Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

Abstract

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

1. Introduction

Let 𝐸 be a real Banach space and 𝐶 a nonempty subset of 𝐸. Let 𝐸^∗ be the dual space of 𝐸. We denote the value of 𝑥^∗∈𝐸^∗ at 𝑥∈𝐸 by ⟨𝑥^∗,𝑥⟩. Let 𝑇∶𝐶→𝐶 be a nonlinear mapping. We denote by 𝐹(𝑇) the fixed points set of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. Let 𝐴∶𝐸→2^𝐸∗ be a set-valued mapping. We denote 𝐷(𝐴) by the domain of 𝐴, that is, 𝐷(𝐴)={𝑥∈𝐸∶𝐴𝑥≠∅} and also denote 𝐺(𝐴) by the graph of 𝐴, that is, 𝐺(𝐴)={(𝑥,𝑥^∗)∈𝐸×𝐸^∗∶𝑥^∗∈𝐴𝑥}. A set-valued mapping 𝐴 is said to be monotone if ⟨𝑥^∗−𝑦^∗,𝑥−𝑦⟩≥0 whenever (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝐺(𝐴). It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operators on 𝐸. It is known that if 𝐴 is maximal monotone, then the set 𝐴⁻¹(0^∗)={𝑧∈𝐸∶0^∗∈𝐴𝑧} is closed and convex.

The problem of finding a zero point of maximal monotone operators plays an important role in optimizations. This is because it can be reformulated to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in various spaces (see, e.g., [1–16]). Initiated by Martinet [17], in a real Hilbert space 𝐻, Rockafellar [18] introduced the following iterative scheme: 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝐽_𝜆𝑛𝑥_𝑛,∀𝑛≥1,(1.1) where {𝜆_𝑛}⊂(0,∞), 𝐽_𝜆 is the resolvent of 𝐴 defined by 𝐽_𝜆∶=𝐽_𝜆𝐴=(𝐼+𝜆𝐴)⁻¹ for all 𝜆>0, and 𝐴 is a maximal monotone operator on 𝐻. Such an algorithm is called the proximal point algorithm. It was proved that the sequence {𝑥_𝑛} generated by (1.1) converges weakly to an element in 𝐴⁻¹(0) provided that liminf_𝑛→∞𝜆_𝑛>0. Recently, Kamimura and Takahashi [19] introduced the following iteration in a real Hilbert space: 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝜆𝑛𝑥_𝑛,∀𝑛≥1,(1.2) where {𝛼_𝑛}⊂[0,1] and {𝜆_𝑛}⊂(0,∞). The weak convergence theorems are also established in a real Hilbert space under suitable conditions imposed on {𝛼_𝑛} and {𝜆_𝑛}.

In 2004, Kamimura et al. [20] extended the above iteration process to a much more general setting. In fact, they proposed the following algorithm: 𝑥₁∈𝐸 and 𝑥_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝐽_𝜆𝑛𝑥_𝑛,∀𝑛≥1,(1.3) where {𝛼_𝑛}⊂[0,1], {𝜆_𝑛}⊂(0,∞), and 𝐽_𝜆∶=𝐽_𝜆𝐴=(𝐽+𝜆𝐴)⁻¹𝐽 for all 𝜆>0. They proved, in a uniformly smooth and uniformly convex Banach space, a weak convergence theorem.

Let 𝐹∶𝐶×𝐶→ℝ, where ℝ is the set of real numbers, be a bifunction. The equilibrium problem is to find 𝑥∈𝐶 such that 𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.4) The solutions set of (1.4) is denoted by EP(𝐹).

For solving the equilibrium problem, we assume that(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶,(A2)𝐹 is monotone, that is 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,(A3) for all 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),(A4) for all 𝑥∈𝐶,𝐹(𝑥,⋅) is convex and lower semi-continuous.

Recently, Takahashi and Zembayashi [21] introduced the following iterative scheme for a relatively nonexpansive mapping 𝑇∶𝐶→𝐶 in a uniformly smooth and uniformly convex Banach space: 𝑥₁∈𝐶 and 𝑦𝐶₁=𝐶,_𝑛𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,∈𝐶suchthat𝐹𝑢_𝑛,𝑦+1𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0∀𝑦∈𝐶,_𝑛+1=𝑥𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1,(1.5) where {𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). Such an algorithm is called the shrinking projection method which was introduced by Takahashi et al. [22]. They proved that the sequence {𝑥_𝑛} converges strongly to an element in 𝐹(𝑇)∩EP(𝐹) under appropriate conditions. The equilibrium problem has been intensively studied by many authors (see, e.g., [23–31]).

