1−𝑦_𝑛‖=0. From (3.2), we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=≤‖‖‖‖𝑃_𝐶𝐼−𝜖_𝑛𝜖𝛾_𝑛𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑃_𝐶𝜖_𝑛−1𝛾𝑓𝑥_𝑛−1+𝐼−𝜖_𝑛−1𝐴𝑊_𝑛𝑦_𝑛−1‖‖𝐴𝑊_𝑛𝑦_𝑛−𝑊_𝑛𝑦_𝑛−1−𝜖_𝑛−𝜖_𝑛−1𝐴𝑊_𝑛𝑦_𝑛−1≤+𝛾1𝜖−_𝑛𝜖𝑓_𝑛𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾𝜖_𝑛−𝜖_𝑛−1𝑓𝑥_𝑛−1‖‖+𝛾||𝜖_𝑛𝛾‖‖𝑦_𝑛−𝑦_𝑛−1‖‖+||𝜖_𝑛−𝜖_𝑛−1||‖‖𝐴𝑊_𝑛𝑦_𝑛‖‖+𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖−𝜖_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖.(3.13) Since 𝐽_𝑀,𝜆 and 𝐼−𝜆𝐵 are nonexpansive, we also have ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖=≤‖‖‖‖𝐽𝑢_{𝑀,𝜆𝑛}𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆𝑢_𝑛−1−𝜆𝐵𝑢_𝑛−1‖‖≤‖‖𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝑢_𝑛−1−𝜆𝐵𝑢_𝑛−1‖‖−𝑢_𝑛−1‖‖.(3.14) On the other hand, from 𝑢_𝑛−1=𝜏^𝑁𝑟_{𝑘,𝑛−1}𝑥_𝑛−1 and 𝑢_𝑛=𝜏^𝑁𝑟_𝑘,𝑛𝑥_𝑛, it follows that 𝐹𝑢_𝑛−1,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛−1+1𝑟_𝑛−1𝐹𝑢⟨𝑦−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1⟩≥0,∀𝑦∈𝐶,(3.15)_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.16) Substituting 𝑦=𝑢_𝑛 into (3.15) and 𝑦=𝑢_𝑛−1 into (3.16), we get 𝐹𝑢_𝑛−1,𝑢_𝑛+𝜑𝑢_𝑛−𝜑𝑢_𝑛−1+1𝑟_𝑛−1𝐹𝑢⟨𝑢_𝑛−𝑢_𝑛−1,𝑢_n−1−𝑥_𝑛−1⟩≥0,_𝑛,𝑢_𝑛+1+𝜑𝑢_𝑛+1−𝜑𝑢_𝑛+1𝑟_𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢𝑛−𝑥_𝑛≥0.(3.17) From (A2), we obtain 𝑢^𝑛−𝑢_𝑛−1,𝑢^𝑛−1−𝑥_𝑛−1𝑟_𝑛−1−𝑢^𝑛−𝑥_𝑛𝑢𝑟_𝑛≥0,_𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1−𝑟^𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛≥0,(3.18) so, 𝑢^𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑢_𝑛+𝑢_𝑛−𝑥_𝑛−1−𝑟^𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛≥0.(3.19) It follows that 𝑢^𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑢_𝑛+𝑢_𝑛−𝑥_𝑛−𝑟^𝑛−1𝑟_𝑛𝑢_𝑛⟨𝑢_𝑛−𝑥_𝑛≥0,−𝑢_𝑛−1,𝑢_𝑛−1−𝑢_𝑛⟩+𝑢_𝑛−𝑢_𝑛−1,1−𝑟^𝑛−1𝑟_𝑛𝑢^𝑛−𝑥_𝑛≥0.(3.20) Without loss of generality, let us assume that there exists a real number 𝑐 such that 𝑟_𝑛−1>𝑐>0, for all 𝑛∈ℕ. Then, we have ‖‖𝑢_𝑛−𝑢_𝑛−1‖‖²≤𝑢_𝑛−𝑢_𝑛−1,1−𝑟^𝑛−1𝑟_𝑛𝑢^𝑛≤‖‖𝑢_𝑛−𝑥_𝑛−𝑢_𝑛−1‖‖||||1−𝑟^𝑛−1𝑟_𝑛||||‖‖𝑢_𝑛−𝑥_𝑛‖‖,(3.21) and hence ‖‖𝑢_𝑛−𝑢_𝑛−1‖‖≤‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1𝑟_𝑛≤‖‖𝑥_𝑛||𝑟_𝑛−𝑟_𝑛−1||‖‖𝑢_𝑛−𝑥_𝑛‖‖−𝑥_𝑛−1‖‖+𝑀¹𝑐||𝑟^𝑛−𝑟_𝑛−1||,(3.22) where 𝑀₁=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈ℕ}. Substituting (3.22) into (3.14), we have ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖≤‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑀¹𝑐||𝑟^𝑛−𝑟_𝑛−1||.(3.23) Substituting (3.23) into (3.13), we get ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1−𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑀¹𝑐||𝑟^𝑛+𝛾𝜖𝜖_𝑛−𝑟_𝑛−1||+||𝜖_𝑛−𝜖_𝑛−1||‖‖𝐴𝑊_𝑛𝑦_𝑛−1‖‖=1−𝜖_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛾||𝜖_𝑛−𝜖_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖𝛾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝜖_𝑛𝛾𝑀¹𝑐||𝑟^𝑛+𝛾𝜖𝜖_𝑛−𝑟_𝑛−1||+||𝜖_𝑛−𝜖_𝑛−1||‖‖𝐴𝑊_𝑛𝑦_𝑛−1‖‖≤1−‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝛾||𝜖_𝑛−𝜖_𝑛−1||‖‖𝑓𝑥_n−1‖‖𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑀¹𝑐||𝑟^𝑛+𝛾||𝜖_𝑛−𝑟_𝑛−1||+||𝜖_𝑛−𝜖_𝑛−1||‖‖𝐴𝑊_𝑛𝑦_𝑛−1‖‖≤1−−𝜖_𝑛−1||‖‖𝑓𝑥_𝑛−1‖‖𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑀¹𝑐||𝑟^𝑛−𝑟_𝑛−1||+𝑀₂||𝜖_𝑛−𝜖_𝑛−1||,(3.24) where 𝑀₂=sup{max{‖𝐴𝑊_𝑛𝑦_𝑛−1‖,‖𝑓(𝑥_𝑛−1)‖∶𝑛∈ℕ}}. Since conditions (C1)-(C2) and by Lemma 2.5, we have ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. From (3.23), we also have ‖𝑦_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞.Step 3. Next, we show that lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑞‖=0.
