Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Abstract

Let 𝐾 be a closed, convex, and nonempty subset of a real 𝑞-uniformly smooth Banach space 𝐸, which is also uniformly convex. For some 𝜅>0, let 𝑇_𝑖∶𝐾→𝐸𝑖∈ℕ and 𝐴∶𝐾→𝐸 be family of nonexpansive maps and 𝜅-inverse strongly accretive map, respectively. Let 𝐺∶𝐾×𝐾→ℝ be a bifunction satisfying some conditions. Let 𝑃_𝐾 be a nonexpansive projection of 𝐸 onto 𝐾. For some fixed real numbers 𝛿∈(0,1), 𝜆∈(0,(𝑞𝜅/𝑑_𝑞)^1/(𝑞−1)), and arbitrary but fixed vectors 𝑥₁,𝑢∈𝐸, let {𝑥_𝑛} and {𝑦_𝑛} be sequences generated by 𝐺(𝑦_𝑛,𝜂)+(1/𝑟)⟨𝜂−𝑦_𝑛, 𝑗_𝑞(𝑦_𝑛−𝑥_𝑛)⟩≥0, ∀𝜂∈𝐾, 𝑥_𝑛+1=𝛼_𝑛𝑢+(1−𝛿)(1−𝛼_𝑛)𝑥_𝑛+𝛿∑_𝑖≥1𝜎_𝑖𝑛𝑇_𝑖𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛), 𝑛≥1, where 𝑟∈(0,1) is fixed, and {𝛼_𝑛},{𝜎_𝑖,𝑛}⊂(0,1) are sequences satisfying appropriate conditions. If 𝐹∶=[∩^∞_𝑖=1𝐹(𝑇_𝑖)]∩VI(𝐾,𝐴)∩EP(𝐺)≠∅, under some mild conditions, we prove that the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to some element in 𝐹.

1. Introduction

Let 𝐸 be a real normed space and 𝐸^∗ its dual space. For some real number 𝑞(1<𝑞<∞), the generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by 𝐽_𝑞(𝑥)=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖^𝑞,𝑓^∗=‖𝑥‖^𝑞−1, (1) where ⟨⋅,⋅⟩ denotes the pairing between elements of 𝐸 and elements of 𝐸^∗.

For 𝑞=2, 𝐽₂ usually denoted by 𝐽 is called the normalised duality mapping.

Let 𝐸 be a real Banach space; a map 𝐴∶𝐷(𝐴)→𝐸 is said to be accretive if for all 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗_𝑞𝑥−𝑦≥0. (2)

For some real number 𝜅>0, 𝐴 is called 𝜅-inverse strongly accretive if for all 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗_𝑞𝑥−𝑦≥𝜅𝐴𝑥−𝐴𝑦^𝑞. (3) Observe that a 𝜅-inverse strongly accretive map is 1/𝜅-Lipschitzian.

Let 𝐾 be a nonempty, closed, and convex subset of 𝐸, and let 𝐴∶𝐾→𝐸 be an accretive mapping. A variational inequality problem is, searching for 𝑥^∗∈𝐾 such that for some 𝑗_𝑞(𝑣−𝑥^∗)∈𝐽_𝑞(𝑣−𝑥^∗)𝐴𝑥^∗,𝑗_𝑞𝑣−𝑥^∗≥0,∀𝑣∈𝐾. (4) Let 𝐺∶𝐾×𝐾→ℝ be a bifunction on a closed convex nonempty subset 𝐾 of a real Banach space 𝐸; an equilibrium problem is searching for 𝑥^∗∈𝐾 such that 𝐺𝑥^∗,𝑣≥0,∀𝑣∈𝐾. (5)

A set of solutions of the problems (4) and (5) are denoted by VI(𝐾,𝐴) and EP(𝐺), respectively.

Let 𝑃 be a mapping of 𝐸 onto 𝐾. Then, 𝑃 is said to be sunny if 𝑃(𝑃𝑥+𝑡(𝑥−𝑃𝑥))=𝑃𝑥 for all 𝑥∈𝐸 and 𝑡≥0. A mapping 𝑃 of 𝐸 into 𝐸 is said to be a retraction if 𝑃²=𝑃. If a mapping 𝑃 is a retraction, then 𝑃𝑧=𝑧 for every 𝑧∈𝑅(𝑃), range of 𝑃. A subset 𝐾 is said to be sunny nonexpansive retract of 𝐸 if there exists a sunny nonexpansive retraction of 𝐸 onto 𝐾. A retraction 𝑃 is said to be orthogonal if for each 𝑥, 𝑥−𝑃(𝑥) is normal to 𝐾 in the sense of James [1].

It is well known (see [2]) that if 𝐸 is a Banach space; a projection mapping is a sunny nonexpansive retraction 𝑃 of 𝐸 onto 𝐾. If 𝐸 is uniformly smooth and there exists a nonexpansive retraction of 𝐸 onto 𝐾, then there exists a nonexpansive projection of 𝐸 onto 𝐾. If 𝐸 is a real smooth Banach space, then 𝑃 is an orthogonal retraction of 𝐸 onto 𝐾 if and only if 𝑃(𝑥)∈𝐾 and ⟨𝑃(𝑥)−𝑥,𝑗_𝑞(𝑃(𝑥)−𝑦)⟩≤0 for all 𝑦∈𝐾. It then follows that, for 𝑥,𝑦∈𝐾, we have ⟨𝑃(𝑥)−𝑥,𝑗_𝑞(𝑃(𝑥)−𝑃(𝑦))⟩≤0 and ⟨𝑃(𝑦)−𝑦,𝑗_𝑞(𝑃(𝑦)−𝑃(𝑥))⟩≤0 which implies 𝑃(𝑥)−𝑃𝑦^𝑞≤𝑥−𝑦,𝑗_𝑞𝑃(𝑥)−𝑃𝑦. (6)

An accretive mapping 𝐴 is said to be maximal if its graph GF(𝐴) is not contained in the graph of any other accretive map. Equivalently, 𝐴 is maximal accretive if for every (𝑣₀,𝑤₀)∈𝐸×𝐸 such that ⟨𝑤−𝑤₀,𝑗_𝑝(𝑣−𝑣₀)⟩≥0 holds for all 𝑤∈𝐴𝑣,𝑣∈𝐾; then, 𝑤₀∈𝐴𝑣₀. A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at 𝑝 if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that 𝑥_𝑛⇀𝑥^∗∈𝐷(𝑇) and 𝑇𝑥_𝑛→𝑝; then, 𝑇𝑥^∗=𝑝. The following proposition is known to hold; see, for example, [3].

