Fixed Points Approximation and Solutions of Some Equilibrium and Variational Inequalities Problems

Abstract

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.

1. Introduction

Let 𝐻 be a real Hilbert space; a mapping 𝐵∶𝐷(𝐵)→𝐻 is said to be monotone if for all 𝑥,𝑦∈𝐷(𝐵)

⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0. (1.1) For some 𝜆>0, the mapping 𝐵 is said to be 𝜆-inverse strongly monotone if

⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝜆‖𝐵𝑥−𝐵𝑦‖². (1.2) A 𝜆-inverse strongly monotone map is some time called 𝜆-cocoercive. A map 𝐵 is said to be relaxed 𝜆-cocoercive if there exists a constant 𝜆>0 such that

⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥−𝜆‖𝐵𝑥−𝐵𝑦‖²∀𝑥,𝑦∈𝐾. (1.3)𝐵 is said to be relaxed (𝜆,𝛾)-cocoercive, if there exist 𝜆,𝛾>0 such that

⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥−𝜆‖𝐵𝑥−𝐵𝑦‖²+𝛾‖𝑥−𝑦‖². (1.4) A map 𝐵∶𝐻→𝐻 is said to be 𝜆-Lipschitzian if there exists a real number 𝜆≥0 such that

‖𝐵𝑥−𝐵𝑦‖≤𝜆‖𝑥−𝑦‖∀𝑥,𝑦∈𝐻. (1.5)𝐵 is a contraction map, if in the above inequality 𝜆∈[0,1) and nonexpansive if 𝜆=1.

Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. A variational inequality problem is searched for 𝑥^∗∈𝐾 such that

⟨𝐵𝑥^∗,𝑦−𝑥^∗⟩≥0∀𝑦∈𝐾, (1.6) where 𝐵 is some nonlinear mapping of 𝐾 into 𝐻. Inequality (1.6) is called the variational inequality.

Recall that for each 𝑥∈𝐻 there exists a unique nearest point in 𝐾 to 𝑥 denoted by 𝑃_𝐾𝑥. That is, ‖𝑥−𝑃_𝐾𝑥‖≤‖𝑥−𝑦‖ for all 𝑦∈𝐾. 𝑃_𝐾 is called a metric projection of 𝐻 onto 𝐾. The mapping 𝑃_𝐾 is nonexpansive in this setting, that is, ‖𝑃_𝐾𝑥−𝑃_𝐾𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. It is also known that 𝑃_𝐾 satisfies the following inequality ‖𝑃_𝐾𝑥−𝑃_𝐾𝑦‖²≤⟨𝑥−𝑦,𝑃_𝐾𝑥−𝑃_𝐾𝑦⟩.

The solution set of the problem (1.6) is denoted by VI(𝐾,𝐵). It is well known (see [1]) that 𝑥^∗∈VI(𝐾,𝐵) if and only if

𝑥^∗=𝑃_𝐾(𝑥^∗−𝜆𝐵𝑥^∗),∀𝜆>0. (1.7) A monotone map 𝐵 is said to be maximal if the graph Γ(𝐵) of 𝐵 is not properly contained in the graph of any other monotone map, where Γ(𝐵)={(𝑥,𝑦)∈𝐻×𝐻∶𝑦∈𝐵𝑥} for a multivalued map 𝐵. It is also known that 𝐵 is maximal monotone if and only if for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈Γ(𝐵) implies 𝑓∈𝐵𝑥. Let 𝐵 be a monotone mapping defined from 𝐾 into 𝐻 and let 𝑁_𝐾𝑞 be a normal cone to 𝐾 at 𝑞∈𝐾, that is, 𝑁_𝐾𝑞={𝑝∈𝐻∶⟨𝑞−𝑢,𝑝⟩≥,forall𝑢∈𝐾}. Define a map 𝑀 by

𝑀𝑞=𝐵𝑞+𝑁_𝐾𝑞,𝑞∈𝐾,∅,𝑞∉𝐾. (1.8) Then, 𝑀 is maximal monotone and 𝑥^∗∈𝑀⁻¹(0)⇔𝑥^∗∈VI(𝐾,𝐵), see, for example, [2].

Let 𝐺∶𝐾×𝐾→ℝ be a bifunction on a closed convex nonempty subset 𝐾 of a real Hilbert space 𝐻. An equilibrium problem is searched for 𝑥^∗∈𝐾 such that

𝐺(𝑥^∗,𝑦)≥0∀𝑦∈𝐾. (1.9) The set of solutions of the equilibrium problem above is denoted by EP(𝐺).

Several physical problems (such as the theories of lubrications, filterations and flows, moving boundary problems, see, e.g., [1, 3]) can be reduced to variational inequality or equilibrium problems. Consequently, these problems have solutions as the solutions of these resultant variational inequality or equilibrium problems.

Maingé [4] introduced a Halpern-type scheme and proved a strong convergence theorem for family of nonexpansive mappings in Hilbert space.

Recently, S. Takahashi and W. Takahashi [5] introduced an iterative scheme which they used to study the problem of approximating a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

More recently, Kumam and Katchang [6], W. Kumam and P. Kumam [7], Li and Su [8] and many others (see, e.g. [9–12] and the references contained in them) studied the problem of fixed point approximations and solutions of some equilibrium and, or solutions of some variational inequalities problems.