For 𝑞∈𝜃 hence 𝑞=𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞). By (3.6) and (3.9), we get ‖‖𝑦_𝑛−𝑞‖‖²=‖‖𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞)‖‖²≤‖‖𝑢_𝑛−𝜆𝐵𝑢_𝑛−(𝑞−𝜆𝐵𝑞)‖‖²≤‖‖𝑢_𝑛−𝑞‖‖²+𝜆(𝜆−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖²≤‖‖𝑥_𝑛−𝑞‖‖²+𝜆(𝜆−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖².(3.25) It follows that ‖‖𝑥_𝑛+1−𝑞‖‖²=‖‖𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑃_𝐶(𝑞)‖‖²≤‖‖𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞‖‖²≤𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖+1−𝜖_𝑛𝛾‖‖𝑦_𝑛−𝑞‖‖²≤𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+1−𝜖_𝑛𝛾‖‖𝑦_𝑛−𝑞‖‖²≤+𝜖_𝑛2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖+‖‖1𝛾𝑓−𝜖𝑥_𝑛𝑛−𝐴𝑞‖‖²+2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖≤𝜖_𝑛𝛾‖‖𝑥_𝑛−𝑞‖‖²+𝜆(𝜆−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖²+‖‖‖‖𝑥𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞_𝑛‖‖−𝑞‖‖²+1−𝜖_𝑛𝛾𝜆(𝜆−2𝛽)‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖².(3.26) So, we obtain 1−𝜖_𝑛𝛾𝜆(2𝛽−𝜆)‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖²≤𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑞‖‖+‖‖𝑥_𝑛+1−𝑞‖‖+𝜉_𝑛,(3.27) where 𝜉_𝑛=2𝜖_𝑛(1−𝜖_𝑛𝛾)‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖‖𝑦_𝑛−𝑞‖. By conditions (C1), (C3) and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, then, we obtain that ‖𝐵𝑢_𝑛−𝐵𝑞‖→0 as 𝑛→∞.Step 4. We show the following: (i)lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0; (ii)lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0; (iii)lim_𝑛→∞‖𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖=0. Since 𝐾_𝑟𝑛(𝑥) is firmly nonexpansive and (2.3), we observe that ‖‖𝑢_𝑛−𝑞‖‖²=‖‖𝜏^𝑁𝑟_𝑛,𝑛𝑥_𝑛−𝜏^𝑁𝑟_𝑛,𝑛𝑞‖‖²≤=1⟨𝑥_𝑛−𝑞,𝑢_𝑛−𝑞⟩2‖‖𝑥^𝑛≤1−𝑞‖‖²+‖‖𝑢_𝑛−𝑞‖‖²−‖‖𝑥_𝑛−𝑞−𝑢_𝑛−𝑞‖‖²2‖‖𝑥^𝑛−𝑞‖‖²+‖‖𝑢_𝑛−𝑞‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖²,(3.28) it follows that ‖‖𝑢_𝑛−𝑞‖‖²≤‖‖𝑥_𝑛−𝑞‖‖²−‖‖𝑥_𝑛−𝑢_𝑛‖‖².(3.29)
Since 𝐽_𝑀,𝜆 is 1-inverse-strongly monotone and by (2.3), we compute ‖‖𝑦_𝑛−𝑞‖‖²=‖‖𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞)‖‖²≤=1𝑢_𝑛−𝜆𝐵𝑢_𝑛−(𝑞−𝜆𝐵𝑞),𝑦_𝑛−𝑞2‖‖𝑢^𝑛−𝜆𝐵𝑢_𝑛−(𝑞−𝜆𝐵𝑞)‖‖²+‖‖𝑦_𝑛−𝑞‖‖²≤−1‖‖𝑢_𝑛−𝜆𝐵𝑢_𝑛−(𝑞−𝜆𝐵𝑞)−𝑦_𝑛−𝑞‖‖²2‖‖𝑢^𝑛=1−𝑞‖‖²+‖‖𝑦_𝑛−𝑞‖‖²−‖‖𝑢_𝑛−𝑦_𝑛−𝜆𝐵𝑢_𝑛−𝐵𝑞‖‖²2‖‖𝑢^𝑛−𝑞‖‖²+‖‖𝑦_𝑛−𝑞‖‖²−‖‖𝑢_𝑛−𝑦_𝑛‖‖²+2𝜆⟨𝑢_𝑛−𝑦_𝑛,𝐵𝑢_𝑛−𝐵𝑞⟩−𝜆²‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖²,(3.30) which implies that ‖‖𝑦_𝑛−𝑞‖‖²≤‖‖𝑢_𝑛−𝑞‖‖²−‖‖𝑢_𝑛−𝑦_𝑛‖‖²+2𝜆‖‖𝑢_𝑛−𝑦_𝑛‖‖‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖.(3.31) Substituting (3.31) into (3.26), we have ‖‖𝑥_𝑛+1−𝑞‖‖²≤≤𝜖𝜖_𝑛𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+‖‖𝑦_𝑛−𝑞‖‖²+2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖+2‖‖𝜖𝛾_𝑛𝑓𝑥_𝑛−𝐴𝑞‖‖²+‖‖𝑢_𝑛−𝑞‖‖²−‖‖𝑢_𝑛−𝑦_𝑛‖‖²+2𝜆_𝑛‖‖𝑢_𝑛−𝑦_𝑛‖‖‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖.(3.32) Then, we derive ‖‖𝑥_𝑛−𝑢_𝑛‖‖²+‖‖𝑢_𝑛−𝑦_𝑛‖‖²≤𝜖_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+‖‖𝑥_𝑛−𝑞‖‖²−‖‖𝑥_𝑛+1−𝑞‖‖²=+𝜖_𝑛2𝜆‖‖𝑢_𝑛−𝑦_𝑛‖‖‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖+2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖.+2‖‖𝜆𝛾𝑓‖‖𝑢_𝑛𝑥_𝑛−𝐴𝑞‖‖²+‖‖𝑥_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛−𝑞‖‖+‖‖𝑥_𝑛+1−𝑞‖‖−𝑦_𝑛‖‖‖‖𝐵𝑢_𝑛−𝐵𝑞‖‖+2𝜖_𝑛1−𝜖_𝑛𝛾‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖‖‖𝑦_𝑛−𝑞‖‖.(3.33) By condition (C1), lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑞‖=0.