Proposition 1. Let 𝐴∶𝐾→𝐸 be a 𝜅-inverse strongly accretive map. Let 𝑀 be defined by 𝑀𝑣=𝐴𝑣+𝑁_𝐾𝑣,𝑣∈𝐾,∅,𝑣∉𝐾, (7) where 𝑁_𝐾𝑣={𝑤∈𝐸∶⟨𝑤,𝑗_𝑞(𝑣−𝑢)⟩≥0, for all 𝑢∈𝐾}; then, 𝑀 is maximal accretive and 𝑢∈𝑀⁻¹(0) if and only if 𝑢∈VI(𝐾,𝐴).

Recently, Maingé [4] studied the Halpern-type scheme for approximation of a common fixed point of countable infinite family of nonexpansive mappings in a real Hilbert space.

The present author [3] proved a strong convergence theorem for family of nonexpansive maps and solution of variational inequality problems. Kumam and Jaiboon [5] studied a hybrid iterative method for mixed equilibrium problem and variational inequality problem in the framework of a real Hilbert space.

Various numerous authors studied the problem of approximating solutions of equilibrium and fixed point problems in the framework of a real Hilbert space; see, for example, [5–18] and the references contained therein. In [19], Ceng et al. studied this problem in the framework of a uniformly smooth and uniformly convex Banach space.

Takahashi and Zembayashi [20] (see also [21–23]) studied the problem of approximating solutions of equilibrium problems and fixed points of some nonlinear maps in the framework of real Banach spaces. It is our purpose in this paper to introduce a new hybrid iterative method for approximating a common element in the intersection of the set of fixed points of countable infinite family of nonexpansive mappings, the set of solutions of variational inequality problem, and the set of solutions of equilibrium problem in Banach spaces. Our theorems extend and improve some recent important results, and our method of proof in this paper is of independent interest.

2. Preliminaries

Let 𝑆∶={𝑥∈𝐸∶‖𝑥‖=1} denote a unit sphere of the real Banach space 𝐸. 𝐸 is said to have a Gâteaux differentiable norm if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (8) exists for each 𝑥,𝑦∈𝑆; 𝐸 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆, the limit is attained uniformly for 𝑥∈𝑆. Let 𝐸 be a normed space with dim𝐸≥2. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝜏)+∶=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1;𝑦=𝜏. (9)

The space 𝐸 is called uniformly smooth if and only if lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0. For some constant 𝑞>1, 𝐸 is called 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌_𝐸(𝑡)≤𝑐𝑡^𝑞, 𝑡>0.

The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)=inf1−𝑥+𝑦2∶‖𝑥‖=𝑦=1,𝜖=𝑥−𝑦. (10)𝐸 is called uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2].

A Banach space 𝐸 is said to be strictly convex if ‖𝑥−𝑦‖/2<1 for 𝑥,𝑦∈𝐸 with ‖𝑥‖=1=‖𝑦‖ and 𝑥≠𝑦.

It is well known that if 𝐸 is smooth then the duality mapping is singled valued, and if 𝐸 has uniformly Gâteaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded subset of 𝐸. Also, every 𝑞-uniformly smooth space is uniformly smooth and has a uniformly Gâteaux differentiable norm, and every uniformly convex space is strictly convex.

In the sequel, we will make use of the following results.

Lemma 2 (see Petryshyn [24]). Let 𝐸 be a real normed linear space. Then, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖².+2𝑦,𝑗𝑥+𝑦∀𝑥,𝑦∈𝐸,𝑗𝑥+𝑦∈𝐽𝑥+𝑦 (11)

Theorem 3 (see Goebel and Kirk [25]). Let 𝐸 be a real uniformly convex Banach space, 𝐾 a closed convex subset of 𝐸, and 𝑇∶𝐾→𝐸 a nonexpansive mapping. Then, (𝐼−𝑇) is demiclosed at zero, where 𝐼 denotes the identity map.

Lemma 4 (see Suzuki [26]). 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 that 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 5 (see Xu [27]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝜎_𝑛+𝛾_𝑛,𝑛≥0, (12) 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→∞.

Lemma 6 (see Xu [28]). Let 𝐸 be a real q-uniformly smooth Banach space for some 𝑞>1; then, there exists some positive constant 𝑑_𝑞 such that 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝑗_𝑞(𝑥)+𝑑_𝑞𝑦^𝑞 (13) for all 𝑥,𝑦∈𝐸 and 𝑗_𝑞(𝑥)∈𝐽_𝑞(𝑥).

Lemma 7 (see Kamimura and Takahashi [29]). Let 𝐸 be a real smooth and uniformly convex Banach space, and let 𝑅>0. Then, there exists a strictly increasing, continuous, and convex function 𝑔∶[0,2𝑅]→ℝ such that 𝑔(0)=0 and 𝑔(‖𝑥−𝑦‖)≤‖𝑥‖²−2⟨𝑥,𝑗𝑦⟩+‖𝑦‖² for all 𝑥,𝑦∈𝐵_𝑅.

The following conditions are required on the bifunction 𝐺∶𝐾×𝐾→ℝ for solving equilibrium problems with respect to 𝐺:(A1)𝐺(𝑥,𝑥)=0 for all 𝑥∈𝐾;(A2)𝐺 is monotone; that is, 𝐺(𝑥,𝑦)+𝐺(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐾;(A3) for all 𝑥,𝑦,𝑧∈𝐾,limsup_𝑡→0+𝐺(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐺(𝑥,𝑦);(A4) for all 𝑥∈𝐾,𝐺(𝑥,⋅) is convex and lower semicontinuous.

