Strong Convergence to a Solution of a Variational Inequality Problem in Banach Spaces

Abstract

We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.

1. Introduction

Let ℕ and ℝ be the set of all positive integers and the set of all real numbers, respectively. Throughout this paper, 𝐸 is a real Banach space with norm ‖⋅‖ and 𝐸^∗ is the dual of 𝐸. For 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, let ⟨𝑥,𝑥^∗⟩ be the value of 𝑥^∗ at 𝑥. Suppose that 𝐶 is a nonempty closed convex subset of 𝐸 and 𝐴 is a monotone operator of 𝐶 into 𝐸^∗; that is, ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥0 holds for all 𝑥,𝑦∈𝐶. Then, we consider the variational inequality problem [1], that is, the problem of finding an element 𝑧∈𝐶 such that ⟨𝑥−𝑧,𝐴𝑧⟩≥0∀𝑥∈𝐶. (1)

The set of all solutions to the variational inequality problem for 𝐴 is denoted by VI(𝐶,𝐴). For 𝛼>0, we say that 𝐴 is 𝛼-inverse strongly monotone [2–5] if 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼𝐴𝑥−𝐴𝑦²∀𝑥,𝑦∈𝐶. (2)

Haugazeau [6] introduced a sequence {𝑥_𝑛} generated by the hybrid method by the following way. Let {𝑇_𝑛} be a family of mappings of a real Hilbert space 𝐻 into itself with ⋂^∞_𝑛=0𝐹(𝑇_𝑛)≠0/, where 𝐹(𝑇_𝑛) is the set of all fixed points of 𝑇_𝑛. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦=𝑥∈𝐻,_𝑛=𝑇_𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐻∶𝑥_𝑛−𝑦_𝑛,𝑦_𝑛,𝑄−𝑧≥0_𝑛=𝑧∈𝐻∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}(𝑥) (3) for each 𝑛∈ℕ, where 𝑃_{𝐶𝑛∩𝑄𝑛} is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛. He proved a strong convergence theorem when 𝑇_𝑛=𝑃_{(𝑛mod𝑚)+1} for every 𝑛∈ℕ, where 𝑃_𝑖 is the metric projection of 𝐻 onto a nonempty closed convex subset 𝐶_𝑖 of 𝐻 for each 𝑖=1,2,…,𝑚 with ⋂^𝑚_𝑖=1𝐶_𝑖≠0/. Later, Solodov and Svaiter [7], Bauschke and Combettes [8], Nakajo and Takahashi [9], and many researchers studied the hybrid method in a real Hilbert space. In a real Banach space, Kamimura and Takahashi [10], Ohsawa and Takahashi [11], Kohsaka and Takahashi [12], Matsushita and Takahashi [13], Matsushita et al. [14], Nakajo et al. [15], and several researchers studied the hybrid method.

In a real Hilbert space 𝐻, Iiduka et al. [16] considered a sequence {𝑥_𝑛} generated by the following hybrid method: 𝑥₁𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 (4) for each 𝑛∈ℕ, where 𝐴 is an 𝛼-inverse strongly monotone operator of 𝐶 into 𝐻 with VI(𝐶,𝐴)≠0/, 𝑃_𝐶 is the metric projection of 𝐻 onto a nonempty closed convex subset 𝐶 of 𝐻, and {𝜆_𝑛}⊂[0,2𝛼]. They proved that {𝑥_𝑛} converges strongly to 𝑃_{VI(𝐶,𝐴)}𝑥; see also [17, 18]. In a 2-uniformly convex and uniformly smooth Banach space 𝐸, Iiduka and Takahashi [19] proved the following.

Theorem 1 (Iiduka and Takahashi [19]). Let 𝐴 be an 𝛼-inverse strongly monotone operator of 𝐸 into 𝐸^∗ with 𝐴⁻¹0≠0/ and {𝜆_𝑛}⊂[𝑎,𝑐₁𝛼] for some 𝑎∈]0,𝑐₁𝛼[, where 𝑐₁ is a positive constant satisfying that ‖𝑥+𝑦‖²≥‖𝑥‖²+2⟨𝑦,𝐽𝑥⟩+𝑐₁‖𝑦‖² for every 𝑥,𝑦∈𝐸. Let {𝑥_𝑛} be a sequence generated by 𝑥₁𝑦=𝑥∈𝐸,_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐸∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 (5) for each 𝑛∈ℕ, where Π_{𝐶𝑛∩𝑄𝑛} is the generalized projection of 𝐸 onto 𝐶_𝑛∩𝑄_𝑛 and 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖² for 𝑥,𝑦∈𝐸. Then, {𝑥_𝑛} converges strongly to Π_𝐴⁻¹₀𝑥.

Motivated by [19], we propose a new family of operators and prove strong convergence theorems of the sequence generated by these mappings. Using these results, we get several additional results for the problem of variational inequalities and the proximal point algorithm.

2. Preliminaries

Throughout this paper, we write 𝑥_𝑛⇀𝑥 to indicate that a sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 will symbolize strong convergence. We denote by 𝑆_𝐸 the unit sphere of a Banach space 𝐸; that is, 𝑆_𝐸={𝑥∈𝐸∶‖𝑥‖=1}.

We define the modulus 𝛿_𝐸 of convexity of 𝐸 as follows: 𝛿_𝐸 is a function of [0,2] into [0,1] such that 𝛿_𝐸(𝜖)=inf1−𝑥+𝑦2∶𝑥,𝑦∈𝑆_𝐸,𝑥−𝑦≥𝜖 (6) for every 𝜖∈[0,2]. 𝐸 is said to be uniformly convex if 𝛿_𝐸(𝜖)>0 for each 𝜖>0. Let 𝑝>1. 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿_𝐸(𝜖)≥𝑐𝜖^𝑝 for every 𝜖∈[0,2]. It is obvious that a 𝑝-uniformly convex Banach space is uniformly convex. 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝑆_𝐸 with 𝑥≠𝑦. We know that a uniformly convex Banach space is strictly convex and reflexive. The duality mapping 𝐽∶𝐸→2^𝐸∗ of 𝐸 is defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓² (7) for every 𝑥∈𝐸. It is also known that if 𝐸 is strictly convex and reflexive, then the duality mapping 𝐽 of 𝐸 is bijective and 𝐽⁻¹∶𝐸^∗→2^𝐸 is the duality mapping of 𝐸^∗. 𝐸 is said to be smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (8) exists for every 𝑥,𝑦∈𝑆_𝐸. The norm of 𝐸 is said to be uniformly Gâteaux differentiable if, for each 𝑦∈𝑆_𝐸, the limit (8) is attained uniformly for 𝑥∈𝑆_𝐸. 𝐸 is said to be uniformly smooth if the limit (8) is attained uniformly for (𝑥,𝑦)∈𝑆_𝐸×𝑆_𝐸. We know that the duality mapping 𝐽 of 𝐸 is single-valued if and only if 𝐸 is smooth. It is also known that if 𝐸 is uniformly smooth, then the duality mapping 𝐽 is uniformly continuous on bounded subsets of 𝐸 and if the norm of 𝐸 is uniformly Gâteaux differentiable, then 𝐽 is norm-to-weak^∗ uniformly continuous on bounded subsets of 𝐸; see [20, 21] for more details. The following is proved by Xu [22]; see also [23].