So, we have ‖𝑥_𝑛−𝑢_𝑛‖→0,‖𝑢_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. It follows that ‖‖𝑥_𝑛−𝑦_𝑛‖‖≤‖‖𝑥_𝑛−𝑢_𝑛‖‖+‖‖𝑢_𝑛−𝑦_𝑛‖‖⟶0,as𝑛⟶∞.(3.34) From (3.2), we have ‖‖𝑥_𝑛−𝑊_𝑛𝑦_𝑛‖‖≤≤‖‖‖‖𝑃𝑥_𝑛𝐶−𝑊_𝑛𝑦_𝑛−1‖‖+‖‖𝑊_𝑛𝑦_𝑛−1−𝑊_𝑛𝑦_𝑛‖‖≤𝜖_𝑛−1𝜖_𝑛−1𝛾𝑓𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝐴𝑊_𝑛𝑦_𝑛−1−𝑃_𝐶𝑊_𝑛𝑦_𝑛−1‖‖+‖‖𝑦_𝑛−1−𝑦_𝑛‖‖‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑊_𝑛𝑦_𝑛−1‖‖+‖‖𝑦_𝑛−1−𝑦_𝑛‖‖.(3.35) By condition (C1) and lim_𝑛→∞‖𝑦_𝑛−1−𝑦_𝑛‖=0, we obtain that ‖𝑥_𝑛−𝑊_𝑛𝑦_𝑛‖→0 as 𝑛→∞.
Hence, we have ‖‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑊_𝑛𝑦_𝑛‖‖+‖‖𝑊_𝑛𝑦_𝑛−𝑊_𝑛𝑥_𝑛‖‖−𝑊_𝑛𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖.(3.36) By (3.34) and lim_𝑛→∞‖𝑥_𝑛−𝑊_𝑛𝑦_𝑛‖=0, we obtain ‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖→0 as 𝑛→∞.
Moreover, we also have ‖‖𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖‖≤‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑊_𝑛𝑦_𝑛‖‖.(3.37) By (3.34) and lim_𝑛→∞‖𝑥_𝑛−𝑊_𝑛𝑦_𝑛‖=0, we obtain ‖𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖→0 as 𝑛→∞.Step 5. We show that 𝑞∈𝜃∶=⋂^∞𝑛=1𝐹(𝑇_𝑛)∩(⋂^𝑁𝑘=1SMEP(𝐹_𝑘))∩𝐼(𝐵,𝑀) and limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛−𝑞⟩≤0. It is easy to see that 𝑃_𝜃(𝛾𝑓+(𝐼−𝐴)) is a contraction of 𝐻 into itself.
Indeed, since 0<𝛾<𝛾/𝜖, we have ‖‖𝑃_𝜃(𝛾𝑓+(𝐼−𝐴))𝑥−𝑃_𝜃(𝛾𝑓+(𝐼−𝐴))𝑦‖‖≤≤𝛾𝛾𝜖‖‖𝑓𝑥(𝑥−)𝑦−‖𝑓+(𝑦)1‖−+‖𝐼−𝐴‖‖𝑥−y‖≤1−𝛾‖𝑥−𝑦‖𝛾+𝛾𝜖‖𝑥−𝑦‖.(3.38) Since 𝐻 is complete, then there exists a unique fixed point 𝑞∈𝐻 such that 𝑞=𝑃_𝜃(𝛾𝑓+(𝐼−𝐴))(𝑞). By Lemma 2.2, we obtain that ⟨(𝛾𝑓−𝐴)𝑞,𝑤−𝑞⟩≤0 for all 𝑤∈𝜃.
Next, we show that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛−𝑞⟩≤0, where 𝑞=𝑃_𝜃(𝛾𝑓+𝐼−𝐴)(𝑞) is the unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑤−𝑞⟩≥0 for all 𝑤∈𝜃. We can choose a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛−𝑞⟩=lim_𝑖→∞(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛𝑖−𝑞.(3.39) As {𝑦_𝑛𝑖} is bounded, there exists a subsequence {𝑦_𝑛𝑖𝑗} of {𝑦_𝑛𝑖} which converges weakly to 𝑤. We may assume without loss of generality that 𝑦_𝑛𝑖⇀𝑤.
Next we claim that 𝑤∈𝜃. Since ‖𝑦_𝑛−𝑊_𝑛𝑦_𝑛‖→0,‖𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖→0, and ‖𝑥_𝑛−𝑦_𝑛‖→0, and by Lemma 2.6, we have 𝑤∈⋂^∞𝑛=1𝐹(𝑇_𝑛).
Next, we show that 𝑤∈⋂^∞𝑘=1SMEP(𝐹_𝑘). Since 𝑢_𝑛=𝜏^𝑁𝑟_𝑘,𝑛𝑥_𝑛, for 𝑘=1,2,3,…,𝑁, we know that 𝐹_𝑘𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶.(3.40) It follows by (A2) that 𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹_𝑘𝑦,𝑢_𝑛,∀𝑦∈𝐶.(3.41) Hence, for 𝑘=1,2,3,…,𝑁, we get 𝜑(𝑦)−𝜑𝑢_𝑛𝑖+1𝑟_𝑛𝑖𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝐹_𝑘𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶.(3.42) For 𝑡∈(0,1] and 𝑦∈𝐻, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. From (3.42), we have 0≥𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−1𝑟_𝑛𝑖𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖+𝐹_𝑘𝑦_𝑡,𝑢_𝑛𝑖.(3.43)
Since ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0, from (A4) and the weakly lower semicontinuity of 𝜑, (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑤. From (A1) and (A4), we have 0=≤𝐹𝑡𝐹_𝑘𝑘𝑦_𝑡,𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑦_𝑡≤𝑡𝐹_𝑘𝑦_𝑡,𝑦+(1−𝑡)𝐹_𝑘𝑦_𝑡,𝑤+𝑡𝜑(𝑦)+(1−𝑡)𝜑(𝑤)−𝜑𝑦_𝑡𝑦_𝑡,𝑦+𝜑(𝑦)−𝜑𝑦_𝑡.(3.44) Dividing by 𝑡, we get 𝐹_𝑘𝑦_𝑡,𝑦+𝜑(𝑦)−𝜑𝑦_𝑡≥0.(3.45) The weakly lower semicontinuity of 𝜑 for 𝑘=1,2,3,…,𝑁, we get 𝐹_𝑘(𝑤,𝑦)+𝜑(𝑦)≥𝜑(𝑤).(3.46) So, we have 𝐹_𝑘(𝑤,𝑦)+𝜑(𝑦)−𝜑(𝑤)≥0,∀𝑘=1,2,3,…,𝑁.(3.47) This implies that 𝑤∈⋂^𝑁𝑘=1SMEP(𝐹_𝑘).