Lemma 8 (see Blum and Oettli [30]). Let 𝐸 be a real smooth, strictly convex, and reflexive Banach space. Let 𝐺∶𝐾×𝐾→ℝ be a bifunction satisfying (A1)–(A4), and let 𝑥∈𝐸, 𝑟>0. Then, there exists 𝑧∈𝐾 such that 𝐺+1𝑧,𝑦𝑟𝑦−𝑧,𝑗(𝑧−𝑥)≥0,∀𝑦∈𝐾. (14)

Lemma 9. Let 𝐾 be a closed convex nonempty subset of a real uniformly smooth and strictly convex Banach space 𝐸. Let 𝐺∶𝐾×𝐾→ℝ be a bifunction satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, define a map 𝑇_𝑟∶𝐸→𝐾 by 𝑇_𝑟+1𝑥=𝑧∈𝐾∶𝐺𝑧,𝑦𝑟𝑦−𝑧,𝑗(𝑧−𝑥)≥0,∀𝑦∈𝐾. (15)
Then, the following hold:(i)𝑇_𝑟 is single-valued;(ii)𝐹𝑖𝑥(𝑇_𝑟)=𝐸𝑃(𝐺); (iii)if 𝑇_𝑟 is firmly nonexpansive-type, that is, for 𝑥,𝑦∈𝐸, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑗𝑇_𝑟𝑥−𝑇_𝑟𝑦≤𝑥−𝑦,𝑗𝑇_𝑟𝑥−𝑇_𝑟𝑦, (16)then 𝐸𝑃(𝐺) is closed and convex.

Proof.   (i) Let 𝑧₁,𝑧₂∈𝑇_𝑟, then 𝐺𝑧₁,𝑧₂+1𝑟𝑧₂−𝑧₁,𝑗𝑧₁−𝑧₂𝐺𝑧≥0,₂,𝑧₁+1𝑟𝑧₁−𝑧₂,𝑗𝑧₂−𝑧₁≥0. (17) Adding these inequalities and using (A2), we get 𝑧₂−𝑧₁,𝑗𝑧₁−𝑧₂≥0, (18) which implies 𝑧₁=𝑧₂. Consider
(ii) 𝑧∈𝐹𝑇_𝑟⟺𝑧=𝑇_𝑟𝑧+1⟺𝐺𝑧,𝑦𝑟((𝐺).𝑦−𝑧,𝑗𝑧−𝑧)≥0,∀𝑦∈𝐾,⟺𝐺𝑧,𝑦≥0,∀𝑦∈𝐾,⟺𝑧∈𝐸𝑃 (19)
(iii) EP(𝐺) is closed and convex follows from (ii) and the fact that every firmly nonexpansive map is nonexpansive and the fixed point set of nonexpansive map is closed and convex.

Let 𝐸 be a real 𝑞-uniformly smooth Banach space, and for some 𝜆>0, let 𝐼∶𝐾→𝐾 and 𝐴∶𝐾→𝐸 be the identity and 𝜅-inverse strongly accretive mappings, respectively. Then, for the map (𝐼−𝜆𝐴)∶𝐾→𝐸, we have the following estimates: (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦^𝑞=𝑥−𝑦−𝜆𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝑞𝜆𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦+𝑑_𝑞𝜆^𝑞𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝜆𝑞𝜅−𝑑_𝑞𝜆^𝑞−1𝐴𝑥−𝐴𝑦^𝑞. (20) If 𝜆 is chosen such that 0≤𝜆≤(𝑞𝜅/𝑑_𝑞)^1/(𝑞−1), we then have (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦≤𝑥−𝑦, (21) and so (𝐼−𝜆𝐴) become a nonexpansive mapping of 𝐾 into 𝐸.

For 𝐿_𝑝(1<𝑝<∞) spaces, we have the following relation: if 𝜆∈(0,2𝜅/(𝑝−1)), (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²=𝑥−𝑦−𝜆𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²+𝜆−2𝜆𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦𝑝−1²𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−𝜆2𝜅−𝑝−1𝜆𝐴𝑥−𝐴𝑦²≤𝑥−𝑦². (22)

Also if 𝐸 is a Hilbert space and we choose 𝜆∈(0,2𝜅), then (𝐼−𝜆𝐴) is nonexpansive.

3. Path Convergence Theorems

In the sequel, we assume for each 𝑡∈(0,1) that the sequence {𝜎_𝑖,𝑡}_𝑖 satisfies ∑_𝑖≥1𝜎_𝑖,𝑡=1−𝑡 and the sequences {𝛼_𝑛},{𝜎_𝑖,𝑛}_𝑖⊂(0,1), satisfy ∑_𝑖≥1𝜎_𝑖,𝑛=1−𝛼_𝑛.

For a countable family of nonexpansive mappings {𝑇_𝑖} of 𝐸, we denote a set 𝒩_ℐ∶={𝑖∈ℕ∶𝑇_𝑖≠𝐼} (𝐼 being the identity mapping on 𝐸).

Let 𝐾 be a nonempty closed and convex subset of a real 𝑞-uniformly smooth Banach space 𝐸 and 𝑃_𝐾 a nonexpansive projection of 𝐸 onto 𝐾. For some real number 𝜅>0, let 𝐴∶𝐾→𝐸 be a 𝜅-inverse strongly accretive mapping. For some real numbers 𝛿∈(0,1), 𝜆∈(0,(𝑞𝜅/𝑑_𝑞)^1/(𝑞−1)), and  𝑟>0 arbitrarily chosen but fixed and for each 𝑡∈(0,1), define a map 𝑇_𝑡∶𝐸→𝐸 by 𝑢∈𝐸, arbitrary and fixed 𝐺+1𝑦,𝜂𝑟𝑇𝜂−𝑦,𝑗𝑦−𝑥≥0,∀𝜂∈𝐾,_𝑡𝑥=𝑡𝑢+(1−𝛿)(1−𝑡)𝑥+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾𝑦−𝜆𝐴𝑦,∀𝑥∈𝐸. (23)

Then, 𝑇_𝑡 is a strict contraction on 𝐸.

For 𝑥¹,𝑥²∈𝐸, 𝑦¹=𝑇_𝑟𝑥¹, 𝑦²=𝑇_𝑟𝑥², we have 𝑇_𝑡𝑥¹−𝑇_𝑡𝑥²≤(1−𝑡)(1−𝛿)𝑥¹−𝑥²+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾𝑦¹−𝜆𝐴𝑦¹−𝑇_𝑖𝑃_𝐾𝑦²−𝜆𝐴𝑦²=(1−𝑡)(1−𝛿)𝑥¹−𝑥²+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥¹−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥²≤(1−𝑡)𝑥¹−𝑥². (24)

Thus, for each 𝑡∈(0,1), there exists a unique 𝑧_𝑡∈𝐸 such that 𝐺+1𝑦,𝜂𝑟𝜂−𝑦,𝑗𝑦−𝑧_𝑡𝑧≥0,∀𝜂∈𝐾,_𝑡=𝑡𝑢+(1−𝛿)(1−𝑡)𝑧_𝑡+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾𝑦−𝜆𝐴𝑦. (25)

Lemma 10. Let 𝐸 be a real 𝑞-uniformly smooth Banach space which is also uniformly convex. Let 𝐾 be a closed, convex, and nonempty subset of 𝐸. For 𝑡∈(0,1), let {𝑧_𝑡} be a net satisfying (25), and assume that ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟)≠∅. Then, {𝑧_𝑡} is bounded and admits at most one accumulation point in ℱ as 𝑡→0.