Theorem 2 (Xu [22]). Let 𝐸 be a smooth Banach space. Then, the following are equivalent. (i)𝐸is 2-uniformly convex.(ii)There exists a constant 𝑐₁>0 such that ‖𝑥+𝑦‖²≥‖𝑥‖²+2⟨𝑦,𝐽𝑥⟩+𝑐₁‖𝑦‖² holds for each 𝑥,𝑦∈𝐸.

Remark 3. In the case where 𝐸 is a real Hilbert space, 𝐽 is the identity mapping and we can choose 𝑐₁=1.

Let 𝐸 be a smooth Banach space. The function 𝜙∶𝐸×𝐸→ℝ is defined by 𝜙𝑦,𝑥=𝑦²−2𝑦,𝐽𝑥+‖𝑥‖² (9) for every 𝑥,𝑦∈𝐸. It is obvious that (‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)² for each 𝑥,𝑦∈𝐸 and 𝜙(𝑧,𝑥)+𝜙(𝑥,𝑦)=𝜙(𝑧,𝑦)+2⟨𝑥−𝑧,𝐽𝑥−𝐽𝑦⟩ for all 𝑥,𝑦,𝑧∈𝐸. It is also known that if 𝐸 is strictly convex and smooth, then, for 𝑥,𝑦∈𝐸, 𝜙(𝑦,𝑥)=0 if and only if 𝑥=𝑦; see also [13]. We have the following result from Theorem 2.

Lemma 4. Let 𝐸 be a 2-uniformly convex and smooth Banach space. Then, for each 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)≥𝑐₁‖𝑥−𝑦‖² holds, where 𝑐₁ is a constant in Theorem 2.

Proof. Let 𝑥,𝑦∈𝐸. By Theorem 2, we have 𝜙𝑥,𝑦=‖𝑥‖²−𝑦²−2𝑥−𝑦,𝐽𝑦≥𝑐₁𝑥−𝑦², (10) which is the desired result.

Let 𝐶 be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space 𝐸 and let 𝑥∈𝐸. Then, there exists a unique element 𝑦₀∈𝐶 such that 𝜙𝑦₀,𝑥=inf_𝑦∈𝐶𝜙𝑦,𝑥. (11)

We denote 𝑦₀ by Π_𝐶𝑥 and call Π_𝐶 the generalized projection of 𝐸 onto 𝐶; see [10, 24, 25]. We have the following well-known results [10, 24, 25] for the generalized projection.

Lemma 5. Let 𝐶 be a nonempty convex subset of a smooth Banach space E, 𝑥∈𝐸, and 𝑦₀∈𝐶. Then, 𝜙(𝑦₀,𝑥)=inf_𝑦∈𝐶𝜙(𝑦,𝑥) if and only if ⟨𝑦₀−𝑧,𝐽𝑥−𝐽𝑦₀⟩≥0 for all 𝑧∈𝐶.

Let 𝐶 be a nonempty closed convex subset of a strictly convex and reflexive Banach space 𝐸 and let 𝑥∈𝐸. Then, there exists a unique element 𝑦₀∈𝐶 such that ‖𝑦₀−𝑥‖=inf_𝑦∈𝐶‖𝑦−𝑥‖. Putting 𝑦₀=𝑃_𝐶𝑥, we call 𝑃_𝐶 the metric projection of 𝐸 onto 𝐶; see [26]. We have the following result for the metric projection; see [20] for more details.

Lemma 6. Let 𝐶 be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space 𝐸, 𝑥∈𝐸, and 𝑦₀∈𝐶. Then, 𝑦₀=𝑃_𝐶𝑥 if and only if ⟨𝑦₀−𝑧,𝐽(𝑥−𝑦₀)⟩≥0 for all 𝑧∈𝐶.

An operator 𝑇∶𝐸→2^𝐸∗ is said to be monotone if ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩≥0 for every (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝑇. Notice that we often identify a set-valued operator with its graph; 𝑥^∗∈𝑇𝑥 if and only if (𝑥,𝑥^∗)∈𝑇.

A monotone operator 𝑇⊂𝐸×𝐸^∗ is said to be maximal if the graph of 𝑇 is not properly contained in the graph of any other monotone operator. It is easy to see that a monotone operator 𝑇⊂𝐸×𝐸^∗ is maximal if and only if, for (𝑢,𝑢^∗)∈𝐸×𝐸^∗, ⟨𝑥−𝑢,𝑥^∗−𝑢^∗⟩≥0 for every (𝑥,𝑥^∗)∈𝑇 implies that (𝑢,𝑢^∗)∈𝑇. We know the following result.

Theorem 7 (Rockafellar [27]; see also [28]). Let 𝐸 be a strictly convex, reflexive, and smooth Banach space and let 𝑇 be a monotone operator of 𝐸 into 𝐸^∗. Then, 𝑇 is maximal if and only if 𝑅(𝐽+𝑟𝑇)=𝐸^∗ for all 𝑟>0, where 𝑅(𝐽+𝑟𝑇) is the range of 𝐽+𝑟𝑇.

From this fact, we also know that if 𝐸 is a strictly convex, reflexive, and smooth Banach space and 𝑇 is a maximal monotone operator of 𝐸 into 𝐸^∗, then, for any 𝑥∈𝐸 and 𝑟>0, there exists a unique element 𝑥_𝑟∈𝐷(𝑇) such that 𝐽(𝑥_𝑟−𝑥)+𝑟𝑇𝑥_𝑟∋0, where 𝐷(𝑇) is the domain of 𝑇. We define 𝐽_𝑟∶𝐸→𝐸 by 𝐽_𝑟𝑥=𝑥_𝑟 for every 𝑥∈𝐸 and 𝑟>0, and such 𝐽_𝑟 is called the resolvent of 𝑇; see [21, 29] for more details.

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space 𝐸 and {𝐴_𝑛}_𝑛∈ℕ a family of operators of 𝐶 into 𝐸^∗ satisfying the following:(i)𝐹=⋂^∞_𝑛=1VI(𝐶,𝐴_𝑛)≠0/;(ii)⟨𝑥−𝑧,𝐴_𝑛𝑥−𝐴_𝑛𝑧⟩≥0 for all 𝑛∈ℕ, 𝑥∈𝐶, and 𝑧∈𝐹;(iii)there exists a sequence {𝛼_𝑛}_𝑛∈ℕ in ]0,∞[ such that 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<∞ and ⟨𝑥−𝑧,𝐴_𝑛𝑥⟩≥𝛼_𝑛‖𝐴_𝑛𝑥−𝐴_𝑛𝑧‖² for every 𝑛∈ℕ, 𝑥∈𝐶, and 𝑧∈𝐹;(iv)for all 𝑧∈𝐹, sup_𝑛∈ℕ‖𝐴_𝑛𝑧‖<∞;(v)for every bounded sequence {𝑧_𝑛}⊂𝐶, 𝑧∈𝐹, and {𝑟_𝑛}⊂]0,∞[ with inf_𝑛∈ℕ𝑟_𝑛>0, if lim_𝑛→∞‖𝑧_𝑛+1−𝑧_𝑛‖=lim_𝑛→∞‖𝑧_𝑛−Π_𝐶𝐽⁻¹(𝐽𝑧_𝑛−𝑟_𝑛𝐴_𝑛𝑧_𝑛)‖=lim_𝑛→∞‖𝐴_𝑛𝑧_𝑛−𝐴_𝑛𝑧‖=0, then there exists a subsequence {𝑧_𝑛𝑖} of {𝑧_𝑛} such that 𝑧_𝑛𝑖⇀𝑧∈𝐹.