Motivated by the previous results, we introduce a hybrid-iterative scheme for finding a zero point of maximal monotone operators 𝐴_𝑖∶𝐸→2^𝐸∗ (𝑖=1,2,…,𝑁) which is also a common element in the solutions set of an equilibrium problem for 𝐹 and in the fixed points set of weak relatively nonexpansive mappings 𝑇_𝑖∶𝐶→𝐶  (𝑖=1,2,…). Using the projection technique, we also prove that the sequence generated by a constructed algorithm converges strongly to an element in [⋂^𝑁𝑖=1𝐴^𝑖−1(0^∗)]∩[⋂^∞𝑖=1𝐹(𝑇_𝑖)]∩EP(𝐹) in a uniformly smooth and uniformly convex Banach space. Finally, we apply our results to a system of convex minimization problems.

2. Preliminaries and Lemmas

In this section, we give some useful preliminaries and lemmas which will be used in the sequel.

Let 𝐸 be a real Banach space and let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦∈𝑈, 𝑥≠𝑦implies‖𝑥+𝑦‖<2.(2.1) A Banach space 𝐸 is said to be uniformly convex if, for each 𝜀∈(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦∈𝑈, ‖𝑥−𝑦‖≥𝜀implies‖𝑥+𝑦‖<2(1−𝛿).(2.2) It is known that a uniformly convex Banach space is reflexive and strictly convex. The function 𝛿∶[0,2]→[0,1] which is called the modulus of convexity of 𝐸 is defined as follows: 𝛿(𝜀)=inf1−‖‖‖𝑥+𝑦2‖‖‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀.(2.3) Then 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2]. A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.4) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for 𝑥,𝑦∈𝑈. The duality mapping 𝐽∶𝐸→2^𝐸∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²(2.5) for all 𝑥∈𝐸. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸 (see [32] for more details).

Let 𝐸 be a smooth Banach space. The function 𝜙∶𝐸×𝐸→ℝ is defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²(2.6) for all 𝑥,𝑦∈𝐸. From the definition of 𝜙, we see that 𝜙(𝑥(‖,𝑦𝑥)‖=−𝜙‖(𝑦𝑥‖,𝑧)²)≤+𝜙𝜙((𝑧𝑥,,𝑦𝑦))+≤2(‖⟨𝑥𝑥‖−+𝑧,‖𝐽𝑦𝑧‖−)²,𝐽𝑦⟩(2.7) for all 𝑥,𝑦,𝑧∈𝐸.

Let 𝐶 be a closed and convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [33] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 is said to be relatively nonexpansive [33, 34] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑝∈𝐹(𝑇) and 𝑥∈𝐶. A point 𝑝 in 𝐶 is said to be a strong asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛} which converges strongly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of strong asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 is said to be weak relatively nonexpansive [35] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑝∈𝐹(𝑇) and 𝑥∈𝐶. It is obvious by definition that the class of weak relatively nonexpansive mappings contains the class of relatively nonexpansive mappings. Indeed, for any mapping 𝑇∶𝐶→𝐶, we see that 𝐹(𝑇)⊂𝐹(𝑇)⊂𝐹(𝑇). Therefore, if 𝑇 is a relatively nonexpansive mapping, then 𝐹(𝑇)=𝐹(𝑇)=𝐹(𝑇).

Nontrivial examples of weak relatively nonexpansive mappings which are not relatively nonexpansive can be found in [36].

Let 𝐸 be a reflexive, strictly convex and smooth Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [37], is a mapping Π_𝐶∶𝐸→𝐶, that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the function 𝜙(𝑦,𝑥), that is, Π_𝐶(𝑥)=𝑥⁻, where 𝑥⁻ is the solution to the minimization problem 𝜙𝑥⁻,𝑥=min{𝜙(𝑦,𝑥)∶𝑦∈𝐶}.(2.8) In a Hilbert space, Π_𝐶 is coincident with the metric projection denoted by 𝑃_𝐶.