In this paper we introduce a new iterative scheme for approximation of a common element in the intersection of the set of fixed points of some countable family of nonexpansive mappings, the set of solutions of some equilibrium problem, and the set of solutions of some variational inequality problem and prove a new theorem. Our theorem generalizes and improves some recent results.

2. Preliminaries

For a sequence {𝑥_𝑛} the notation 𝑥_𝑛→𝑥^∗ and 𝑥_𝑛⇀𝑥^∗ means that the sequence {𝑥_𝑛} converges strongly and weakly to 𝑥^∗, repectively. A Banach space 𝐸 is said to satisfy an Opial's condition (or in other words is an Opial's space) if for a sequence {𝑥_𝑛} in 𝐸 with 𝑥_𝑛⇀𝑥^∗, then

liminf_𝑛→∞‖𝑥_𝑛−𝑥^∗‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖forany𝑦∈𝐸𝑦≠𝑥^∗. (2.1) It is well known that Hilbert spaces are Opial’s spaces (see [13]).

In the sequel we shall make use of the following results.

Lemma 2.1 (see [14]). Let 𝐾 be a nonempty closed convex subset of 𝐻 and let 𝐺 be a bifunction of 𝐾×𝐾 into ℝ satisfying(𝐴1)𝐺(𝑥,𝑥)=0forall𝑥∈𝐾;(𝐴2)𝐺 is monotone, i.e. 𝐺(𝑥,𝑦)+𝐺(𝑦,𝑥)≤0forall𝑥,𝑦∈𝐾;(𝐴3)forall𝑥,𝑦,𝑧∈𝐾,limsup_𝑡→0+𝐺(𝑡_𝑧+(1+𝑡)𝑥,𝑦)≤𝐺(𝑥,𝑦);(𝐴4)forall𝑥∈𝐾,𝐺(𝑥,.) is convex and lower semicontinuous.Let 𝑟>0 and 𝑥∈𝐻. Then there exists 𝑧∈𝐾 such that 1𝐺(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐾. (2.2)

Lemma 2.2 (see [15]). Let 𝐾 be a nonempty closed convex subset of 𝐻 and let 𝐺 be a bifunction of 𝐾×𝐾 into ℝ satisfying (𝐴1)-(𝐴4). For 𝑟>0 and 𝑥∈𝐻 define a map 𝑇_𝑟∶𝐻→𝐾 by 𝑇_𝑟1𝑥={𝑧∈𝐾∶𝐺(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐾}. (2.3) Then, the following holds:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩; (2.4)(3)Fix(𝑇_𝑟)=EP(𝐺);(4)EP(𝐺) is closed and convex.

Lemma 2.3 (see [16]). Let 𝐻 be a real inner product space. Then, the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩∀𝑥,𝑦∈𝐻. (2.5)

Lemma 2.4 (see [17]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝐸 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf𝛽_𝑛≤limsup𝛽_𝑛<1. Suppose 𝑥_𝑛+1=𝛽_𝑛𝑦_𝑛+(1−𝛽_𝑛)𝑥_𝑛 for all integers 𝑛≥0 and limsup(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.5 (see [18]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤(1−𝛼_𝑛)𝑎_𝑛+𝛼_𝑛𝜎_𝑛+𝛾_𝑛,𝑛≥0, (2.6) Where (i){𝛼_𝑛}⊂[0,1], ∑𝛼_𝑛=∞; (ii)limsup𝜎_𝑛≤0; (iii)𝛾_𝑛≥0;(𝑛≥0),∑𝛾_𝑛<∞. Then, 𝑎_𝑛→0 as 𝑛→∞.

3. Main Results

In the sequel we assume that the sequences {𝛼_𝑛},{𝜎_𝑖,𝑛}_𝑖⊂(0,1) satisfy ∑_𝑖≥1𝜎_𝑖,𝑛=1−𝛼_𝑛.

Theorem 3.1. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐺 be a bifunction from 𝐾×𝐾 to ℝ which satisfies conditions (𝐴1)-(𝐴4). Let {𝑇_𝑖,𝑖=1,2,3,…} be family of nonexpansive mappings of 𝐾 into 𝐻 and let 𝐵 be a 𝜇-Lipschitzian, relaxed (𝜆,𝛾)-cocoercive map of 𝐾 into 𝐻 such that 𝐹∶=⋂^∞_𝑖=1Fix(𝑇_𝑖)∩EP(𝐺)∩VI(𝐾,𝐵)≠∅. For an arbitrary but fixed 𝛿∈(0,1), let {𝑥_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁,𝑢∈𝐻,𝐺(𝑦_𝑛1,𝜂)+𝑟_𝑛⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝜂∈𝐾,_𝑛+1=𝛼_𝑛𝑢+(1−𝛿)(1−𝛼_𝑛)𝑥_𝑛+𝛿_𝑖≥1𝜎_𝑖𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛,𝑛≥1, (3.1) where {𝛼_𝑛} and {𝜎_𝑖𝑛} are sequences in [0,1] and {𝑟_𝑛} and {𝑠_𝑛} are sequences in [0,∞) satisfying (𝐶1)lim_𝑛→∞𝛼_𝑛=0,(𝐶2)∑^∞_𝑛=1𝛼_𝑛=∞,(𝐶3){𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 satisfying 0≤𝑎≤𝑏≤2(𝛾−𝜆𝜇²)/𝜇²,(𝐶4)lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0, lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, lim_𝑛→∞∑_𝑖≥1|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|=0,(𝐶5)liminf_𝑛→∞𝑟_𝑛>0.Then, both {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑃_𝐹𝑢.