Let us observe some properties of the mappings and the subsets deduced from the assumptions above.

First, we know that, for any 𝑛∈ℕ, the image of 𝐹 by 𝐴_𝑛 is a singleton. Indeed, for 𝑧₁,𝑧₂∈𝐹, we have ⟨𝑧₁−𝑧₂,𝐴_𝑛𝑧₁⟩≥𝛼_𝑛‖𝐴_𝑛𝑧₁−𝐴_𝑛𝑧₂‖² by the condition (iii). On the other hand, since 𝑧₁∈VI(𝐶,𝐴_𝑛), it follows that ⟨𝑧₂−𝑧₁,𝐴_𝑛𝑧₁⟩≥0. Thus, we get 𝐴_𝑛𝑧₁=𝐴_𝑛𝑧₂ for all 𝑧₁,𝑧₂∈𝐹.

Next, if we assume ⋂^∞_𝑛=1𝐴_𝑛⁻¹0≠0/, then we have ⋂^∞_𝑛=1𝐴_𝑛⁻¹0=⋂^∞_𝑛=1VI(𝐶,𝐴_𝑛). Indeed, the inclusion ⋂^∞_𝑛=1𝐴_𝑛⁻¹0⊂⋂^∞_𝑛=1VI(𝐶,𝐴_𝑛) is trivial. To show the opposite inclusion, let 𝑧∈⋂^∞_𝑛=1VI(𝐶,𝐴_𝑛) and 𝑢∈⋂^∞_𝑛=1𝐴_𝑛⁻¹0. By the condition (iii), we have 0=⟨𝑢−𝑧,𝐴_𝑛𝑢⟩≥𝛼_𝑛‖𝐴_𝑛𝑢−𝐴_𝑛𝑧‖²=𝛼_𝑛‖𝐴_𝑛𝑧‖², which implies 𝐴_𝑛𝑧=0; that is, 𝑧∈𝐴_𝑛⁻¹0 for all 𝑛∈ℕ. Hence, we get ⋂^∞_𝑛=1𝐴_𝑛⁻¹0⊃⋂^∞_𝑛=1VI(𝐶,𝐴_𝑛).

We also know that 𝐹 is closed and convex. Indeed, for 𝑧₁,𝑧₂∈𝐹 and 𝛽∈]0,1[, let 𝑧=𝛽𝑧₁+(1−𝛽)𝑧₂. By the condition (iii), ⟨𝑧−𝑧₁,𝐴_𝑛𝑧⟩≥𝛼_𝑛‖𝐴_𝑛𝑧−𝐴_𝑛𝑧₁‖² and ⟨𝑧−𝑧₂,𝐴_𝑛𝑧⟩≥𝛼_𝑛‖𝐴_𝑛𝑧−𝐴_𝑛𝑧₂‖² hold for all 𝑛∈ℕ. Thus we get 0=𝑧−𝛽𝑧₁+1−𝛽𝑧₂,𝐴_𝑛𝑧=𝛽𝑧−𝑧₁,𝐴_𝑛𝑧+1−𝛽𝑧−𝑧₂,𝐴_𝑛𝑧≥𝛼_𝑛𝛽𝐴_𝑛𝑧−𝐴_𝑛𝑧₁²+1−𝛽𝐴_𝑛𝑧−𝐴_𝑛𝑧₂²≥0, (12) which implies that 𝐴_𝑛𝑧=𝐴_𝑛𝑧₁=𝐴_𝑛𝑧₂ for each 𝑛∈ℕ. Since ⟨𝑥−𝑧₁,𝐴_𝑛𝑧⟩=⟨𝑥−𝑧₁,𝐴_𝑛𝑧₁⟩≥0 and ⟨𝑥−𝑧₂,𝐴_𝑛𝑧⟩=⟨𝑥−𝑧₂,𝐴_𝑛𝑧₂⟩≥0 for every 𝑛∈ℕ and 𝑥∈𝐶, we have 𝑥−𝑧,𝐴_𝑛𝑧=𝛽𝑥−𝑧₁,𝐴_𝑛𝑧+1−𝛽𝑥−𝑧₂,𝐴_𝑛𝑧≥0 (13) for all 𝑛∈ℕ and 𝑥∈𝐶; that is, 𝑧∈VI(𝐶,𝐴_𝑛) for each 𝑛∈ℕ. Hence, 𝐹 is convex.

To see 𝐹 being closed, let {𝑧_𝑚} be a sequence in 𝐹 such that 𝑧_𝑚→𝑧. Since we have ⟨𝑧−𝑧_𝑚,𝐴_𝑛𝑧⟩≥𝛼_𝑛‖𝐴_𝑛𝑧−𝐴_𝑛𝑧_𝑚‖² for every 𝑚,𝑛∈ℕ from the condition (iii), we get lim_𝑚→∞‖𝐴_𝑛𝑧−𝐴_𝑛𝑧_𝑚‖=0 for all 𝑛∈ℕ. Since ⟨𝑥−𝑧_𝑚,𝐴_𝑛𝑧_𝑚⟩≥0 for each 𝑚,𝑛∈ℕ and 𝑥∈𝐶, we obtain ⟨𝑥−𝑧,𝐴_𝑛𝑧⟩≥0 for every 𝑛∈ℕ and 𝑥∈𝐶; that is, 𝑧∈VI(𝐶,𝐴_𝑛) for all 𝑛∈ℕ. Therefore, 𝐹 is closed.

Now, we get the following result by the hybrid method using the generalized projections.

Theorem 8. Let 𝐶 be a nonempty closed convex subset of a 2-uniformly convex Banach space 𝐸 whose norm is uniformly Gâteaux differentiable, and let {𝐴_𝑛} be a sequence of operators of 𝐶 into 𝐸^∗ satisfying the conditions (i)–(v). Let {𝜆_𝑛} be a sequence in ]0,∞[ such that inf_𝑛∈ℕ𝜆_𝑛>0 and inf_𝑛∈ℕ(2𝑐₁𝛼_𝑛−𝜆_𝑛)>0, where 𝑐₁ is the constant in Theorem 2. Let 𝑥∈𝐶 and {𝑥_𝑛} a sequence in 𝐶 generated by 𝑥₁𝑦=𝑥,_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 (14)
for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to Π_𝐹𝑥.