Lastly, we show that 𝑤∈𝐼(𝐵,𝑀). In fact, since B is 𝛽-inverse strongly monotone, hence 𝐵 is a monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that 𝑀+𝐵 is a maximal monotone. Let (𝑣,𝑔)∈𝐺(𝑀+𝐵), since 𝑔−𝐵𝑣∈𝑀(𝑣). Again since 𝑦_𝑛𝑖=𝐽_𝑀,𝜆(𝑢_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖), we have 𝑢_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖∈(𝐼+𝜆𝑀)(𝑦_𝑛𝑖), that is, (1/𝜆)(𝑢_𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖)∈𝑀(𝑦_𝑛𝑖). By virtue of the maximal monotonicity of 𝑀+𝐵, we have 𝑣−𝑦^𝑛_𝑖,𝑔−𝐵𝑣−𝜆1𝑢^𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖≥0,(3.48) and hence 𝑣−𝑦_𝑛𝑖,𝑔≥=𝑣𝑣−−𝑦𝑦_{𝑛𝑖𝑛𝑖},𝐵𝑣+𝜆1𝑢^𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖,𝐵𝑣−𝐵𝑦_𝑛𝑖+𝑣−𝑦_𝑛𝑖,𝐵𝑦_𝑛𝑖−𝐵𝑢_𝑛𝑖+𝑣−𝑦_𝑛𝑖,1𝜆𝑢^𝑛𝑖−𝑦_𝑛𝑖.(3.49) It follows from lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0, we have lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑦_𝑛‖=0 and 𝑦_𝑛𝑖⇀𝑤, it follows that limsup_𝑛→∞⟨𝑣−𝑦_𝑛,𝑔⟩=⟨𝑣−𝑤,𝑔⟩≥0.(3.50) It follows from the maximal monotonicity of 𝐵+𝑀 that 𝜃∈(𝑀+𝐵)(𝑤), that is, 𝑤∈𝐼(𝐵,𝑀). Therefore, 𝑤∈𝜃. We observe that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛−𝑞⟩=lim_𝑖→∞(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛𝑖−𝑞=⟨(𝛾𝑓−𝐴)𝑞,𝑤−𝑞⟩≤0.(3.51)Step 6. Finally, we prove 𝑥_𝑛→𝑞. By using (3.2) and together with Schwarz inequality, we have ‖‖𝑥_𝑛+1−𝑞‖‖²=‖‖𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑃_𝐶(𝑞)‖‖²≤‖‖𝜖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞‖‖²≤𝐼−𝜖_𝑛𝐴²‖‖𝑊_𝑛𝑦_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²≤+12𝜖−_𝑛𝜖_𝑛𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞𝛾²‖‖𝑦_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²≤+12𝜖−_𝑛𝜖_𝑛𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞−2𝜖^2𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞+2𝜖_𝑛𝛾²‖‖𝑥_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2𝜖_𝑛𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞)≤1−𝜖⟨_𝑛𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩−2𝜖^2𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞+2𝜖_𝑛𝛾²‖‖𝑥_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2𝜖_𝑛‖‖𝑊_𝑛𝑦_𝑛−𝑞‖‖‖‖𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞)‖‖≤1−𝜖⟨_𝑛𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩−2𝜖^2𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞+2𝜖_𝑛𝛾²‖‖𝑥_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2𝛾𝜖𝜖_𝑛‖‖𝑦_𝑛−𝑞‖‖‖‖𝑥_𝑛−𝑞‖‖≤1−𝜖⟨_𝑛𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩−2𝜖^2𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞𝛾²‖‖𝑥_𝑛−𝑞‖‖²+𝜖^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑞‖‖²≤+2𝜖1_𝑛−⟨𝑊𝜖_𝑛𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩−2𝜖^2𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞𝛾²+2𝛾𝜖𝜖_𝑛‖‖𝑥^𝑛−𝑞‖‖²+𝜖_𝑛−2𝜖𝜖_𝑛𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2⟨𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩=1−2‖‖𝐴𝑊_𝑛𝑦_𝑛−𝑞‖‖‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑞‖‖²+𝜖_𝑛−2𝜖𝜖_𝑛‖‖_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖²+2⟨𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩+𝜖_𝑛‖‖𝐴𝑊_𝑛𝑦_𝑛−𝑞‖‖‖‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖𝛾²‖‖𝑥_𝑛−𝑞‖‖².(3.52)
Since {𝑥_𝑛} is bounded, where 𝜂≥‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖²−2‖𝐴(𝑊_𝑛𝑦_𝑛−𝑞)‖‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖+𝛾²‖𝑥_𝑛−𝑞‖² for all 𝑛≥0. It follows that ‖‖𝑥_𝑛+1−𝑞‖‖²≤1−2𝛾−𝛾𝜖𝜖_𝑛‖‖𝑥_𝑛−𝑞‖‖²+𝜖_𝑛𝛿_𝑛,(3.53) where 𝛿_𝑛=2⟨𝑊_𝑛𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩+𝜂𝛼_𝑛. Since limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑊_𝑛𝑦_𝑛−𝑞⟩≤0, we get limsup_𝑛→∞𝛿_𝑛≤0. Applying Lemma 2.5, we can conclude that 𝑥_𝑛→𝑞. This completes the proof.