Proof. Let 𝑥^∗∈ℱ. Then, using (25), we have 𝑧_𝑡−𝑥^∗2=⟨𝑡𝑢−𝑥^∗+(1−𝑡)(1−𝛿)𝑧_𝑡−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾𝑇_𝑟𝑧_𝑡−𝜆𝐴𝑇_𝑟𝑧_𝑡−𝑥^∗,𝑗𝑧_𝑡−𝑥^∗⟩≤𝑡𝑢−𝑥^∗,𝑗𝑧_𝑡−𝑥^∗+(1−𝑡)(1−𝛿)𝑧_𝑡−𝑥^∗2+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑧_𝑡−𝑥^∗2=𝑡𝑢−𝑥^∗,𝑗𝑧_𝑡−𝑥^∗+(1−𝑡)𝑧_𝑡−𝑥^∗2, (26) which implies 𝑧_𝑡−𝑥^∗≤𝑢−𝑥^∗. (27) Thus, {𝑧_𝑡} is bounded.
Now, assume for the sake of contradiction that 𝑥^′ and 𝑥^∗ are two distinct accumulation points of {𝑧_𝑡} in ℱ; then, there exists a subnet {𝑧_𝑡𝑠} of {𝑧_𝑡} such that 𝑧_𝑡𝑠→𝑥^′ as 𝑠→∞, and so we have the following estimates: 𝑧_𝑡𝑠−𝑥^∗2=𝑡_𝑠𝑢−𝑥^∗+1−𝑡_𝑠(1−𝛿)𝑧_𝑡𝑠−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑡𝑇_𝑖𝑃_𝐾𝑧_𝑡𝑠−𝜆𝐴𝑧_𝑡𝑠−𝑥^∗,𝑗𝑧_𝑡𝑠−𝑥^∗≤𝑡_𝑠𝑢−𝑥^∗,𝑗𝑧_𝑡𝑠−𝑥^∗+1−𝑡_𝑠𝑧_𝑡𝑠−𝑥^∗2 (28) so that 𝑧_𝑡𝑠−𝑥^∗2≤𝑢−𝑥^∗,𝑗𝑧_𝑡𝑠−𝑥^∗, (29) and since 𝑧_𝑡𝑠→𝑥^′ as 𝑠→∞, we get from (29) 𝑥^′−𝑥^∗2≤𝑢−𝑥^∗,𝑗𝑥^′−𝑥^∗. (30)
Applying similar argument to 𝑥^∗ as an accumulation point of {𝑧_𝑡} in ℱ, we also get 𝑥^∗−𝑥^′2≤𝑢−𝑥^′,𝑗𝑥^∗−𝑥^′. (31)
Adding these last two inequalities, we get 2𝑥^∗−𝑥^′2≤𝑥^∗−𝑥^′2, (32) a contradiction, and thus 𝑥^′=𝑥^∗. This completes the proof.

Lemma 11. Let 𝐸 be a real 𝑞-uniformly smooth Banach space which is also uniformly convex. Let 𝐾 be a closed, convex, and nonempty subset of 𝐸. Let 𝑟∈(0,1) be fixed and {𝑡_𝑛}⊂(0,1) such that lim_𝑛→∞𝑡_𝑛=0 and lim_𝑛→∞(𝑡_𝑛/𝜎_𝑖,𝑛)=0 for all 𝑖∈𝒩_ℐ. Let {𝑧_𝑡𝑛} be a sequence satisfying (25), and let ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟)≠∅. Then, lim_𝑛→∞‖𝑧_𝑡𝑛−(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴))𝑇_𝑟𝑧_𝑡𝑛‖=0, for all 𝑖∈ℕ.

Proof. For 𝑖∈ℕ and 𝑥^∗∈ℱ, we have the following estimates (using Lemma 7 establishing the existence of 𝑔): 𝑔𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛=𝑔𝑥^∗−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟z_𝑡𝑛−𝑥^∗−𝑧_𝑡𝑛≤𝑥^∗−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛²−2𝑥^∗−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛,𝑗𝑥^∗−𝑧_𝑡𝑛+𝑥^∗−𝑧_𝑡𝑛²≤𝑥^∗−𝑧_𝑡𝑛²×𝑥−2_∗−𝑧_𝑡𝑛+𝑧_𝑡𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛,𝑗𝑥^∗−𝑧_𝑡𝑛+𝑥^∗−𝑧_𝑡𝑛²=2𝑧_𝑡𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗. (33) Using (25), we have 𝑧_𝑡𝑛−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗=𝑡_𝑛𝑢−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗+1−𝑡_𝑛(1−𝛿)𝑧_𝑡𝑛−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑛×𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛+𝑧_𝑡𝑛−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗=𝑡_𝑛𝑢−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗+1−𝑡_𝑛𝑧_𝑡𝑛−𝑥^∗,𝑗𝑧_𝑡𝑛−𝑥^∗ (34) which implies 𝛿_𝑖≥1𝜎_𝑖,𝑛𝑧_𝑡𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗=𝑡_𝑛𝑢−𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗. (35)
Using this and (33), we get 𝛿2_𝑖≥1𝜎_𝑖,𝑛𝑔𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛≤𝑡_𝑛𝑢−𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗. (36) Thus, 𝛿2𝑔𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛≤𝑡_𝑛𝜎_𝑖,𝑛𝑢−𝑧_𝑡𝑛,𝑗𝑧_𝑡𝑛−𝑥^∗,∀𝑖∈ℕ. (37) Since {𝑧_𝑡𝑛} is bounded and 𝑡_𝑛/𝜎_𝑖,𝑛→0 as 𝑛→∞, we have lim_𝑛→∞𝑔(‖(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴))𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛‖)=0 for all 𝑖∈ℕ. By property of 𝑔,  lim_𝑛→∞‖(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴))𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛‖=0 for all 𝑖∈ℕ. This completes the proof.