Proof. It is obvious that 𝑄_𝑛 is closed and convex for every 𝑛∈ℕ. Since 𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛)+2𝜆_𝑛⟨𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛⟩ if and only if 2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑦_𝑛⟩+‖𝑦_𝑛‖²−‖𝑥_𝑛‖²−2𝜆_𝑛⟨𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛⟩≤0, we have that 𝐶_𝑛 is closed and convex for all 𝑛∈ℕ. Next, we show that, for 𝑛∈ℕ, 𝑥_𝑛∈𝐶 implies that 𝐹⊂𝐶_𝑛. Let 𝑧∈𝐹. We have 𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧=𝜙𝑧,𝑦_𝑛−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧=−𝜙𝑦_𝑛,𝑥_𝑛+2𝑦_𝑛−𝑧,𝐽𝑦_𝑛−𝐽𝑥_𝑛+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧 (15) for every 𝑛∈ℕ. Since 𝑦_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛) and by Lemma 5, we get ⟨𝑦_𝑛−𝑧,𝐽𝑥_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛−𝐽𝑦_𝑛⟩≥0. Further, by Lemma 4, we obtain 𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧≤−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑥_𝑛+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧=−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑥_𝑛+𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧. (16)
Using 𝑥_𝑛∈𝐶 and the condition (iii), we have ⟨𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛⟩≥𝛼_𝑛‖𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧‖² and thus 𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧≤−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑥_𝑛+𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧²+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧=−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧²−2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧≤−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧²+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧≤−𝑐₁𝑦_𝑛−𝑥_𝑛²−2𝜆_𝑛𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧²+2𝜆_𝑛𝛽2𝑦_𝑛−𝑥_𝑛²+1𝐴2𝛽^𝑛𝑥_𝑛−𝐴_𝑛𝑧²=𝛽𝜆_𝑛−𝑐₁𝑦_𝑛−𝑥_𝑛²+𝜆_𝑛1𝛽−2𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧² (17) for any 𝛽>0. Since a sequence {𝜆_𝑛} satisfies 0<𝜆_𝑛<2𝑐₁𝛼_𝑛 for all 𝑛∈ℕ, limsup_𝑛→∞𝜆_𝑛<∞, and inf_𝑛∈ℕ(𝑐₁/𝜆_𝑛−1/(2𝛼_𝑛))>0, we can choose a positive sequence {𝛽_𝑛} such that 𝛽_𝑛𝜆_𝑛−𝑐₁<0,𝜆_𝑛1𝛽_𝑛−2𝛼_𝑛<0∀𝑛∈ℕ,limsup_𝑛→∞𝛽_𝑛𝜆_𝑛−𝑐₁<0,limsup_𝑛→∞𝜆_𝑛1𝛽_𝑛−2𝛼_𝑛<0. (18)
So, we obtain 𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧≤0. (19)
Since 𝑧∈VI(𝐶,𝐴_𝑛), we have ⟨𝑦_𝑛−𝑧,𝐴_𝑛𝑧⟩≥0. Using 𝑥_𝑛∈𝐶 and the condition (ii), we have ⟨𝑥_𝑛−𝑧,𝐴_𝑛𝑧⟩≤⟨𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛⟩. Thus, we get 𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛; (20) that is, 𝐹⊂𝐶_𝑛. From this fact, we get that 𝐹⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛∈ℕ and {𝑥_𝑛} is well defined. Indeed, 𝑥₁=𝑥∈𝐶 is given and since 𝑄₁=𝐶, we have 𝐹⊂𝐶₁∩𝑄₁. Assume that 𝑥_𝑘 is well defined and 𝐹⊂𝐶_𝑘∩𝑄_𝑘 for some 𝑘∈ℕ. There exists a unique element 𝑥_𝑘+1=Π_{𝐶𝑘∩𝑄𝑘}𝑥 and we get ⟨𝑥_𝑘+1−𝑧,𝐽𝑥−𝐽𝑥_𝑘+1⟩≥0 for all 𝑧∈𝐶_𝑘∩𝑄_𝑘 by Lemma 5. Since 𝐹⊂𝐶_𝑘∩𝑄_𝑘, we have ⟨𝑥_𝑘+1−𝑧,𝐽𝑥−𝐽𝑥_𝑘+1⟩≥0 for every 𝑧∈𝐹; that is, 𝐹⊂𝑄_𝑘+1. Since 𝑥_𝑘+1∈𝐶_𝑘∩𝑄_𝑘⊂𝐶, we have 𝐹⊂𝐶_𝑘+1. Hence, we obtain 𝐹⊂𝐶_𝑘+1∩𝑄_𝑘+1. By mathematical induction, we get 𝐹⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛∈ℕ and {𝑥_𝑛} is well defined. Since 𝑥_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 and 𝐹⊂𝐶_𝑛∩𝑄_𝑛, we have 𝜙(𝑥_𝑛+1,𝑥)≤𝜙(Π_𝐹𝑥,𝑥) for every 𝑛∈ℕ, which implies that {𝑥_𝑛} is bounded. Further, since 𝑥_𝑛+1∈𝑄_𝑛, we have 𝜙𝑥_𝑛+1,𝑥_𝑛+𝜙𝑥_𝑛𝑥,𝑥=𝜙_𝑛+1,𝑥+2𝑥_𝑛−𝑥_𝑛+1,𝐽𝑥_𝑛𝑥−𝐽𝑥≤𝜙_𝑛+1,𝑥 (21) for all 𝑛∈ℕ. Thus, there exists lim_𝑛→∞𝜙(𝑥_𝑛,𝑥) and lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0. (22)
By Lemma 4, we get lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (23)
Using 𝑥_𝑛+1∈𝐶_𝑛, we have 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑥_𝑛+1,𝐴_𝑛𝑥_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑥_𝑛+1𝐴_𝑛𝑥_𝑛 (24) for all 𝑛∈ℕ. From the condition (iii), we have 𝑥_𝑛−𝑧𝐴_𝑛𝑥_𝑛≥𝛼_𝑛𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧²≥𝛼_𝑛𝐴_𝑛𝑥_𝑛²−2𝐴_𝑛𝑥_𝑛𝐴_𝑛𝑧, (25) which implies that 𝑥_𝑛−𝑧≥𝛼_𝑛𝐴_𝑛𝑥_𝑛−2𝛼_𝑛𝐴_𝑛𝑧 (26) for every 𝑛∈ℕ and 𝑧∈𝐹. Since 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<∞, using the condition (iv) and the boundedness of {𝑥_𝑛}, we get that {𝐴_𝑛𝑥_𝑛} is bounded. Since (22)–(24) hold and {𝜆_𝑛} is bounded, we have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑦_𝑛)=0, which implies that lim_𝑛→∞𝑥_𝑛+1−𝑦_𝑛=0 (27)
by Lemma 4. From (23) and (27), we also have lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. Using the facts that 𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧=2𝑧,𝐽𝑥_𝑛−𝐽𝑦_𝑛+𝑦_𝑛²−𝑥_𝑛²+2𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑧 (28) for all 𝑛∈ℕ, lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, the condition (iv) holds, {𝑥_𝑛}, {𝑦_𝑛}, and {𝜆_𝑛} are bounded, and the duality mapping 𝐽 is norm-to-weak^∗ uniformly continuous on bounded subset of 𝐸, we obtain lim_𝑛→∞𝜙𝑧,𝑦_𝑛+2𝜆_𝑛𝑦_𝑛−𝑧,𝐴_𝑛𝑧−𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑧=0. (29)
Since (17) holds for every 𝑛∈ℕ, it follows from (29) and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 that lim_𝑛→∞𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧=0. (30)
From the condition (v), there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑧∈𝐹. Since the norm of 𝐸 is weakly lower semicontinuous, we get 𝜙(𝑧,𝑥)=‖𝑧‖²−2⟨𝑧,𝐽𝑥⟩+‖𝑥‖²≤liminf_𝑖→∞𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥+‖𝑥‖²=liminf_𝑖→∞𝜙𝑥_𝑛𝑖,𝑥=lim_𝑛→∞𝜙𝑥_𝑛,𝑥≤𝜙Π_𝐹𝑥,𝑥, (31) which implies 𝑧=Π_𝐹𝑥 and lim_𝑛→∞𝜙𝑥_𝑛,𝑥=𝜙Π_𝐹𝑥,𝑥. (32)
Using 𝑥_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥, 𝐹⊂𝐶_𝑛∩𝑄_𝑛, and Lemma 5, we have 0≥𝑥_𝑛+1−Π_𝐹𝑥,𝐽𝑥_𝑛+1=1−𝐽𝑥2𝜙Π_𝐹𝑥,𝑥_𝑛+1+𝜙𝑥_𝑛+1,𝑥−𝜙Π_𝐹𝑥,𝑥 (33) which implies that 𝜙Π_𝐹𝑥,𝑥≥𝜙Π_𝐹𝑥,𝑥_𝑛+1+𝜙𝑥_𝑛+1,𝑥 (34) for all 𝑛∈ℕ. From (32), we get lim_𝑛→∞𝜙(Π_𝐹𝑥,𝑥_𝑛+1)=0 and by Lemma 4, we obtain 𝑥_𝑛→Π_𝐹𝑥, which is the desired result.