Corollary 3.2. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Let 𝐵 be 𝛽-inverse-strongly monotone and 𝜑∶𝐶→ℛ a convex and lower semicontinuous function. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1), 𝑀∶𝐻→2^𝐻 a maximal monotone mapping, and {𝑇_𝑛} a family of nonexpansive mappings of 𝐻 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐼(𝐵,𝑀)≠0.(3.54) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢_𝑛∈𝐶 arbitrarily: 𝑢_𝑛𝑥=𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝑊_𝑛𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,(3.55) for all 𝑛=0,1,2,…, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝑓+𝐼)(𝑞) which solves the following variational inequality: ⟨(𝑓−𝐼)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃.(3.56)

Proof. Putting 𝐴≡𝐼 and 𝛾≡1 in Theorem 3.1, we can obtain the desired conclusion immediately.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Let 𝐵 be 𝛽-inverse-strongly monotone, 𝜑∶𝐶→ℛ a convex and lower semicontinuous function, and 𝑀∶𝐻→2^𝐻 a maximal monotone mapping. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐻 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐼(𝐵,𝑀)≠0.(3.57) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢∈𝐶 and 𝑢_𝑛∈𝐶: u_𝑛=𝑥𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝑢+𝐼−𝜖_𝑛𝑊_𝑛𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛,(3.58) for all 𝑛=0,1,2,…, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝑞) which solves the following variational inequality: ⟨𝑢−𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃.(3.59)