Theorem 12. Let 𝐸 be a real 𝑞-uniformly smooth Banach space which is also uniformly convex. Let 𝐾 be a closed, convex, and nonempty subset of 𝐸. Let 𝑟∈(0,1) be fixed and {𝑡_𝑛}⊂(0,1) such that lim_𝑛→∞𝑡_𝑛=0 and lim_𝑛→∞(𝑡_𝑛/𝜎_𝑖,𝑛)=0 for all 𝑖∈𝒩_ℐ. Let {𝑧_𝑡𝑛} be a sequence satisfying (25), and let ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟)≠∅. If the duality mapping 𝑗 of 𝐸 is weakly sequentially continuous, then {𝑧_𝑡𝑛} converges strongly to an element in ℱ.

Proof. Since {𝑧_𝑡𝑛} is bounded, there exists a subsequence say {𝑧_𝑡𝑛𝑘} of {𝑧_𝑡𝑛} that converges weakly to some point 𝑧∈𝐾. Using demiclosedness property of [𝐼−(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴))𝑇_𝑟] at 0 for 𝑖∈ℕ, and the fact that lim_𝑘→∞‖(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴))𝑇_𝑟𝑧_𝑡𝑛𝑘−𝑧_𝑡𝑛𝑘‖=0, we get that 𝑧 is a point in ℱ. We also observe from (33) that 𝑧_𝑡𝑛𝑘−z²=𝑡_𝑛𝑘(𝑢−𝑧)+1−𝑡_𝑛𝑘(1−𝛿)𝑧_𝑡𝑛𝑘−𝑧+𝛿_𝑖≥1𝜎_{𝑖,𝑛𝑘}𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛𝑘−𝑧,𝑗𝑧_𝑡𝑛𝑘−𝑧≤𝑡_𝑛𝑘𝑢−𝑧,𝑗𝑧_𝑡𝑛𝑘−𝑧+1−𝑡_𝑛𝑘(1−𝛿)𝑧_𝑡𝑛𝑘−𝑧²+𝛿_𝑖≥1𝜎_{𝑖,𝑛𝑘}𝑧_𝑡𝑛𝑘−𝑧²=𝑡_𝑛𝑘𝑢−𝑧,𝑗𝑧_𝑡n𝑘−𝑧+1−𝑡_𝑛𝑘𝑧_𝑡𝑛𝑘−𝑧² (38) which implies 𝑧_𝑡𝑛𝑘−𝑧²≤𝑢−𝑧,𝑗𝑧_𝑡𝑛𝑘−𝑧. (39)
Since 𝑗 admits weak sequential continuity, the last inequality implies that the subsequence {𝑧_𝑡𝑛𝑘} converges strongly to 𝑧, and since {𝑧_𝑡𝑛} admits unique accumulation point in ℱ, then it converges strongly to 𝑧. This completes the proof.

The following corollary follows from Theorem 12.

Corollary 13. Let 𝐸 be a real 𝐿_𝑝 space, (1<𝑝<∞). Let 𝐾, {𝑡_𝑛}, ℱ, and {𝑧_𝑡𝑛} be as in Theorem 12. Then, {𝑧_𝑡𝑛} converges strongly to an element of ℱ.

Theorem 14. Let 𝐸 be a real 𝑞-uniformly smooth Banach space which is also uniformly convex. Let 𝐾 be a closed, convex, and nonempty subset of 𝐸. Let 𝑟∈(0,1) be fixed and {𝑡_𝑛} a sequence in (0,1) such that lim_𝑛→∞𝑡_𝑛=0 and lim_𝑛→∞(𝑡_𝑛/𝜎_𝑖,𝑛)=0 for all 𝑖∈𝒩_ℐ. Let {𝑧_𝑡𝑛} be a sequence satisfying (25), and let ℱ∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟)≠∅. If for at least one 𝑖 in ℕ, 𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 is demicompact, then {𝑧_𝑡𝑛} converges strongly to an element of ℱ.

Proof. For some fixed 𝑗₀∈ℕ, let 𝑇_𝑗0𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 be demicompact. Since lim_𝑛→∞‖𝑇_𝑗0𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑧_𝑡𝑛‖=0, there exists a subsequence say {𝑧_𝑡𝑛𝑘} of {𝑧_𝑡𝑛} that converges strongly to some point 𝑧∈𝐸. By continuity of 𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 for all 𝑖∈ℕ, we have that 𝑧∈ℱ. But the sequence {𝑧_𝑡𝑛} admits unique accumulation point in ℱ; so, it converges strongly to 𝑧.

The following corollaries follow from Theorem 14.

Corollary 15. Let 𝐸 be a real 𝐿_𝑝 space, (1<𝑝<∞). Let 𝐾, {𝑡_𝑛}, ℱ, and {𝑧_𝑡𝑛} be as in Theorem 14. If for at least one 𝑖∈ℕ, the map 𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 is demicompact, then {𝑧_𝑡𝑛} converges strongly to an element of ℱ.

Corollary 16. Let 𝐾 be a closed, convex, and nonempty subset of a real Hilbert space 𝐻. Let {𝑡_𝑛}, ℱ, and {𝑧_𝑡𝑛} be as in Theorem 14. Then, {𝑧_𝑡𝑛} converges strongly to an element of ℱ.

4. Iterative Convergence Theorem

We now state and prove the following theorem.

Theorem 17. Let 𝐸 be a real 𝑞-uniformly smooth Banach space which is also uniformly convex. Let 𝐾 be a closed, convex, and nonempty subset of 𝐸. For some 𝜅>0, let 𝑇_𝑖∶𝐾→𝐸𝑖∈ℕ and 𝐴∶𝐾→𝐸 be a family of nonexpansive maps and a 𝜅-inverse strongly accretive map, respectively. Let 𝐺∶𝐾×𝐾→ℝ be a bifunction satisfying (A1)–(A4). Let 𝑃_𝐾 be a nonexpansive projection of 𝐸 onto 𝐾. For some fixed real numbers 𝑟,𝛿∈(0,1) and 𝜆∈(0,(𝑞𝜅/𝑑_𝑞)^1/(𝑞−1)), define a sequence {𝑥_𝑛} iteratively by 𝑥₁,𝑢∈𝐸 and 𝑛∈ℕ as 𝐺𝑦_𝑛+1,𝜂𝑟𝜂−𝑦_𝑛,𝑗𝑦_𝑛−𝑥_𝑛𝑥≥0,∀𝜂∈𝐾,_𝑛+1=𝛼_𝑛𝑢+(1−𝛿)1−𝛼_𝑛𝑥_𝑛+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾×𝑦_𝑛−𝜆𝐴𝑦_𝑛, (40) where {𝛼_𝑛},{𝜎_𝑖,𝑛}⊂(0,1) are sequences satisfying the following conditions:(i)lim_𝑛→∞𝛼_𝑛=0,(ii)∑^∞_𝑛=1𝛼_𝑛=∞,(iii)lim_𝑛→∞∑_𝑖≥1|𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛|=0.
Let 𝐹∶=[⋂^∞_𝑖=1𝐹(𝑇_𝑖)]⋂EP(𝐺)⋂VI(𝐾,𝐴)≠∅. If either the duality map 𝑗 of 𝐸 is weakly sequentially continuous or for at least one 𝑖∈ℕ, 𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 is demicompact, then {𝑥_𝑛} converges strongly to some element in 𝐹.