Next, we have the following result by the hybrid method using the metric projections.

Theorem 9. Assume that 𝐸, 𝐶, {𝐴_𝑛}, {𝜆_𝑛}, and 𝑐₁ are the same as in Theorem 8. Let 𝑥∈𝐶 and {𝑥_𝑛} a sequence in 𝐶 generated by 𝑥₁𝑦=𝑥,_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 (35) for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥.

Proof. 𝑄_𝑛 is closed convex for every 𝑛∈ℕ. As in the proof of Theorem 8, we have that 𝐶_𝑛 is closed and convex. We also get that, for 𝑛∈ℕ, 𝑥_𝑛∈𝐶 implies that 𝐹⊂𝐶_𝑛. By this fact, we obtain 𝐹⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛∈ℕ and {𝑥_𝑛} is well defined. Indeed, 𝑥₁=𝑥∈𝐶 is given and 𝐹⊂𝐶₁∩𝑄₁ since 𝑄₁=𝐶. Assume that 𝑥_𝑘 is well defined and 𝐹⊂𝐶_𝑘∩𝑄_𝑘 for some 𝑘∈ℕ. There exists a unique element 𝑥_𝑘+1=𝑃_{𝐶𝑘∩𝑄𝑘}𝑥 and we get ⟨𝑥_𝑘+1−𝑧,𝐽(𝑥−𝑥_𝑘+1)⟩≥0 for all 𝑧∈𝐶_𝑘∩𝑄_𝑘 by Lemma 6. Since 𝐹⊂𝐶_𝑘∩𝑄_𝑘, we have ⟨𝑥_𝑘+1−𝑧,𝐽(𝑥−𝑥_𝑘+1)⟩≥0 for every 𝑧∈𝐹; that is, 𝐹⊂𝑄_𝑘+1. Since 𝑥_𝑘+1∈𝐶_𝑘∩𝑄_𝑘⊂𝐶, we also have 𝐹⊂𝐶_𝑘+1. Thus, we obtain 𝐹⊂𝐶_𝑘+1∩𝑄_𝑘+1. By mathematical induction, we get 𝐹⊂𝐶_𝑛∩𝑄_𝑛 for every 𝑛∈ℕ and {𝑥_𝑛} is well defined.
Since 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 and 𝐹⊂𝐶_𝑛∩𝑄_𝑛, we have 𝑥_𝑛+1−𝑥≤𝑥−𝑃_𝐹𝑥 (36) for every 𝑛∈ℕ and, hence, {𝑥_𝑛} is bounded. Using 𝑥_𝑛+1∈𝑄_𝑛 and Theorem 2, we have 𝑥_𝑛+1−𝑥²−𝑥_𝑛−𝑥²≥2𝑥_𝑛+1−𝑥_𝑛,𝐽𝑥_𝑛−𝑥+𝑐₁𝑥_𝑛+1−𝑥_𝑛²≥𝑐₁𝑥_𝑛+1−𝑥_𝑛² (37) for each 𝑛∈ℕ, which implies that there exists lim_𝑛→∞‖𝑥_𝑛−𝑥‖ and lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (38)
Since 𝑦_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛), by Lemma 5, we have 𝑦_𝑛−𝑥_𝑛,𝐽𝑥_𝑛−𝐽𝑦_𝑛−𝜆_𝑛𝐴_𝑛𝑥_𝑛≥0, (39) which implies that −𝑐₁𝑥_𝑛−𝑦_𝑛²≥12−𝜙𝑦_𝑛,𝑥_𝑛−𝜙𝑥_𝑛,𝑦_𝑛=𝑦_𝑛−𝑥_𝑛,𝐽𝑥_𝑛−𝐽𝑦_𝑛≥𝜆_𝑛𝑦_𝑛−𝑥_𝑛,𝐴_𝑛𝑥_𝑛≥−𝜆_𝑛𝑦_𝑛−𝑥_𝑛𝐴_𝑛𝑥_𝑛 (40) from Lemma 4. As in the proof of Theorem 8, {𝐴_𝑛𝑥_𝑛} is bounded. Thus, we get that {‖𝑦_𝑛−𝑥_𝑛‖} is also bounded by the boundedness of {𝜆_𝑛} and so is {𝑦_𝑛}. Since 𝑥_𝑛+1∈𝐶_𝑛, we have 𝜙(𝑥_𝑛+1,𝑦_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)+2𝜆_𝑛⟨𝑥_𝑛−𝑥_𝑛+1,𝐴_𝑛𝑥_𝑛⟩; that is, 𝜙𝑥_𝑛,𝑦_𝑛≤2𝑥_𝑛−𝑥_𝑛+1,𝐽𝑥_𝑛−𝐽𝑦_𝑛+2𝜆_𝑛𝑥_𝑛−𝑥_𝑛+1,𝐴_𝑛𝑥_𝑛 (41) for all 𝑛∈ℕ. By the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, {𝜆_𝑛}, and {𝐴_𝑛𝑥_𝑛} with (38) and Lemma 4, we have lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (42)
As in the proof of Theorem 8, using (17) and (29), we have lim_𝑛→∞𝐴_𝑛𝑥_𝑛−𝐴_𝑛𝑧=0. (43)
From the condition (v), there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑧∈𝐹. Since the norm of 𝐸 is weakly lower semicontinuous and (36) holds, we get ‖𝑧−𝑥‖≤liminf_𝑖→∞𝑥_𝑛𝑖−𝑥=lim_𝑛→∞𝑥_𝑛−𝑥≤𝑃_𝐹𝑥−𝑥 (44) which implies that 𝑧=𝑃_𝐹𝑥 and lim_𝑛→∞𝑥_𝑛−𝑥=𝑃_𝐹𝑥−𝑥. (45)
Using the facts that 𝑥_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 and 𝐹⊂𝐶_𝑛∩𝑄_𝑛 and by Theorem 2, we obtain 𝑥−𝑃_𝐹𝑥²≥𝑥−𝑥_𝑛+1²+2𝑥_𝑛+1−𝑃_𝐹𝑥,𝐽𝑥−𝑥_𝑛+1+𝑐₁𝑥_𝑛+1−𝑃_𝐹𝑥²≥𝑥−𝑥_𝑛+1²+𝑐₁𝑥_𝑛+1−𝑃_𝐹𝑥² (46) for all 𝑛∈ℕ. By (45), we have lim_𝑛→∞𝑥_𝑛+1−𝑃_𝐹𝑥=0, (47) which is the desired result.