Proof. First, we show that (𝐼−𝑠_𝑛𝐵) is nonexpansive, actually, using the property of 𝐵 we have for 𝑥,𝑦∈𝐻, ‖(𝐼−𝑠_𝑛𝐵)𝑥−(𝐼−𝑠_𝑛𝐵)𝑦‖²=‖𝑥−𝑦−𝑠_𝑛(𝐵𝑥−𝐵𝑦)‖²=‖𝑥−𝑦‖²−2𝑠_𝑛⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩+𝑠²_𝑛‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²−2𝑠_𝑛[−𝜆‖𝐵𝑥−𝐵𝑦‖²+𝛾‖𝑥−𝑦‖²]+𝑠²_𝑛‖𝐵𝑥−𝐵𝑦‖²≤‖𝑥−𝑦‖²+2𝑠_𝑛𝜇²𝜆‖𝑥−𝑦‖²−2𝑠_𝑛𝛾‖𝑥−𝑦‖²+𝜇²𝑠²_𝑛‖𝑥−𝑦‖²=(1+2𝑠_𝑛𝜇²𝜆−2𝑠_𝑛𝛾+𝜇²𝑠²_𝑛)‖𝑥−𝑦‖²≤‖𝑥−𝑦‖², (3.2) and thus (𝐼−𝑠_𝑛𝐵) is nonexpansive.
Let 𝑥^∗∈𝐹; since 𝑦_𝑛=𝑇_𝑟𝑛𝑥_𝑛, 𝑛∈ℕ and the fact that 𝑇_𝑟𝑛 is firmly nonexpansive (and hence nonexpansive) we have the following: ‖𝑦_𝑛−𝑥^∗‖=‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖≤‖𝑥_𝑛−𝑥^∗‖. (3.3) We claim that {𝑥_𝑛} satisfies ‖𝑥_𝑛−𝑥^∗‖≤max{‖𝑢−𝑥^∗‖,‖𝑥₁−𝑥^∗‖}∀𝑛≥1. (3.4) We prove this by induction. Cearly the result is true for 𝑛=1. Assume that the result holds for 𝑛=𝑘 for some 𝑘∈ℕ. Then, for 𝑛=𝑘+1 we have ‖𝑥_𝑘+1−𝑥^∗‖=‖𝛼_𝑘(𝑢−𝑥^∗)+(1−𝛿)(1−𝛼_𝑘)(𝑥_𝑘−𝑥^∗)+𝛿_𝑖≥1𝜎_𝑖𝑘𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑘𝐵)(𝑦_𝑘−𝑥^∗)‖≤𝛼_𝑘‖𝑢−𝑥^∗‖+(1−𝛿)(1−𝛼_𝑘)‖𝑥_𝑘−𝑥^∗‖+𝛿_𝑖≥1𝜎_𝑖𝑘‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑘𝐵)𝑦_𝑘−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑘𝐵)𝑥^∗‖≤𝛼_𝑘‖𝑢−𝑥^∗‖+(1−𝛼_𝑘)‖𝑥_𝑘−𝑥^∗‖≤max{‖𝑢−𝑥^∗‖,‖𝑥_𝑘−𝑥^∗‖}. (3.5) Hence the result, and so {𝑥_𝑛} is bounded. Furthermore, {𝑦_𝑛}, {𝑇_𝑖𝑃_𝐾(𝑦_𝑛−𝑠_𝑛𝐵𝑦_𝑛)} and {𝐵𝑦_𝑛} are each bounded.
We now show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Note that 𝑦_𝑛=𝑇_𝑟𝑛𝑥_𝑛, 𝑦_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1, so that 𝐺(𝑦_𝑛1,𝜂)+𝑟_𝑛⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛⟩≥0∀𝜂∈𝐾,𝐺(𝑦_𝑛+11,𝜂)+𝑟_𝑛+1⟨𝜂−𝑦_𝑛+1,𝑦_𝑛+1−𝑥_𝑛+1⟩≥0∀𝜂∈𝐾. (3.6) Using (3.6) and (𝐴2), we have ⟨𝑦_𝑛+1−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑟_𝑛−𝑦_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1⟩≥0. (3.7) which implies that ⟨𝑦_𝑛+1−𝑦_𝑛,𝑦_𝑛−𝑦_𝑛+1+𝑦_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1(𝑦_𝑛+1−𝑥_𝑛+1)⟩≥0. (3.8) from which we get ‖𝑦_𝑛+1−𝑦_𝑛‖²≤‖𝑦_𝑛+1−𝑦_𝑛‖[‖𝑥_𝑛+1−𝑥_𝑛𝑟‖+|1−^𝑛𝑟_𝑛+1|‖𝑦_𝑛+1−𝑥_𝑛+1‖]. (3.9) If, without loss of generality 𝑀, 𝑚 are real numbers such that 𝑟_𝑛>𝑚>0 for all 𝑛 and 𝑀∶=sup_𝑛,𝑖{‖𝑦_𝑛−𝑥_𝑛‖,‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛‖,‖𝐵𝑦_𝑛‖} we then have ‖𝑦_𝑛+1−𝑦_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛𝑟‖+|1−^𝑛𝑟_𝑛+1|‖𝑦_𝑛+1−𝑥_𝑛+1‖≤‖𝑥_𝑛+1−𝑥_𝑛𝑀‖+𝑚|𝑟_𝑛+1−𝑟_𝑛|. (3.10) Now, define two sequences {𝛽_𝑛} and {𝑧_𝑛} by 𝛽_𝑛∶=(1−𝛿)𝛼_𝑛+𝛿 and 𝑧_𝑛∶=(𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑥_𝑛)/𝛽_𝑛. Then, 𝑧_𝑛=𝛼_𝑛𝑢+𝛿∑_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛𝛽_𝑛. (3.11) Observe that {𝑧_𝑛} is bounded and that ‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛𝛼‖≤|^𝑛+1𝛽_𝑛+1−𝛼_𝑛𝛽_𝑛+|‖𝑢‖𝛿(1−𝛼_𝑛+1)𝛽_𝑛+1‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖+𝛿𝛽_{𝑛+1𝑖≥1}|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛+1𝐵)𝑦_𝑛‖+𝛿𝛽_𝑛+1𝛽_𝑛|𝛽_𝑛−𝛽_𝑛+1|_𝑖≥1𝜎_𝑖,𝑛+1‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛+1𝐵)𝑦_𝑛‖+𝛿(1−𝛼_𝑛)‖𝐵𝑦_𝑛‖𝛽_𝑛|𝑠_𝑛−𝑠_𝑛+1|. (3.12) Using (3.10) we have that ‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛𝛼‖≤|^𝑛+1𝛽_𝑛+1−𝛼_𝑛𝛽_𝑛|‖𝑢‖+|𝛿(1−𝛼_𝑛+1)𝛽_𝑛+1−1|‖𝑥_𝑛+1−𝑥_𝑛‖+𝛿(1−𝛼_𝑛+1)𝑀𝛽_𝑛+1𝑚|𝑟_𝑛+1−𝑟_𝑛|+𝛿𝑀𝛽_{𝑛+1𝑖≥1}|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|+𝛿𝑀𝛽_𝑛+1𝛽_𝑛|𝛽_𝑛−𝛽_𝑛+1|+𝛿(1−𝛼_𝑛)𝑀𝛽_𝑛|𝑠_𝑛−𝑠_𝑛+1|. (3.13) This