Proof. Let 𝑥^∗∈ℱ then, we claim that ‖𝑥_𝑛−𝑥^∗‖≤max{‖𝑢−𝑥^∗‖,‖𝑥₁−𝑥^∗‖} for all 𝑛≥1. It is clear that the claim is true for 𝑛=1. Assume that it is true for 𝑛=𝑘 for some 𝑘≥1,𝑘∈ℕ. Then, 𝑥_𝑘+1−𝑥^∗≤𝛼_𝑘𝑢−𝑥^∗+1−𝛼_𝑘(1−𝛿)𝑥_𝑘−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑘𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑘−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥^∗≤𝛼_𝑘𝑢−𝑥^∗+1−𝛼_𝑘𝑥_𝑘−𝑥^∗≤max𝑢−𝑥^∗,𝑥₁−𝑥^∗. (41) Hence, the result, and so {𝑥_𝑛} is bounded. Furthermore, {𝑦_𝑛}, {𝑇_𝑖𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛)}, and {𝐴𝑦_𝑛} are each bounded.
We now show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Note that 𝑦_𝑛=𝑇_𝑟𝑥_𝑛, 𝑦_𝑛+1=𝑇_𝑟𝑥_𝑛+1, so that 𝑦_𝑛+1−𝑦_𝑛=𝑇_𝑟𝑥_𝑛+1−𝑇_𝑟𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (42)
Define two sequences {𝛽_𝑛} and {𝑤_𝑛} by 𝛽_𝑛∶=(1−𝛿)𝛼_𝑛+𝛿 and 𝑤_𝑛∶=(𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛𝑥_𝑛)/𝛽_𝑛. Then, 𝑤_𝑛=𝛼_𝑛𝑢+𝛿∑_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾𝑦_𝑛−𝜆𝐴𝑦_𝑛𝛽_𝑛. (43) Observe that {𝑤_𝑛} is bounded and that 𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+1𝛽_𝑛+1−𝛼_𝑛𝛽_𝑛𝑢‖+𝛿‖1−𝛼_𝑛+1𝛽_𝑛+1−1𝑥_𝑛+1−𝑥_𝑛+𝛿𝑀𝛽_{𝑛+1𝑖≥1}𝜎_𝑖,𝑛+1−𝜎_𝑖,𝑛+𝛿𝑀𝛽_𝑛+1𝛽_𝑛𝛽_𝑛−𝛽_𝑛+1, (44) for some positive real number 𝑀. This implies limsup_𝑛→∞𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0, (45) and by Lemma 4, lim_𝑛→∞||𝑤_𝑛−𝑥_𝑛||=0. Hence, 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛𝑤_𝑛−𝑥_𝑛→0as𝑛→∞. (46)
From (42) and (46), we have 𝑦_𝑛+1−𝑦_𝑛→0as𝑛→∞. (47)
From (40), we have 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛(𝑢−𝑥_𝑛)+𝛿∑_𝑖≥1𝜎_𝑖,𝑛[𝑇_𝑖𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑥_𝑛] which implies 𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑢−𝑥_𝑛, (48) and thus lim_𝑛→∞‖∑_𝑖≥1𝜎_𝑖,𝑛[𝑇_𝑖𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛)−𝑥_𝑛]‖=0. Let {𝑡_𝑛} be a real sequence in (0,1) satisfying the following conditions: lim_𝑛→∞𝑡_𝑛=0,_𝑖≥1𝜎_𝑖,𝑛=1−𝑡_𝑛,lim_𝑛→∞∑_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥_𝑛𝑡_𝑛=0. (49)
Let 𝑧_𝑡𝑛∈𝐾 be the unique fixed point satisfying (25) for each 𝑛∈ℕ, and let 𝑧_𝑡𝑛→𝑧∈ℱ as 𝑛→∞. Using (25) and Lemma 2, we have the following estimates: 𝑧_𝑡𝑛−𝑥_𝑛²≤(1−𝛿)1−𝑡_𝑛𝑧_𝑡𝑛−𝑥_𝑛+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧_𝑡𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛+𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑥_𝑛²+2𝑡_𝑛𝑢−𝑥_𝑛,𝑗𝑧_𝑡𝑛−𝑥_𝑛≤(1−𝛿)1−𝑡_𝑛𝑧_𝑡𝑛−𝑥_𝑛+𝛿1−𝑡_𝑛𝑧_𝑡𝑛−𝑥_𝑛+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑥_𝑛²+2𝑡_𝑛𝑢−𝑥_𝑛,𝑗𝑧_𝑡𝑛−𝑥_𝑛=1−𝑡_𝑛𝑧_𝑡𝑛−𝑥_𝑛+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑥_𝑛²+2𝑡_𝑛𝑢−𝑥_𝑛,𝑗𝑧_𝑡𝑛−𝑥_𝑛. (50)
This implies 𝑢−𝑧_𝑡𝑛,𝑗𝑥_𝑛−𝑧_𝑡𝑛≤𝑡_𝑛2𝑧_𝑡𝑛−𝑥_𝑛²+1−𝑡_𝑛𝑧_𝑡n−𝑥_𝑛×𝛿∑_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑥_𝑛𝑡_𝑛+𝛿²∑_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑥_𝑛²2𝑡_𝑛, (51) and, hence, limsup_𝑛→∞𝑢−𝑧_𝑡𝑛,𝑗𝑥_𝑛−𝑧_𝑡𝑛≤0. (52) Moreover, 𝑢−𝑧_𝑡𝑛,𝑗𝑥_𝑛−𝑧_𝑡𝑛=𝑢−𝑧,𝑗𝑥_𝑛+𝑥−𝑧𝑢−𝑧,𝑗_𝑛−𝑧_𝑡𝑛−𝑗𝑥_𝑛+−𝑧𝑧−𝑧_𝑡𝑛,𝑗𝑥_𝑛−𝑧_𝑡𝑛, (53) and since 𝑗 is norm-to-weak* uniformly continuous on bounded sets, we have limsup_𝑛→∞𝑢−𝑧,𝑗𝑥_𝑛−𝑧≤0. (54)
From the recursion formula (40) and Lemma 2, we have the following: 𝑥_𝑛+1−𝑧²=𝛼_𝑛(𝑢−𝑧)+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑧+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧²≤1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑧+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑥_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟𝑧²+2𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1−𝑧≤1−𝛼_𝑛𝑥_𝑛−𝑧²+2𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1−𝑧, (55) and by Lemma 5, we get that {𝑥_𝑛} converges strongly to 𝑧∈ℱ.
To complete the proof, we show that 𝑧∈EP(𝐺)∩VI(𝐾,𝐴).
We start by showing that 𝑧∈EP(𝐺).