Remark 10. Even though we replace the definition of 𝐶_𝑛 in Theorems 8 and 9 with 𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛𝐴_𝑛𝑧=0∀𝑧∈𝐹,𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐴_𝑛𝑥_𝑛𝐴_𝑛𝑧≠0forsome𝑧∈𝐹, (48) the theorems are still valid.

4. The Variational Inequality Problem for Monotone Operators

Let 𝐼 be a countable set and 𝑖∶ℕ→𝐼 a mapping. Nakajo et al. [30] propose the condition (NST) as follows: 𝑖 satisfies the condition (NST) if there exists a subsequence {𝑛_𝑘} of ℕ such that, for any 𝑗∈𝐼, there is 𝑀_𝑗∈ℕ with 𝑗∈{𝑖(𝑛_𝑘),𝑖(𝑛_𝑘+1),…,𝑖(𝑛_𝑘+𝑀_𝑗−1)} for all sufficiently large 𝑘∈ℕ. Using the condition (NST), we get the following result for the variational inequality problem by Theorem 8.

Theorem 11. Let 𝐶 be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸, 𝐼 a countable set, and {𝐵_𝑖}_𝑖∈𝐼 a family of operators of 𝐶 into 𝐸^∗ such that (i)𝐹=⋂_𝑖∈𝐼𝑉𝐼(𝐶,𝐵_𝑖)≠0/;(ii)𝐵_𝑖 is an inverse strongly monotone operator for each 𝑖∈𝐼; that is, there exists {𝛽_𝑖∶𝑖∈𝐼}⊂]0,∞[ such that for every 𝑖∈𝐼 and 𝑥,𝑦∈𝐶, the inequality ⟨𝑥−𝑦,𝐵_𝑖𝑥−𝐵_𝑖𝑦⟩≥𝛽_𝑖‖𝐵_𝑖𝑥−𝐵_𝑖𝑦‖² holds;(iii)for all 𝑧∈𝐹, sup_𝑖∈𝐼‖𝐵_𝑖𝑧‖<∞.
Suppose that the index mapping 𝑖∶ℕ→𝐼 satisfies the condition (NST) and 0<liminf_𝑛→∞𝛽_𝑖(𝑛)≤limsup_𝑛→∞𝛽_𝑖(𝑛)<∞. Let {𝜆_𝑛} be a sequence in ]0,∞[ such that inf_𝑛∈ℕ𝜆_𝑛>0 and inf_𝑛∈ℕ(2𝑐₁𝛽_𝑖(𝑛)−𝜆_𝑛)>0, where 𝑐₁ is the constant in Theorem 2. Let 𝑥∈𝐶 and {𝑥_𝑛} a sequence in 𝐶 generated by 𝑥₁𝑦=𝑥,_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵_𝑖(𝑛)𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐵_𝑖(𝑛)𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 (49) for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to Π_𝐹𝑥.

Proof. We apply Theorem 8 with 𝐴_𝑛=𝐵_𝑖(𝑛) for all 𝑛∈ℕ. Then, the conditions (i)–(iv) are satisfied, and we will verify the condition (v). Let {𝑧_𝑛} be a bounded sequence in 𝐶 with lim_𝑛→∞𝑧_𝑛+1−𝑧_𝑛=lim_𝑛→∞𝑧_𝑛−Π_𝐶𝐽⁻¹𝐽𝑧_𝑛−𝑟_𝑛𝐴_𝑛𝑧_𝑛=0, (50){𝑟_𝑛}⊂]0,∞[ with liminf_𝑛→∞𝑟_𝑛>0, and 𝑧∈𝐹 such that lim_𝑛→∞‖𝐴_𝑛𝑧_𝑛−𝐴_𝑛𝑧‖=0. By the condition (NST), there exists a weakly convergent subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} such that, for any 𝑖∈𝐼, there is 𝑀_𝑖∈ℕ with 𝑖∈{𝑖(𝑛_𝑘),𝑖(𝑛_𝑘+1),…,𝑖(𝑛_𝑘+𝑀_𝑖−1)} for all sufficiently large 𝑘∈ℕ. Let 𝑧_𝑛𝑘⇀𝑢 and fix 𝑖∈𝐼. There exists 𝑗_𝑘∈{0,1,…,𝑀_𝑖−1} such that 𝑖(𝑛_𝑘+𝑗_𝑘)=𝑖 for every sufficiently large 𝑘∈ℕ. We consider a subsequence of {𝑛_𝑘+𝑗_𝑘} for all 𝑘∈{𝑘∈ℕ∶𝑛_𝑘+𝑗_𝑘<𝑛_𝑘+1+𝑗_𝑘+1} and denote it by {𝑛_𝑘+𝑗_𝑘} again. We have 𝑧_{𝑛𝑘+𝑗𝑘}−𝑧_𝑛𝑘≤^{𝑛𝑘+𝑀𝑖−1}_{𝑙=𝑛𝑘}𝑧_𝑙+1−𝑧_𝑙 (51) for all 𝑘∈ℕ, which implies that 𝑧_{𝑛𝑘+𝑗𝑘}⇀𝑢. Let 𝑦_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑧_𝑛−𝑟_𝑛𝐴_𝑛𝑧_𝑛). By Lemma 5, we have 𝑦_{𝑛𝑘+𝑗𝑘}−𝑦𝐽𝑧_{𝑛𝑘+𝑗𝑘}−𝐽𝑦_{𝑛𝑘+𝑗𝑘}≥𝑦_{𝑛𝑘+𝑗𝑘}−𝑦,𝐽𝑧_{𝑛𝑘+𝑗𝑘}−𝐽𝑦_{𝑛𝑘+𝑗𝑘}≥𝑟_{𝑛𝑘+𝑗𝑘}𝑦_{𝑛𝑘+𝑗𝑘}−𝑦,𝐴_{𝑛𝑘+𝑗𝑘}𝑧_{𝑛𝑘+𝑗𝑘} (52) for every sufficiently large 𝑘∈ℕ and 𝑦∈𝐶. Since 𝐴_{𝑛𝑘+𝑗𝑘}=𝐵_{𝑖(𝑛𝑘+𝑗𝑘)}=𝐵_𝑖 for each sufficiently large 𝑘∈ℕ, 𝑦_{𝑛𝑘+𝑗𝑘}⇀𝑢, ‖𝑧_{𝑛𝑘+𝑗𝑘}−𝑦_{𝑛𝑘+𝑗𝑘}‖→0, liminf_𝑛→∞𝑟_𝑛>0, lim_𝑘→∞‖𝐵_𝑖𝑧_{𝑛𝑘+𝑗𝑘}−𝐵_𝑖𝑧‖=0, and the duality mapping 𝐽 is uniformly continuous on bounded subset of 𝐸, we have 𝑦−𝑢,𝐵_𝑖𝑧≥0 (53) for all 𝑦∈𝐶. Since 𝐵_𝑖 is inverse strongly monotone, we have 𝑧_{𝑛𝑘+𝑗𝑘}−𝑢,𝐵_𝑖𝑧_{𝑛𝑘+𝑗𝑘}−𝐵_𝑖𝑢≥𝛽_𝑖𝐵_𝑖𝑧_{𝑛𝑘+𝑗𝑘}−𝐵_𝑖𝑢² (54) for every 𝑘∈ℕ, which implies that 0=𝑢−𝑢,𝐵_𝑖𝑧−𝐵_𝑖𝑢≥𝛽_𝑖𝐵_𝑖𝑧−𝐵_𝑖𝑢²; (55) that is, 𝐵_𝑖𝑧=𝐵_𝑖𝑢. From (53), 𝑦−𝑢,𝐵_𝑖𝑢≥0 (56) for all 𝑦∈𝐶. Therefore, 𝑢∈VI(𝐶,𝐵_𝑖) for every 𝑖∈𝐼; that is, 𝑢∈𝐹. Hence, the condition (v) is satisfied. Consequently, we obtain 𝑥_𝑛→Π_𝐹𝑥 by Theorem 8.