Lemma 2.1 (see [38]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛},{𝑦_𝑛} be two sequences in 𝐸. If lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.2 (see [37, 38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space 𝐸, let 𝑥∈𝐸 and let 𝑧∈𝐶. Then 𝑧=Π_𝐶(𝑥) if and only if ⟨𝑦−𝑧,𝐽𝑥−𝐽𝑧⟩≤0 for all 𝑦∈𝐶.

Lemma 2.3 (see [37, 38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Then 𝜙𝑥,Π_𝐶𝑦+𝜙Π_𝐶𝑦,𝑦≤𝜙(𝑥,𝑦)∀𝑥∈𝐶,𝑦∈𝐸.(2.9)

Lemma 2.4 (see [39]). Let 𝐸 be a smooth and strictly convex Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑇 be a mapping from 𝐶 into itself such that 𝐹(𝑇) is nonempty and 𝜙(𝑢,𝑇𝑥)≤𝜙(𝑢,𝑥) for all (𝑢,𝑥)∈𝐹(𝑇)×𝐶. Then 𝐹(𝑇) is closed and convex.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. It is known that 𝐴∶𝐸→2^𝐸∗ is maximal monotone if and only if 𝑅(𝐽+𝜆𝐴)=𝐸^∗ for all 𝜆>0, where 𝑅(𝐵) stands for the range of 𝐵.

Define the resolvent of 𝐴 by 𝐽_𝜆𝐴=(𝐽+𝜆𝐴)⁻¹𝐽 for all 𝜆>0. It is known that 𝐽_𝜆𝐴 is a single-valued mapping from 𝐸 to 𝐷(𝐴) and 𝐴⁻¹(0^∗)=𝐹(𝐽_𝜆𝐴) for all 𝜆>0. For each 𝜆>0, the Yosida approximation of 𝐴 is defined by 𝐴_𝜆(𝑥)=1𝜆𝐽(𝑥)−𝐽𝐽^𝜆𝐴(𝑥)(2.10) for all 𝑥∈𝐸. We know that 𝐴_𝜆(𝑥)∈𝐴(𝐽_𝜆𝐴(𝑥)) for all 𝜆>0 and 𝑥∈𝐸.

Lemma 2.5 (see [5]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐴⊂𝐸×𝐸^∗ be a maximal monotone operator with 𝐴⁻¹(0^∗)≠∅, and let 𝐽_𝜆𝐴=(𝐽+𝜆𝐴)⁻¹𝐽 for each 𝜆>0. Then 𝜙𝑝,𝐽_𝜆𝐴(𝑥)+𝜙𝐽_𝜆𝐴(𝑥),𝑥≤𝜙(𝑝,𝑥)(2.11) for all 𝜆>0, 𝑝∈𝐴⁻¹(0^∗), and 𝑥∈𝐸.

Lemma 2.6 (see[40]). Let 𝐶 be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.12)

Lemma 2.7 (see [41]). Let 𝐶 be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For all 𝑟>0 and 𝑥∈𝐸, define the mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑧∈𝐶∶𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.13) Then, the following holds:(1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is a firmly nonexpansive-type mapping [42], that is, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(2.14)(3)𝐹(𝑇_𝑟)=EP(𝐹);(4)EP(𝐹) is closed and convex.

Lemma 2.8 (see [41]). Let 𝐶 be a closed and convex subset of a smooth, strictly, and reflexive Banach space 𝐸, let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), let 𝑟>0. Then 𝜙𝑝,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙(𝑝,𝑥),(2.15) for all 𝑥∈𝐸 and 𝑝∈𝐹(𝑇_𝑟).

3. Strong Convergence Theorems

In this section, we are now ready to prove our main theorem.