Let 𝑥^∗∈𝐹; then, 𝑦_𝑛−𝑥^∗2=𝑇_𝑟𝑥_𝑛−𝑇_𝑟𝑥^∗2≤𝑇_𝑟𝑥_𝑛−𝑇_𝑟𝑥^∗,𝑗𝑥_𝑛−𝑥^∗=𝑦_𝑛−𝑥^∗,𝑗𝑥_𝑛−𝑥^∗≤12𝑥_𝑛−𝑥^∗2+𝑦_𝑛−𝑥^∗2−𝑔𝑥_𝑛−𝑦_𝑛; (56) thus, 𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑔𝑥_𝑛−𝑦_𝑛. (57) Using (40) and (57), we have 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑦_𝑛−𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗2+𝛿1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗2+𝛿1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−𝑔𝑥_𝑛−𝑦_𝑛. (58) This implies 𝛿1−𝛼_𝑛𝑔𝑥_𝑛−𝑦_𝑛≤𝛼_𝑛𝑢−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗, (59) and thus lim_𝑛→∞𝛿(1−𝛼_𝑛)𝑔(‖𝑥_𝑛−𝑦_𝑛‖)=0. Using property of 𝑔, we get lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (60) From (60), we have 𝑦_𝑛→𝑧 and 𝑗(𝑥_𝑛−𝑦_𝑛)→0 as 𝑛→∞. Since 𝑦_𝑛=𝑇_𝑟𝑥_𝑛, we have 𝐺𝑦_𝑛+1,𝜂𝑟𝜂−𝑦_𝑛,𝑗𝑦_𝑛−𝑥_𝑛≥0,∀𝜂∈𝐾, (61) It follows from (A2) that 𝜂−𝑦_𝑛,𝑗𝑦_𝑛−𝑥_𝑛𝑟≥𝐺𝜂,𝑦_𝑛, (62) and so using (A4), we have 𝐺(𝜂,𝑧)≤0 for all 𝜂∈𝐾. For real number 𝑡, 0<𝑡≤1, and 𝜂∈𝐾, let 𝜂_𝑡=𝑡𝜂+(1−𝑡)𝑧. Clearly, 𝜂_𝑡∈𝐾. So, using (A1) and (A4), we have 0=𝐺𝜂_𝑡,𝜂_𝑡≤𝑡𝐺𝜂_𝑡,𝜂+(1−𝑡)𝐺𝜂_𝑡,𝑧≤𝑡𝐺𝜂_𝑡,𝜂. (63)
This implies 𝐺(𝜂_𝑡,𝜂)≥0, and using this and (A3), we have that 𝐺(𝑧,𝜂)≥0 for all 𝜂∈𝐾; hence, 𝑧∈EP(𝐺).
Next, we show that 𝑧∈VI(𝐾,𝐵).
Let 𝑥^∗∈𝐹 and 𝑏_𝑛∶=𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛); then, 𝑏_𝑛−𝑥^∗𝑞=𝑃_𝐾𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐾𝑥^∗−𝜆𝐴𝑥^∗𝑞≤𝑦_𝑛−𝑥^∗−𝜆𝐴𝑦_𝑛−𝐴𝑥^∗𝑞≤𝑦_𝑛−𝑥^∗𝑞+𝜆𝑑_𝑞𝜆^𝑞−1−𝑞𝜅𝐴𝑦_𝑛−𝐴𝑥^∗𝑞. (64)
Using recursion formula (40), we have the following estimates: 𝑥_𝑛+1−𝑥^∗𝑞=𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑏_𝑛−𝑥^∗𝑞≤𝛼_𝑛𝑢−𝑥^∗𝑞+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗𝑞+1−𝛼_𝑛𝛿𝑏_𝑛−𝑥^∗𝑞≤𝛼_𝑛𝑢−𝑥^∗𝑞+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗𝑞+1−𝛼_𝑛𝛿×𝑥_𝑛−𝑥^∗𝑞+𝜆𝑑_𝑞𝜆^𝑞−1−𝑞𝜅𝐴𝑦_𝑛−𝐴𝑥^∗𝑞≤𝛼_𝑛𝑢−𝑥^∗𝑞+𝑥_𝑛−𝑥^∗𝑞+1−𝛼_𝑛𝛿𝜆𝑑_𝑞𝜆^𝑞−1×−𝑞𝜅𝐴𝑦_𝑛−𝐴𝑥^∗𝑞, (65) which implies, by Mean Value Theorem, that −1−𝛼_𝑛𝛿𝜆𝑑_𝑞𝜆^𝑞−1−𝑞𝜅𝐴𝑦_𝑛−𝐴𝑥^∗𝑞≤𝛼_𝑛𝑢−𝑥^∗𝑞+𝑥_𝑛−𝑥^∗𝑞−𝑥_𝑛+1−𝑥^∗𝑞≤𝛼_𝑛𝑢−𝑥^∗𝑞+𝑞𝜎_𝑛^𝑞−1𝑥_𝑛−𝑥^∗−𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝑢−𝑥^∗𝑞+𝑞𝜎_𝑛^𝑞−1𝑥_𝑛−𝑥_𝑛+1, (66) where 𝜎_𝑛 is some nonnegative real number between ‖𝑥_𝑛−𝑥^∗‖ and ‖𝑥_𝑛+1−𝑥^∗‖ for each 𝑛. Since {𝑥_𝑛} is bounded, 𝛼_𝑛→0, and ‖𝑥_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞, we have ‖𝐴𝑦_𝑛−𝐴𝑥^∗‖→0 as 𝑛→∞.
We also have the following: 𝑏_𝑛−𝑥^∗2=𝑃_𝐾𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑃_𝐾𝑥^∗−𝜆𝐴𝑥^∗2≤𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥^∗−𝜆𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗=𝑦_𝑛−𝑥^∗,𝑗𝑏_𝑛−𝑥^∗−𝜆𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗≤12𝑏_𝑛−𝑥^∗2+𝑦_𝑛−𝑥^∗2−𝑔𝑏_𝑛−𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗ (67) so that 𝑏_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑔𝑏_𝑛−𝑦_𝑛−2𝜆𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗. (68)
We then have 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗+𝛿_𝑖≥1𝜎_𝑖,𝑛𝑇_𝑖𝑏_𝑛−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝛿𝑏_𝑛−𝑥^∗2≤𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛(1−𝛿)𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝛿𝑥_𝑛−𝑥^∗2−𝑔𝑏_𝑛−𝑥_𝑛−2𝜆𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗=𝛼_𝑛𝑢−𝑥^∗2+1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛×𝛿𝑔𝑏_𝑛−𝑦_𝑛−2𝛿𝜆1−𝛼_𝑛𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗, (69) and so 1−𝛼_𝑛𝛿𝑔𝑏_𝑛−𝑦_𝑛≤𝛼_𝑛𝑢−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2−2𝛿𝜆1−𝛼_𝑛𝐴𝑦_𝑛−𝐴𝑥^∗,𝑗𝑏_𝑛−𝑥^∗. (70)
As 𝛼_𝑛→0,‖𝑥_𝑛−𝑥_𝑛+1‖→0 and ‖𝐴𝑥_𝑛−𝐴𝑥^∗‖→0 as 𝑛→∞, we get lim_𝑛→∞1−𝛼_𝑛𝛿𝑔𝑏_𝑛−𝑦_𝑛=0, (71) which implies lim_𝑛→∞𝑏_𝑛−𝑦_𝑛=0. (72)
Let 𝑀𝑣=𝐴𝑣+𝑁_𝐾𝑣,𝑣∈𝐾,∅,𝑣∉𝐾. (73)
Then, 𝑀 is maximal accretive. Let GF(𝑀) denote the graph of 𝑀.
Let (𝑣,𝑤)∈GF(𝑀). Since 𝑤−𝐴𝑣∈𝑁_𝐾𝑣 and 𝑏_𝑛∈𝐾, we have ⟨𝑤−𝐴𝑣,𝑗_𝑞(𝑣−𝑏_𝑛)⟩≥0 by definition of 𝑁_𝐾𝑣. Also, as 𝑏_𝑛=𝑃_𝐾(𝑦_𝑛−𝜆𝐴𝑦_𝑛) (using property of the projection 𝑃_𝐾), we have 𝑏_𝑛−𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛≥0 (74) and, hence, 𝑏_𝑛−𝑦_𝑛𝜆+𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛≥0. (75)
Using this, we obtain the following estimates: 𝑤,𝑗_𝑞𝑣−𝑏_𝑛≥𝐴𝑣,𝑗_𝑞𝑣−𝑏_𝑛≥𝐴𝑣,𝑗_𝑞𝑣−𝑏_𝑛−𝑏_𝑛−𝑦_𝑛𝜆+𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛=𝑏𝐴𝑣−^𝑛−𝑦_𝑛𝜆−𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛=𝐴𝑣−𝐴𝑏_𝑛,𝑗_𝑞𝑣−𝑏_𝑛+𝐴𝑏_𝑛−𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛−𝑏_𝑛−𝑦_𝑛𝜆,𝑗_𝑞𝑣−𝑏_𝑛≥𝐴𝑏_𝑛−𝐴𝑦_𝑛,𝑗_𝑞𝑣−𝑏_𝑛−𝑏_𝑛−𝑦_𝑛𝜆,𝑗_𝑞𝑣−𝑏_𝑛, (76) which implies ⟨𝑤,𝑗_𝑞(𝑣−𝑧)⟩≥0 (letting 𝑛→∞).
Since 𝑀 is maximal accretive, we obtained that 𝑧∈𝑀⁻¹(0), and, hence, 𝑧∈VI(𝐾,𝐴). This completes the proof.