Proof. Putting 𝑓(𝑥)≡𝑢, for all 𝑥∈𝐶 in Corollary 3.2, we can obtain the desired conclusion immediately.

Corollary 3.4. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻, and let 𝐵 be 𝛽-inverse-strongly monotone mapping and 𝐴 a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and {𝑇_𝑛} be a family of nonexpansive mappings of 𝐻 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩VI(𝐶,𝐵)≠∅.(3.60) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily: 𝑥_𝑛+1=𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛𝑃_𝐶𝑥_𝑛−𝜆𝐵𝑥_𝑛,(3.61) for all 𝑛=0,1,2,…, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃.(3.62)

Proof. Taking 𝐹≡0, 𝜑≡0,𝑢_𝑛=𝑥_𝑛, and 𝐽_𝑀,𝜆=𝑃_𝐶 in Theorem 3.1, we can obtain the desired conclusion immediately.

Remark 3.5. Corollary 3.4 generalizes and improves the result of Klin-Eam and Suantai [45].

4. Applications

In this section, we apply the iterative scheme (1.25) for finding a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping.

Definition 4.1. A mapping 𝑆∶𝐶→𝐶 is called a strictly pseudocontraction if there exists a constant 0≤𝜅<1 such that ‖𝑆𝑥−𝑆𝑦‖²≤‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦‖²,∀𝑥,𝑦∈𝐶.(4.1) If 𝜅=0, then 𝑆 is nonexpansive. In this case, we say that 𝑆∶𝐶→𝐶 is a 𝜅-strictly pseudocontraction. Putting 𝐵=𝐼−𝑆. Then, we have ‖(𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦‖²≤‖𝑥−𝑦‖²+𝜅‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶.(4.2) Observe that ‖(𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦‖²=‖𝑥−𝑦‖²+‖𝐵𝑥−𝐵𝑦‖²−2⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩,∀𝑥,𝑦∈𝐶.(4.3) Hence, we obtain ⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩≥1−𝜅2‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶.(4.4) Then, 𝐵 is a ((1−𝜅)/2)-inverse-strongly monotone mapping.

Using Theorem 3.1, we first prove a strongly convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontraction.

Theorem 4.2. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻, and let 𝐵 be an 𝛽-inverse-strongly monotone, 𝜑∶𝐶→ℛ a convex and lower semicontinuous function, and 𝑓∶𝐶→𝐶 a contraction with coefficient 𝛼(0<𝛼<1), and let 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐻 into itself, and let 𝑆 be a 𝜅-strictly pseudocontraction of 𝐶 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐹(𝑆)≠0.(4.5) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢_𝑛∈𝐶 arbitrarily: 𝑢𝑥_𝑛=𝐾^𝐹𝑁_{𝑟𝑛,𝑛}⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑊_𝑛(1−𝜆)𝑥_𝑛+𝜆𝑆𝑥_𝑛,(4.6) for all 𝑛=0,1,2,…, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃,(4.7) which is the optimality condition for the minimization problem min_𝑞∈𝜃12⟨𝐴𝑞,𝑞⟩−ℎ(𝑞),(4.8) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Put 𝐵≡𝐼−𝑇, then 𝐵 is (1−𝜅)/2 inverse-strongly monotone and 𝐹(𝑆)=𝐼(𝐵,𝑀), and 𝐽_𝑀,𝜆(𝑥_𝑛−𝜆𝐵𝑥_𝑛)=(1−𝜆)𝑥_𝑛+𝜆𝑇𝑥_𝑛. So by Theorem 3.1, we obtain the desired result.