As in the proof of Theorem 11, we get the following result for the variational inequality problem by Theorem 9.

Theorem 12. Assume that 𝐸, 𝐶, 𝐼, {𝐵_𝑖}, 𝐹, 𝑐₁, 𝑖, {𝛽_𝑖}, and {𝜆_𝑛} are the same as Theorem 11. Let 𝑥∈𝐶 and {𝑥_𝑛} a sequence in 𝐶 generated by 𝑥₁𝑦=𝑥,_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵_𝑖(𝑛)𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+2𝜆_𝑛𝑥_𝑛−𝑧,𝐵_𝑖(𝑛)𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 (57) for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥.

Remark 13. In Theorems 11 and 12, under the assumption that ⋂_𝑖∈𝐼𝐵_𝑖⁻¹0≠0/, we have VI(𝐶,𝐵_𝑖)=𝐵_𝑖⁻¹0 for all 𝑖∈𝐼. Indeed, 𝐵_𝑖⁻¹0⊂VI(𝐶,𝐵_𝑖) is trivial. Let 𝑖∈𝐼, 𝑢∈VI(𝐶,𝐵_𝑖), and 𝑧∈𝐵_𝑖⁻¹0. From the condition (ii), we have ⟨𝑧−𝑢,𝐵_𝑖𝑧−𝐵_𝑖𝑢⟩≥𝛽_𝑖‖𝐵_𝑖𝑧−𝐵_𝑖𝑢‖² which implies that −⟨𝑧−𝑢,𝐵_𝑖𝑢⟩≥𝛽_𝑖‖𝐵_𝑖𝑢‖². On the other hand, ⟨𝑧−𝑢,𝐵_𝑖𝑢⟩≥0 from 𝑢∈VI(𝐶,𝐵_𝑖). So, we obtain 𝑢∈𝐵_𝑖⁻¹0; that is, VI(𝐶,𝐵_𝑖)⊂𝐵_𝑖⁻¹0. Therefore, 𝐵_𝑖⁻¹0=VI(𝐶,𝐵_𝑖) for all 𝑖∈𝐼. Now suppose that 𝐹=⋂_𝑖∈𝐼𝐵_𝑖⁻¹0≠0/ instead of the condition (i) and 𝐶_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛)}. By the argument mentioned above and Remark 10, Theorems 11 and 12 hold under the conditions (i) and (ii) and we get the result of [19].

Remark 14. We know that, for a continuously Fréchet differentiable and convex functional 𝑓 on a Banach space 𝐸, if ∇𝑓 is Lipschitz continuous with constant 1/𝛼, then ∇𝑓 is 𝛼-inverse strongly monotone operator; see [2, 19]. So, we can apply Theorems 11 and 12 and Remark 13 to such a functional; see [19].

5. The Proximal Point Algorithm

Let 𝐸 be a strictly convex, reflexive, and smooth Banach space, 𝑇⊂𝐸×𝐸^∗ a maximal monotone operator with 𝑇⁻¹0≠0/, 𝑟>0, and 𝐴𝑥=𝐽(𝑥−𝐽_𝑟𝑥) for all 𝑥∈𝐸, where 𝐽_𝑟 is the resolvent of 𝑇. Then, 𝐴 is well defined as a mapping of 𝐸 into 𝐸^∗ for all 𝑟>0. We also have VI(𝐸,𝐴)=𝑇⁻¹⟨⟩≥‖‖0,𝑥−𝑢,𝐴𝑥𝐴𝑥²∀𝑥∈𝐸,𝑢∈𝑇⁻¹0. (58)

In fact, VI(𝐸,𝐴)=𝑇⁻¹0 since 𝑢∈𝑇⁻¹0 is equivalent to 𝐽_𝑟𝑢=𝑢 and 𝐴𝑢=0. Let 𝑥∈𝐸 and 𝑢∈𝑇⁻¹0. Since (1/𝑟)𝐽(𝑥−𝐽_𝑟𝑥)∈𝑇𝐽_𝑟𝑥 and 0∈𝑇𝑢, we have ⟨𝐽_𝑟𝑥−𝑢,𝐽(𝑥−𝐽_𝑟𝑥)⟩≥0 which implies that ⟨𝑥−𝑢,𝐴𝑥⟩≥‖𝐴𝑥‖². By Theorem 8 and Remark 10, we get the following result using the index mapping which satisfies the condition (NST).

Theorem 15. Let 𝐼 be a countable set, 𝐸 a 2-uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and {𝑇_𝑖}_𝑖∈𝐼 a family of maximal monotone operators of 𝐸 into 𝐸^∗ such that 𝐹=⋂_𝑖∈𝐼𝑇_𝑖⁻¹0≠0/. Let {𝑟_𝑛} be a sequence in ]0,∞[ with liminf_𝑛→∞𝑟_𝑛>0 and {𝜆_𝑛}_𝑛∈ℕ a sequence in ]0,∞[ such that inf_𝑛∈ℕ𝜆_𝑛>0 and inf_𝑛∈ℕ(2𝑐₁−𝜆_𝑛)>0, where 𝑐₁ is the constant in Theorem 2. Let 𝑥∈𝐸 and {𝑥_𝑛}_𝑛∈ℕ a sequence in 𝐸 generated by 𝑥₁𝑦=𝑥,_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐽𝑥_𝑛−𝐽^{𝑇𝑖(𝑛)}_𝑟𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐸∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 (59) for each 𝑛∈ℕ, where the index mapping 𝑖∶ℕ→𝐼 satisfies the condition (NST) and 𝐽^{𝑇𝑖(𝑛)}_𝑟𝑛 is the resolvent of 𝑇_𝑖(𝑛). Then, {𝑥_𝑛} converges strongly to Π_𝐹𝑥.