implies limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0, (3.14) and by Lemma 2.4, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. Hence, ‖𝑥_𝑛+1−𝑥_𝑛‖=𝛽_𝑛‖𝑧_𝑛−𝑥_𝑛‖→0as𝑛→∞. (3.15) Using (3.10) we have ‖𝑦_𝑛+1−𝑦_𝑛‖→0as𝑛→∞. (3.16) We now have ‖𝑦_𝑛−𝑥^∗‖²=‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗‖²≤⟨𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗,𝑥_𝑛−𝑥^∗⟩=⟨𝑦_𝑛−𝑥^∗,𝑥_𝑛−𝑥^∗⟩=12(‖𝑦_𝑛−𝑥^∗‖²+‖𝑥_𝑛−𝑥^∗‖²−‖𝑥_𝑛−𝑦_𝑛‖²), (3.17) so that ‖𝑦_𝑛−𝑥^∗‖²≤‖𝑥_𝑛−𝑥^∗‖²−‖𝑥_𝑛−𝑦_𝑛‖². (3.18) But, ‖𝑥_𝑛+1−𝑥^∗‖²=‖𝛼_𝑛(𝑢−𝑥^∗)+(1−𝛿)(1−𝛼_𝑛)(𝑥_𝑛−𝑥^∗)+𝛿[_𝑖≥1𝜎_𝑖𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗]‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿_𝑖≥1𝜎_𝑖𝑛‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿(1−𝛼_𝑛)‖𝑦_𝑛−𝑥^∗‖². (3.19) Putting (3.18) in (3.19), we have 𝛿(1−𝛼_𝑛)‖𝑥_𝑛−𝑦_𝑛‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+‖𝑥_𝑛−𝑥^∗‖²−‖𝑥_𝑛+1−𝑥^∗‖²=𝛼_𝑛‖𝑢−𝑥^∗‖²+(‖𝑥_𝑛−𝑥^∗‖−‖𝑥_𝑛+1−𝑥^∗‖)(‖𝑥_𝑛−𝑥^∗‖+‖𝑥_𝑛+1−𝑥^∗‖)≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(‖𝑥_𝑛−𝑥_𝑛+1‖)(‖𝑥_𝑛−𝑥^∗‖+‖𝑥_𝑛+1−𝑥^∗‖), (3.20) and hence, lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. (3.21) Observe also that if 𝑤_𝑛∶=𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛, then ‖𝑤_𝑛−𝑥^∗‖²=‖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗‖²≤‖𝑦_𝑛−𝑥^∗−𝑠_𝑛(𝐵𝑦_𝑛−𝐵𝑥^∗)‖²=‖𝑦_𝑛−𝑥^∗‖²−2𝑠_𝑛⟨𝑦_𝑛−𝑥^∗,𝐵𝑦_𝑛−𝐵𝑥^∗⟩+𝑠²_𝑛‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²≤‖𝑦_𝑛−𝑥^∗‖²−2𝑠_𝑛[−𝜆‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²+𝛾‖𝑦_𝑛−𝑥^∗‖²]+𝑠²_𝑛‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²=‖𝑦_𝑛−𝑥^∗‖²+2𝑠_𝑛𝜆‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²−2𝑠_𝑛𝛾‖𝑦_𝑛−𝑥^∗‖²+𝑠²_𝑛‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²≤‖𝑦_𝑛−𝑥^∗‖²+[2𝑠_𝑛𝜆+𝑠²_𝑛−2𝑠_𝑛𝛾𝜇²]‖𝐵𝑦_𝑛−𝐵𝑥^∗‖². (3.22) Also, ‖𝑥_𝑛+1−𝑥^∗‖²=‖𝛼_𝑛(𝑢−𝑥^∗)+(1−𝛿)(1−𝛼_𝑛)(𝑥_𝑛−𝑥^∗)+𝛿[_𝑖≥1𝜎_𝑖𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗]‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿_𝑖≥1𝜎_𝑖𝑛‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗‖²=𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿_𝑖≥1𝜎_𝑖𝑛‖𝑇_𝑖𝑤_𝑛−𝑥^∗‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿(1−𝛼_𝑛)‖𝑤_𝑛−𝑥^∗‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛿)(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿(1−𝛼_𝑛)[‖𝑦_𝑛−𝑥^∗‖²+[2𝑠_𝑛𝜆+𝑠²_𝑛−2𝑠_𝑛𝛾𝜇²]‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²]≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛿(1−𝛼_𝑛)[2𝑠_𝑛𝜆+𝑠²_𝑛−2𝑠_𝑛𝛾𝜇²]‖𝐵𝑦_𝑛−𝐵𝑥^∗‖², (3.23) which implies −𝛿(1−𝛼_𝑛)[2𝑠_𝑛𝜆+𝑠²_𝑛−2𝑠_𝑛𝛾𝜇²]‖𝐵𝑦_𝑛−𝐵𝑥^∗‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+‖𝑥_𝑛−𝑥^∗‖²−‖𝑥_𝑛+1−𝑥^∗‖² (3.24) so that lim_𝑛→∞‖𝐵𝑦_𝑛−𝐵𝑥^∗‖=0. (3.25)