Corollary 4.3. Let 𝐻 be a real Hilbert space and 𝐶 a closed convex subset of 𝐻, and let B be 𝛽-inverse-strongly monotone and 𝜑∶𝐶→ℛ a convex and lower semicontinuous function. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and 𝑇_𝑛 a nonexpansive mapping of 𝐻 into itself, and let 𝑆 be a 𝜅-strictly pseudocontraction of 𝐶 into itself such that 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐹(𝑆)≠0.(4.9) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily: 𝑢𝑥_𝑛=𝐾^𝐹𝑁_𝑟n,𝑛⋅𝐾^𝐹𝑁−1_{𝑟𝑛−1,𝑛}⋅𝐾^𝐹𝑁−2_{𝑟𝑛−2,𝑛}⋅⋯⋅𝐾^𝐹2_𝑟2,𝑛⋅𝐾^𝐹1_𝑟1,𝑛⋅𝑥_𝑛,∀𝑛∈𝑁_𝑛+1=𝑃_𝐶𝜖_𝑛𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝑊_𝑛(1−𝜆)𝑢_𝑛+𝜆𝑆𝑢_𝑛,(4.10) for all 𝑛=0,1,2,…, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝜃, where 𝑞=𝑃_𝜃(𝑓+𝐼)(𝑞) which solves the following variational inequality: ⟨(𝑓−𝐼)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈𝜃,(4.11) which is the optimality condition for the minimization problem min_𝑞∈𝜃12⟨𝐴𝑞,𝑞⟩−ℎ(𝑞),(4.12) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Put 𝐴≡𝐼 and 𝛾≡1 in Theorem 4.2, we obtain the desired result.

5. Numerical Example

Now, we give a real numerical example in which the condition satisfies the ones of Theorem 3.1 and some numerical experiment results to explain the main result Theorem 3.1 as follows.

Example 5.1. Let 𝐻=𝑅, 𝐶=[−1,1], 𝑇_𝑛=𝐼, 𝜆_𝑛=𝛽∈(0,1), 𝑛∈𝑁, 𝐹_𝑘(𝑥,𝑦)=0, for all 𝑥,𝑦∈𝐶, 𝑟_𝑛,𝑛=1, 𝑘∈{1,2,3,…,𝑁}, 𝜑(𝑥)=0, for all 𝑥∈𝐶,𝐵=𝐴=𝐼,𝑓(𝑥)=(1/5)𝑥, for all 𝑥∈𝐻, 𝜆=1/2 with contraction coefficient 𝛼=1/10, 𝜖_𝑛=1/𝑛 for every 𝑛∈𝑁, and 𝛾=1. Then {𝑥_𝑛} is the sequence generated by 𝑥_𝑛+1=12−130𝑛𝑥^𝑛,(5.1) and 𝑥_𝑛→0 as 𝑛→∞, where 0 is the unique solution of the minimization problem min_𝑥∈𝐶=25𝑥²+𝑞.(5.2)