The following corollaries follow from Theorem 17.

Corollary 18. Let 𝐸=𝐿_𝑝 space (1<𝑝<∞). Let 𝐾, 𝑃_𝐾, 𝛿, 𝐴, 𝑟, and 𝑇_𝑖, 𝑖=1,2,… be as in Theorem 17. Let 𝜆∈(0,(2𝜅/(𝑝−1))), and define sequences {𝑥_𝑛} and {𝑦_𝑛} by (40). Then, {𝑥_𝑛} and {𝑦_𝑛} converge strongly to some element in 𝐹.

Corollary 19. Let 𝐸=𝐿_𝑝  space  (1<𝑝<∞). Let 𝐾, 𝑃_𝐾, 𝛿, 𝐴, 𝑟, and 𝑇_𝑖, 𝑖=1,2,… be as in Theorem 17. Let 𝜆∈(0,(2𝜅/(𝑝−1))), and define sequences {𝑥_𝑛} and {𝑦_𝑛} by (40). If for at least one 𝑖 in ℕ,𝑇_𝑖𝑃_𝐾(𝐼−𝜆𝐴)𝑇_𝑟 is demicompact, then the sequences {𝑥_𝑛} and {𝑦_𝑛} both converge strongly to some element in 𝐹.

Corollary 20. Let 𝐸=𝐻 be a real Hilbert space. Let 𝐾, 𝑃_𝐾, 𝛿, 𝐴, 𝑟, and 𝑇_𝑖, 𝑖=1,2,… be as in Theorem 17. Let 𝜆∈(0,2𝜅), and define sequences {𝑥_𝑛} and {𝑦_𝑛} by (40). Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} both converge strongly to some element in 𝐹.

Remark 21. Prototypes of the sequences {𝛼_𝑛} and {𝜎_𝑖,𝑛} in our theorems are the following: 𝛼_𝑛1∶=𝑛+1,𝜎_𝑖,𝑛𝑛∶=2^𝑖(𝑛+1),∀𝑖∈ℕ. (77)