We go further to prove that for each 𝑖∈ℕ,lim_𝑛→∞‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛)𝐵𝑦_𝑛−𝑦_𝑛‖=0. (3.26) Consider the following estimates: ‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛‖²=‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑥^∗+𝑥^∗−𝑦_𝑛‖²=‖𝑥^∗−𝑦_𝑛‖²+2⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑥^∗,𝑥^∗−𝑦_𝑛⟩+‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑥^∗‖²≤2‖𝑥^∗−𝑦_𝑛‖²+2⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥^∗,𝑥^∗−𝑦_𝑛⟩=2‖𝑦_𝑛−𝑥^∗‖²+2⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛,𝑥^∗−𝑦_𝑛⟩−2‖𝑦_𝑛−𝑥^∗‖²=2⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛,𝑥^∗−𝑦_𝑛⟩. (3.27) Using (3.1), we have ⟨𝑥_𝑛+1−𝑥^∗,𝑦_𝑛−𝑥^∗⟩=𝛼_𝑛⟨𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+𝛿_𝑖≥1𝜎_𝑖,𝑛⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗⟩=𝛼_𝑛⟨𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+𝛿_𝑖≥1𝜎_𝑖,𝑛⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+𝛿(1−𝛼_𝑛)‖𝑦_𝑛−𝑥^∗‖²=𝛼_𝑛⟨𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+𝛿_𝑖≥1𝜎_𝑖,𝑛⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)‖𝑦_𝑛−𝑥^∗‖²+𝛿(1−𝛼_𝑛)‖𝑦_𝑛−𝑥^∗‖², (3.28) which implies 𝛿_𝑖≥1𝜎_𝑖,𝑛⟨(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛,𝑥^∗−𝑦_𝑛⟩=𝛼_𝑛⟨𝑢−𝑥^∗,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)⟨𝑦_𝑛−𝑥^∗,𝑦_𝑛−𝑥^∗⟩−⟨𝑥_𝑛+1−𝑥^∗,𝑦_𝑛−𝑥^∗⟩=𝛼_𝑛⟨𝑢−𝑥_𝑛+1,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)⟨𝑦_𝑛−𝑥_𝑛+1,𝑦_𝑛−𝑥^∗⟩. (3.29) Using this and (3.27), we get 𝛿2_𝑖≥1𝜎_𝑖,𝑛‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛‖²≤𝛼_𝑛⟨𝑢−𝑥_𝑛+1,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)(1−𝛿)⟨𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑥^∗⟩+(1−𝛼_𝑛)⟨𝑦_𝑛−𝑥_𝑛+1,𝑦_𝑛−𝑥^∗⟩. (3.30) Since {𝑥_𝑛}, {𝑦_𝑛} are each bounded and using (3.21), we have that lim_𝑛→∞‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑦_𝑛‖=0∀𝑖∈ℕ. (3.31) Using this and (3.21), we also have lim_𝑛→∞‖(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵))𝑦_𝑛−𝑥_𝑛‖=0∀𝑖∈ℕ. (3.32) Next we show that limsup_𝑛→∞⟨𝑢−𝑤,𝑥_𝑛−𝑤⟩≤0, where 𝑤=𝑃_⋂^∞_{𝑖=1Fix(𝑇𝑖)∩EP(𝐺)∩VI(𝐾,𝐵)}𝑢. Let {𝑥_𝑛𝑗} be a subsequence of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑢−𝑤,𝑥_𝑛−𝑤⟩=lim_𝑗→∞⟨𝑢−𝑤,𝑥_𝑛𝑗−𝑤⟩.
Since {𝑦_𝑛} is bounded, there exists a subsequence {𝑦_𝑛𝑗} of {𝑦_𝑛} such that 𝑦_𝑛𝑗⇀𝑧. Then, 𝑧∈⋂^∞_𝑖=1Fix(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)),otherwise, for 𝑖∈ℕ, we haveliminf_𝑗→∞‖𝑦_𝑛𝑗−𝑧‖<liminf_𝑗→∞‖𝑦_𝑛𝑗−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝑗𝐵)𝑧‖≤liminf_𝑗→∞[‖𝑦_𝑛𝑗−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝑗𝐵)𝑦_𝑛𝑗‖+‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝑗𝐵)𝑦_𝑛𝑗−𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝑗𝐵)𝑧‖]≤liminf_𝑗→∞‖𝑦_𝑛𝑗−𝑧‖, (3.33) which is a contradiction, so 𝑧∈⋂^∞_𝑖=1Fix(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)).
Next we show that 𝑧∈EP(𝐺). Since 𝑦_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝐺(𝑦_𝑛1,𝜂)+𝑟_𝑛⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛⟩≥0,∀𝜂∈𝐾. (3.34) It follows from (𝐴2) that ⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑟_𝑛⟩≥𝐺(𝜂,𝑦_𝑛), (3.35) and so ⟨𝜂−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖⟩≥𝐺(𝜂,𝑦_𝑛𝑖). (3.36) Since (𝑦_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0, 𝑦_𝑛𝑖⇀𝑧 and using (𝐴4), we have 𝐺(𝜂,𝑧)≤0 for all 𝜂∈𝐾. For a real number 𝑡, 0<𝑡≤1 and 𝜂∈𝐾, let 𝜂_𝑡=𝑡𝜂+(1−𝑡)𝑧. Clearly 𝜂_𝑡∈𝐾, so that using (𝐴1) and (𝐴4), we have 0=𝐺(𝜂_𝑡,𝜂_𝑡)≤𝑡𝐺(𝜂_𝑡,𝜂)+(1−𝑡)𝐺(𝜂_𝑡,𝑧)≤𝑡𝐺(𝜂_𝑡,𝜂). (3.37) This implies 𝐺(𝜂_𝑡,𝜂)≥0, and using this and (𝐴3) we have that 𝐺(𝑧,𝜂)≥0 for all 𝜂∈𝐾 and hence 𝑧∈EP(𝐺).
Next we show that 𝑧∈VI(𝐾,𝐵).