Proof. Suppose that 𝐴_𝑛𝑥=𝐽(𝑥−𝐽^{𝑇𝑖(𝑛)}_𝑟𝑛𝑥) for every 𝑛∈ℕ and 𝑥∈𝐸 in Theorem 8. Then, we have that 𝐴_𝑛 is a mapping of 𝐸 into 𝐸^∗ with ⋂_𝑛∈ℕVI(𝐸,𝐴_𝑛)=𝐹≠0/, the condition (iii) is satisfied with 𝛼_𝑛=1 for all 𝑛∈ℕ, and the conditions (ii) and (iv) hold by 𝐴_𝑛𝑧=0 for all 𝑛∈ℕ and all 𝑧∈𝐹. Let {𝑧_𝑛} be a bounded sequence in 𝐸, 𝑧∈𝐹, and {𝑟_𝑛}⊂]0,∞[ with liminf_𝑛→∞𝑟_𝑛>0. Assume that lim_𝑛→∞𝑧_𝑛+1−𝑧_𝑛=lim_𝑛→∞𝑧_𝑛−𝐽⁻¹𝐽𝑧_𝑛−𝑟_𝑛𝐴_𝑛𝑧_𝑛=lim_𝑛→∞𝐴_𝑛𝑧_𝑛−𝐴_𝑛𝑧=0. (60)
By the condition (NST), there exists a subsequence {𝑧_𝑛𝑘} of {𝑧_𝑛} such that, for any 𝑖∈𝐼, there is 𝑀_𝑖∈ℕ with 𝑖∈{𝑖(𝑛_𝑘),𝑖(𝑛_𝑘+1),…,𝑖(𝑛_𝑘+𝑀_𝑖−1)} for all sufficiently large 𝑘∈ℕ. Let 𝑧_𝑛𝑘⇀𝑢 and 𝑖∈𝐼. As in the proof of Theorem 11, there exists 𝑗_𝑘∈{0,1,…,𝑀_𝑖−1} such that 𝑖(𝑛_𝑘+𝑗_𝑘)=𝑖 for every sufficiently large 𝑘∈ℕ and we get 𝑧_{𝑛𝑘+𝑗𝑘}⇀𝑢. Let (𝑣,𝑣^∗)∈𝑇_𝑖. Since 𝐽^𝑇𝑖_{𝑟𝑛𝑘𝑘+𝑗}𝑧_{𝑛𝑘+𝑗𝑘}1−𝑣,𝑟_{𝑛𝑘+𝑗𝑘}𝐽𝑧_{𝑛𝑘+𝑗𝑘}−𝐽^𝑇𝑖_{𝑟𝑛𝑘𝑘+𝑗}𝑧_{𝑛𝑘+𝑗𝑘}−𝑣^∗≥0,lim_𝑘→∞𝑧_{𝑛𝑘+𝑗𝑘}−𝐽^𝑇𝑖_{𝑟𝑛𝑘𝑘+𝑗}𝑧_{𝑛𝑘+𝑗𝑘}=0, (61) we obtain 𝑢−𝑣,−𝑣^∗≥0 (62) for each (𝑣,𝑣^∗)∈𝑇_𝑖. As 𝑇_𝑖 is a maximal monotone operator, 𝑢∈𝑇_𝑖⁻¹0 for every 𝑖∈𝐼. So, we get 𝑢∈𝐹. Therefore, the condition (v) holds. So, we get conclusion by Theorem 8 and Remark 10.

As in the proof of Theorem 15, we get the following result from Theorem 9 and Remark 10.

Theorem 16. Assume that 𝐸, 𝐼, {𝑇_𝑖}, 𝐹, {𝑟_𝑛}, {𝜆_𝑛}, 𝑐₁, 𝑖, and 𝐽^{𝑇𝑖(𝑛)}_𝑟𝑛 are the same as Theorem 15. Let 𝑥∈𝐸 and let {𝑥_𝑛} be a sequence in 𝐸 generated by𝑥₁𝑦=𝑥,_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐽𝑥_𝑛−𝐽^{𝑇𝑖(𝑛)}_𝑟𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐸∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 (63) for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥.

Let 𝑓∶𝐸→]−∞,∞] be a proper, lower semicontinuous, and convex function. Then, it is known that the subdifferential 𝜕𝑓 of 𝑓 defined by 𝜕𝑓(𝑥)=𝑥^∗∈𝐸^∗𝑦(𝑥)+∶𝑓≥𝑓𝑦−𝑥,𝑥^∗∀𝑦∈𝐸 (64) for all 𝑥∈𝐸 is a maximal monotone operator [31, 32]. Moreover, when 𝐸 is strictly convex, reflexive, and smooth, we know that, for the resolvent of 𝜕𝑓, 𝐽_𝑟^𝜕𝑓𝑥=argmin_𝑦∈𝐸𝑓𝑦+12𝑟𝑦−𝑥² (65) for every 𝑟>0 and 𝑥∈𝐸 and 𝜕𝑓⁻¹0=argmin_𝑦∈𝐸𝑓(𝑦); see [21] for more details. Now, we have the following results from Theorems 15 and 16.

Theorem 17. Let 𝐼 be a countable set, 𝐸 a 2-uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and {𝑓_𝑖}_𝑖∈𝐼 a family of proper, lower semicontinuous, and convex functions of 𝐸 into ]−∞,∞] such that 𝐹=⋂_𝑖∈𝐼argmin_𝑦∈𝐸𝑓_𝑖(𝑦)≠0/. Let {𝑟_𝑛} be a sequence in ]0,∞[ with liminf_𝑛→∞𝑟_𝑛>0 and {𝜆_𝑛}_𝑛∈ℕ a sequence in ]0,∞[ such that inf_𝑛∈ℕ𝜆_𝑛>0 and inf_𝑛∈ℕ(2𝑐₁−𝜆_𝑛)>0, where 𝑐₁ is the constant in Theorem 2. Let 𝑥∈𝐸 and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝐸 generated by 𝑥₁𝑢=𝑥,_𝑛=argmin_𝑦∈𝐸𝑓_𝑖(𝑛)𝑦+12𝑟_𝑛𝑦−𝑥_𝑛²,𝑦_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐽𝑥_𝑛−𝑢_𝑛,𝐶_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐸∶𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥 (66)
for each 𝑛∈ℕ, where the index mapping 𝑖∶ℕ→𝐼 satisfies the condition (NST). Then, {𝑥_𝑛} converges strongly to Π_𝐹𝑥.

Theorem 18. Assume that 𝐸, 𝐼, {𝑓_𝑖}, 𝐹, {𝑟_𝑛}, {𝜆_𝑛}, 𝑐₁, and 𝑖 are the same as Theorem 17. Let 𝑥∈𝐸 and let {𝑥_𝑛} be a sequence in 𝐸 generated by 𝑥₁𝑢=𝑥,_𝑛=argmin_𝑦∈𝐸𝑓_𝑖(𝑛)𝑦+12𝑟_𝑛𝑦−𝑥_𝑛²𝑦_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐽𝑥_𝑛−𝑢_𝑛,𝐶_𝑛=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐸∶𝑥_𝑛−𝑧,𝐽𝑥−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥 (67) for each 𝑛∈ℕ. Then, {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥.

At the end of this section, we make a remark about a result of the problem of image recovery due to [33, 34]. Let 𝑖_𝐶 be the indicator function of a nonempty closed convex subset 𝐶 of 𝐸. We know that 𝑖_𝐶∶𝐸→]−∞,∞[ is proper lower semicontinuous and convex and, for a nonempty closed convex subset 𝐶 of a strictly convex, reflexive, and smooth Banach space 𝐸, (𝜕𝑖_𝐶)⁻¹0=𝐶 and 𝐽^𝜕𝑖𝐶_𝑟𝑥=𝑃_𝐶(𝑥) for every 𝑟>0 and 𝑥∈𝐸; see [15]. So, when 𝑓_𝑗=𝑖_𝐶𝑗 for nonempty closed convex subset 𝐶_𝑗 of 𝐸 for every 𝑗∈𝐼 with ⋂_𝑗∈𝐼𝐶_𝑗≠0/ in Theorems 17 and 18, we get the strong convergence to a common point of {𝐶_𝑗}_𝑗∈𝐼.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author is supported by Grant-in-Aid for Scientific Research no. 22540175 from the Japan Society for the Promotion of Science.