Proof. We prove Example 5.1 by Step 1, Step 2, and Step 3. By Step 4, we give two numerical experiment results which can directly explain that the sequence {𝑥_𝑛} strongly converges to 0.
Step 1. We show 𝐾^𝐹𝑁_{𝑟𝑛,𝑛}𝑥=𝑃_𝐶𝑥,∀𝑥∈𝐻,𝐹_𝑁∈{1,2,3,…,𝑁},(5.3) where 𝑃_𝐶𝑥=𝑥|𝑥𝑥,|,𝑥𝑥∈∈𝐻𝐶.⧵𝐶(5.4)
Indeed, since 𝐹_𝑘(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝐶,𝑛∈{1,2,3,…,𝑁}, due to the definition of 𝐾_𝑟(𝑥), for all 𝑥∈𝐻, as Lemma 2.7, we have 𝐾_𝑟(𝑥)={𝑢∈𝐶∶⟨𝑦−𝑢,𝑢−𝑥⟩≥0,∀𝑦∈𝐶}.(5.5)
Also by the equivalent property (2.2) of the nearest projection 𝑃_𝐶 from 𝐻→𝐶, we obtain this conclusion, when we take 𝑥∈𝐶, 𝐾^𝐹𝑁_{𝑟𝑛,𝑛}𝑥=𝑃_𝐶𝑥=𝐼𝑥. By (iii) in Lemma 2.7, we have ^𝑁_𝑘=1SMEP𝐹_𝑘=𝐶.(5.6)Step 2. We show that 𝑊_𝑛=𝐼.(5.7)
Indeed. By (1.23), we have 𝑊𝑊₁=𝑈₁₁=𝜆₁𝑇₁𝑈₁₂+1−𝜆₁𝐼=𝜆₁𝑇₁+1−𝜆₁𝐼,₂=𝑈₂₁==𝜆𝜆₁₁𝑇₁𝑈₂₂+1−𝜆₁𝐼=𝜆₁𝑇₁𝜆₂𝑇₂𝑈₂₃+1−𝜆₂𝐼+1−𝜆₁𝐼𝑊𝜆₂𝑇₁𝑇₂+𝜆₁1−𝜆₂𝑇₁+1−𝜆₁𝐼,₃=𝑈₃₁==𝜆𝜆₁₁𝑇₁𝑈₃₂+1−𝜆₁𝐼=𝜆₁𝑇₁𝜆₂𝑇₂𝑈₃₃+1−𝜆₂𝐼+1−𝜆₁𝐼=𝜆₁𝜆₂𝑇₁𝑇₂𝑈₃₃+𝜆₁1−𝜆₂𝑇₁+1−𝜆₁𝐼,=𝜆₁𝜆₂𝑇₁𝑇₂𝜆₃𝑇₃𝑈₃₄+1−𝜆₃𝐼+𝜆₁1−𝜆₂𝑇₁+1−𝜆₁𝐼,𝜆₂𝜆₃𝑇₁𝑇₂𝑇₃+𝜆₁𝜆₂1−𝜆₃𝑇₁𝑇₂+𝜆₁1−𝜆₂𝑇₁+1−𝜆₁𝐼.(5.8)
Computing in this way by (1.23), we obtain 𝑊_𝑛=𝑈_𝑛1=𝜆₁𝜆₂⋯𝜆_𝑛𝑇₁𝑇₂⋯𝑇_𝑛+𝜆₁𝜆₂⋯𝜆_𝑛−11−𝜆_𝑛𝑇₁𝑇₂⋯𝑇_𝑛−1+𝜆₁𝜆₂⋯𝜆_𝑛−21−𝜆_𝑛−1𝑇₁𝑇₂⋯𝑇_𝑛−2+⋯+𝜆₁1−𝜆₂𝑇₁+1−𝜆₁𝐼.(5.9)
Since 𝑇_𝑛=𝐼,𝜆_𝑛=𝛽,𝑛∈𝑁, thus 𝑊_𝑛=𝛽^𝑛+𝛽^𝑛−1(1−𝛽)+⋯+𝛽(1−𝛽)+(1−𝛽)𝐼=𝐼.(5.10)Step 3. We show that 𝑥_𝑛+1=12−130𝑛𝑥^𝑛,𝑥_𝑛+1⟶0,as𝑛⟶∞,(5.11) where 0 is the unique solution of the minimization problem min_𝑥∈𝐶25𝑥²+𝑞.(5.12)
Indeed, we can see that 𝐴=𝐼 is a strongly position bounded linear operator with coefficient 𝛾=1/2and𝛾 is a real number such that 0<𝛾<𝛾/𝛼, so we can take 𝛾=1. Due to (5.1), (5.4), and (5.7), we can obtain a special sequence {𝑥_𝑛} of (3.2) in Theorem 3.1 as follows: 𝑥_𝑛+1=12−130𝑛𝑥^𝑛.(5.13) Since 𝑇_𝑛=𝐼,𝑛∈𝑁, so, ^∞_𝑛=1𝐹𝑇_𝑛=𝐻,(5.14) combining with (5.6), we have 𝜃∶=^∞_𝑛=1𝐹𝑇_𝑛∩^𝑁_𝑘=1SMEP𝐹_𝑘∩𝐼(𝐵,𝑀)=𝐶=[−1,1].(5.15)
By Lemma 2.5, it is obviously that 𝑧_𝑛→0, 0 is the unique solution of the minimization problem min_𝑥∈𝐶25𝑥²+𝑞,(5.16) where 𝑞 is a constant number.Step 4. We give the numerical experiment results using software Mathlab 7.0 and get Table 1 to Table 2, which show that the iteration process of the sequence {𝑥_𝑛} is a monotone-decreasing sequence and converges to 0, but the more the iteration steps are, the more showily the sequence {𝑥_𝑛} converges to 0.

Table 1 

This table shows the value of sequence {𝑥_𝑛} on each iteration step (initial value 𝑥₁=1).

Table 2 

This table shows the value of sequence {𝑥_𝑛} on each iteration step (initial value 𝑥₁=1/2).

Now, we turn to realizing (3.2) for approximating a fixed point of 𝑇. We take the initial valued 𝑥₁=1 and 𝑥₁ = 1/2, respectively. All the numerical results are given in Tables 1 and 2. The corresponding graph appears in Figures 1(a) and 1(b).

(a)

(b)

Figure 1 

The iteration comparison chart of different initial values. (a) 𝑥1=1 and (b) 𝑥1=1/2.

The numerical results support our main theorem as shown by calculating and plotting graphs using Matlab 7.11.0.

Acknowledgments

The authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the project NRU-CSEC no. 54000267) for financial support. Furthermore, the second author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkut's University of Technology Thonburi (KMUTT) (Grant no. MRG5360044). Finally, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out major errors in the original version of this paper.