Let 𝑥^∗∈𝐹 then, we have the following: ‖𝑤_𝑛−𝑥^∗‖²=‖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑥^∗‖²≤⟨𝑦_𝑛−𝑠_𝑛𝐵𝑦_𝑛−(𝑥^∗−𝑠_𝑛𝐵𝑥^∗),𝑤_𝑛−𝑥^∗⟩=⟨𝑦_𝑛−𝑥^∗,𝑤_𝑛−𝑥^∗⟩−𝑠_𝑛⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩≤12[‖𝑤_𝑛−𝑥^∗‖²+‖𝑦_𝑛−𝑥^∗‖²−‖𝑤_𝑛−𝑦_𝑛‖²]−𝑠_𝑛⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩, (3.38) so that ‖𝑤_𝑛−𝑥^∗‖²≤‖𝑦_𝑛−𝑥^∗‖²−‖𝑤_𝑛−𝑦_𝑛‖²−2𝑠_𝑛⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩. (3.39) We then have ‖𝑥_𝑛+1−𝑥^∗‖²=‖𝛼_𝑛(𝑢−𝑥^∗)+(1−𝛼_𝑛)(1−𝛿)(𝑥_𝑛−𝑥^∗)+𝛿_𝑖≥1𝜎_𝑖,𝑛[𝑇_𝑖𝑤_𝑛−𝑥^∗]‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛼_𝑛)(1−𝛿)‖𝑥_𝑛−𝑥^∗‖²+(1−𝛼_𝑛)𝛿‖𝑤_𝑛−𝑥^∗‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛼_𝑛)(1−𝛿)‖𝑥_𝑛−𝑥^∗‖²+(1−𝛼_𝑛)𝛿[‖𝑦_𝑛−𝑥^∗‖²−‖𝑤_𝑛−𝑦_𝑛‖²−2𝑠_𝑛⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩]≤𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛼_𝑛)(1−𝛿)‖𝑥_𝑛−𝑥^∗‖²+(1−𝛼_𝑛)𝛿[‖𝑥_𝑛−𝑥^∗‖²−‖𝑤_𝑛−𝑦_𝑛‖²−2𝑠_𝑛⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩]=𝛼_𝑛‖𝑢−𝑥^∗‖²+(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²−𝛿(1−𝛼_𝑛)‖𝑤_𝑛−𝑦_𝑛‖²−2𝑠_𝑛𝛿(1−𝛼_𝑛)⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩, (3.40) so that 𝛿(1−𝛼_𝑛)‖𝑤_𝑛−𝑦_𝑛‖²≤𝛼_𝑛‖𝑢−𝑥^∗‖²+‖𝑥_𝑛−𝑥^∗‖²−‖𝑥_𝑛+1−𝑥^∗‖²−2𝛿𝑠_𝑛(1−𝛼_𝑛)⟨𝐵𝑦_𝑛−𝐵𝑥^∗,𝑤_𝑛−𝑥^∗⟩, (3.41) and as 𝛼_𝑛→0, ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and ‖𝐵𝑦_𝑛−𝐵𝑥^∗‖→0 as 𝑛→∞, we get lim_𝑛→∞‖𝑦_𝑛−𝑤_𝑛‖=0. (3.42) Using this and (3.21) we also have lim_𝑛→∞‖𝑥_𝑛−𝑤_𝑛‖=0. (3.43) As 𝐵 is a relaxed (𝜆,𝛾)-cocoercive and using condition (𝐶3) we have for 𝑥,𝑦∈𝐻⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥(−𝜆)‖𝐵𝑥−𝐵𝑦‖²+𝛾‖𝑥−𝑦‖²≥(𝛾−𝜆𝜇²)‖𝑥−𝑦‖²≥0, (3.44) and so 𝐵 is monotone. If 𝑀𝑞=𝐵𝑞+𝑁_𝐾𝑞,𝑞∈𝐾,∅,𝑞∉𝐾, (3.45) then 𝑀 is maximal monotone. Let Γ(𝑀) denote the graph of 𝑀.
Let (𝑞,𝑝)∈Γ(𝑀). Since 𝑝−𝐵𝑞∈𝑁_𝐾𝑞 and 𝑤_𝑛∈𝐾, we have ⟨𝑝−𝐵𝑞,𝑞−𝑤_𝑛⟩≥0 by definition of 𝑁_𝐾𝑞. Also, as 𝑤_𝑛=𝑃_𝐾(𝑦_𝑛−𝑠_𝑛𝐵𝑦_𝑛) (using property of the projection 𝑃_𝐾), we have ⟨𝑤_𝑛−(𝑦_𝑛−𝑠_𝑛𝐵𝑦_𝑛),𝑞−𝑤_𝑛⟩≥0, (3.46) and hence ⟨𝑤_𝑛−𝑦_𝑛𝑠_𝑛+𝐵𝑦_𝑛,𝑞−𝑤_𝑛⟩≥0. (3.47) Using this, we obtain the following estimates: ⟨𝑝,𝑞−𝑤_𝑛𝑖⟩≥⟨𝐵𝑞,𝑞−𝑤_𝑛𝑖⟩≥⟨𝐵𝑞,𝑞−𝑤_𝑛𝑖𝑤⟩−⟨^𝑛𝑖−𝑦_𝑛𝑖𝑠_𝑛𝑖+𝐵𝑦_𝑛𝑖,𝑞−𝑤_𝑛𝑖⟩𝑤=⟨𝐵𝑞−^𝑛𝑖−𝑦_𝑛𝑖𝑠_𝑛−𝐵𝑦_𝑛𝑖,𝑞−𝑤_𝑛𝑖⟩=⟨𝐵𝑞−𝐵𝑤_𝑛𝑖,𝑞−𝑤_𝑛𝑖⟩+⟨𝐵𝑤_𝑛𝑖−𝐵𝑦_𝑛𝑖,𝑞−𝑤_𝑛𝑖𝑤⟩−⟨^𝑛𝑖−𝑦_𝑛𝑖𝑠_𝑛,𝑞−𝑤_𝑛𝑖⟩≥⟨𝐵𝑤_𝑛𝑖−𝐵𝑦_𝑛𝑖,𝑞−𝑤_𝑛𝑖𝑤⟩−⟨^𝑛𝑖−𝑦_𝑛𝑖𝑠_𝑛,𝑞−𝑤_𝑛𝑖⟩, (3.48) which implies ⟨𝑝,𝑞−𝑧⟩≥0 (letting 𝑖→∞).
Since 𝑀 is maximal monotone, we obtained that 𝑧∈𝑀⁻¹(0) and hence 𝑧∈VI(𝐾,𝐵).
We mention here that since we proved that 𝑧∈⋂^∞_𝑖=1Fix(𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)) and 𝑧∈VI(𝐾,𝐵), then 𝑧∈⋂^∞_𝑖=1Fix(𝑇_𝑖). So, clearly 𝑧∈⋂^∞_𝑖=1Fix(𝑇_𝑖)∩EP(𝐺)∩VI(𝐾,𝐵).
Since 𝑤=𝑃_⋂^∞_{𝑖=1Fix(𝑇𝑖)∩EP(𝐺)∩VI(𝐾,𝐵)}𝑢, we have limsup_𝑛→∞⟨𝑢−𝑤,𝑥_𝑛−𝑤⟩=lim_𝑖→∞⟨𝑢−𝑤,𝑥_𝑛𝑖−𝑤⟩=⟨𝑢−𝑤,𝑧−𝑤⟩≤0. (3.49) Hence, limsup_𝑛→∞⟨𝑢−𝑤,𝑥_𝑛−𝑤⟩≤0.
From the recursion formula (3.1) and Lemma 2.3, we have ‖𝑥_𝑛+1−𝑤‖²=‖𝛼_𝑛(𝑢−𝑤)+(1−𝛼_𝑛)(1−𝛿)(𝑥_𝑛−𝑤)+𝛿_𝑖≥1𝜎_𝑖,𝑛[𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑤]‖²≤‖(1−𝛼_𝑛)(1−𝛿)(𝑥_𝑛−𝑤)+𝛿_𝑖≥1𝜎_𝑖,𝑛[𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑤]‖²+2𝛼_𝑛⟨𝑢−𝑤,𝑥_𝑛+1−𝑤⟩≤[(1−𝛼_𝑛)(1−𝛿)‖𝑥_𝑛−𝑤‖+𝛿_𝑖≥1𝜎_𝑖,𝑛‖𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛−𝑤‖]²+2𝛼_𝑛⟨𝑢−𝑤,𝑥_𝑛+1−𝑤⟩≤(1−𝛼_𝑛)‖𝑥_𝑛−𝑤‖²+2𝛼_𝑛⟨𝑢−𝑤,𝑥_𝑛+1−𝑤⟩. (3.50) Using Lemma 2.5, we have that {𝑥_𝑛} and consequently {𝑦_𝑛} converge to 𝑤 and the proof is complete.

The following corollaries follow from Theorem 3.1.

Corollary 3.2. Let 𝐾,𝐺 and 𝐵 be as in Theorem 3.1 and let {𝑇_𝑖,𝑖=1,2,3,…,𝑁} be finite family of nonexpansive mappings of 𝐾 into 𝐻. Let 𝐹∶=⋂^𝑁_𝑖=1Fix(𝑇_𝑖)∩EP(𝐺)∩VI(𝐾,𝐵)≠∅. For an arbitrary but fixed 𝛿∈(0,1), let {𝑥_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁,𝑢∈𝐻,𝐺(𝑦_𝑛1,𝜂)+𝑟_𝑛⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝜂∈𝐾,_𝑛+1=𝛼_𝑛𝑢+(1−𝛿)(1−𝛼_𝑛)𝑥_𝑛+𝛿^𝑁_𝑖=1𝜎_𝑖𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝑠_𝑛𝐵)𝑦_𝑛,𝑛≥1, (3.51) where {𝛼_𝑛} and {𝜎_𝑖𝑛} are sequences in [0,1] and {𝑟_𝑛} and {𝑠_𝑛} are sequences in [0,∞) satisfying (𝐶1)lim_𝑛→∞𝛼_𝑛=0, ∑^𝑁_𝑖=1𝜎_𝑖𝑛=(1−𝛼_𝑛),(𝐶2)∑^∞_𝑛=1𝛼_𝑛=∞,(𝐶3){𝑠_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 satisfying 0≤𝑎≤𝑏≤2(𝛾−𝜆𝜇²)/𝜇²,(𝐶4)lim_𝑛→∞|𝑠_𝑛+1−𝑠_𝑛|=0, lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0, lim_𝑛→∞∑^𝑁_𝑖=1|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|=0,(𝐶5)liminf_𝑛→∞𝑟_𝑛>0.Then, both {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑃_𝐹𝑢.

Corollary 3.3. Let 𝐾 and 𝐺 be as in Theorem 3.1. Let 𝑇 be a nonexpansive map of 𝐾 into 𝐻. Let 𝐹∶=Fix(𝑇)∩EP(𝐺)≠∅. For an arbitrary but fixed 𝛿∈(0,1), let {𝑥_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁,𝑢∈𝐻,𝐺(𝑦_𝑛1,𝜂)+𝑟_𝑛⟨𝜂−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝜂∈𝐾,_𝑛+1=𝛼_𝑛𝑢+(1−𝛿)(1−𝛼_𝑛)𝑥_𝑛+𝛿(1−𝛼_𝑛)𝑇𝑦_𝑛,𝑛≥1, (3.52) where {𝛼_𝑛} is a sequence in [0,1] and {𝑟_𝑛} is a sequences in [0,∞) satisfying (𝐶1)lim_𝑛→∞𝛼_𝑛=0(𝐶2)∑^∞_𝑛=1𝛼_𝑛=∞,(𝐶3)lim_𝑛→∞|𝑟_𝑛+1−𝑟_𝑛|=0,(𝐶4)liminf_𝑛→∞𝑟_𝑛>0.Then, both {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑃_𝐹𝑢.

Remark 3.4. Prototypes of the sequences {𝛼_𝑛} and {𝜎_𝑖,𝑛} in our theorem are the following: 𝛼_𝑛1∶=𝑛+1,𝜎_𝑖,𝑛𝑛∶=2^𝑖(𝑛+1)∀𝑖∈ℕ. (3.53)