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty, closed and convex subset of 𝐸. Let 𝐴_𝑖∶𝐸→2^𝐸∗ (𝑖=1,2,…,𝑁) be maximal monotone operators, let 𝐹∶𝐶×𝐶→ℝ be a bifunction, and let 𝑇_𝑖∶𝐶→𝐶  (𝑖=1,2,…) be weak relatively nonexpansive mappings such that ℱ∶=[⋂^𝑁𝑖=1𝐴^𝑖−1(0^∗)]∩[⋂^∞𝑖=1𝐹(𝑇_𝑖)]∩𝐸𝑃(𝐹)≠∅. Let {𝑒_𝑛}^∞𝑛=1⊂𝐸 be the sequence such that lim_𝑛→∞𝑒_𝑛=0. Define the sequence {𝑥_𝑛}^∞𝑛=1 in 𝐶 as follows: 𝑦𝑥₁∈𝐶₁=𝐶,_𝑛=𝐽_𝜆^𝑁𝑛_𝐴𝑁∘𝐽𝑢_𝜆^𝑛𝑁−1_{𝐴𝑁−1𝑛}∘⋯∘𝐽_𝜆^1𝑛_𝐴1𝑥_𝑛+𝑒_𝑛,𝐶_𝑛+1=𝑇_𝑟𝑛𝑦_𝑛,=𝑧∈𝑥𝐶_𝑛𝑛+1∶sup_𝑖≥1𝜙𝑧,𝑇_𝑖𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝑒_𝑛,=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1.(3.1) If liminf_𝑛→∞𝜆^𝑖𝑛>0 for each 𝑖=1,2,…,𝑁 and liminf_𝑛→∞𝑟_𝑛>0, then the sequence {𝑥_𝑛} converges strongly to 𝑞=Π_ℱ(𝑥₁).

Proof. We split the proof into several steps as follows.
Step  1. ℱ⊂𝐶_𝑛 for all 𝑛≥1.
From Lemma 2.4, we know that ⋂^∞𝑖=1𝐹(𝑇_𝑖) is closed and convex. From Lemma 2.7(4), we also know that EP(𝐹) is closed and convex. On the other hand, since 𝐴_𝑖 (𝑖=1,2,…,𝑁) are maximal monotone, 𝐴^𝑖−1(0^∗) are closed and convex for each 𝑖=1,2,…,𝑁; consequently, ⋂^𝑁𝑖=1𝐴^𝑖−1(0^∗) is closed and convex. Hence ℱ is a nonempty, closed, and convex subset of 𝐶.
We next show that 𝐶_𝑛 is closed and convex for all 𝑛≥1. Obviously, 𝐶₁=𝐶 is closed and convex. Now suppose that 𝐶_𝑘 is closed and convex for some 𝑘∈ℕ. Then, for each 𝑧∈𝐶_𝑘 and 𝑖≥1, we see that 𝜙(𝑧,𝑇_𝑖𝑢_𝑘)≤𝜙(𝑧,𝑥_𝑘) is equivalent to 2⟨𝑧,𝐽𝑥_𝑘⟩−2⟨𝑧,𝐽𝑇_𝑖𝑢_𝑘⟩≤‖‖𝑥_𝑘‖‖²−‖‖𝑇_𝑖𝑢_𝑘‖‖².(3.2) By the construction of the set 𝐶_𝑘+1, we see that 𝐶_𝑘+1==_∞𝑧∈𝐶_𝑘∶sup_𝑖≥1𝜙𝑧,𝑇_𝑖𝑢_𝑘≤𝜙𝑧,𝑥_𝑘_𝑖=1𝑧∈𝐶_𝑘∶𝜙𝑧,𝑇_𝑖𝑢_𝑘≤𝜙𝑧,𝑥_𝑘.(3.3) Hence, 𝐶_𝑘+1 is closed and convex. This shows, by induction, that 𝐶_𝑛 is closed and convex for all 𝑛≥1. It is obvious that ℱ⊂𝐶₁=𝐶. Now, suppose that ℱ⊂𝐶_𝑘 for some 𝑘∈ℕ. For any 𝑝∈ℱ, by Lemmas 2.5 and 2.8, we have 𝜙𝑝,𝑇_𝑖𝑢_𝑘≤𝜙𝑝,𝑢_𝑘≤=𝜙𝜙𝑝𝑝,,𝑇𝑦_𝑟𝑘𝑘𝑦_𝑘=𝜙𝑝,𝐽_{𝜆𝑁𝑘𝐴𝑁}≤𝜙𝑝,𝐽∘_{𝜆𝑘𝑁−1𝐴𝑁−1}𝐽_𝜆^𝑘𝑁−1_𝐴𝑁−1∘⋯∘𝐽_𝜆^1𝑘_𝐴1𝑥_𝑘+𝑒_𝑘≤𝜙⋮𝑝,𝐽_{𝜆2𝑘𝐴2}∘𝐽_𝜆^𝑘𝑁−2_𝐴𝑁−2∘⋯∘𝐽_𝜆^1𝑘_𝐴1𝑥_𝑘+𝑒_𝑘≤𝜙𝑝,𝐽_{𝜆1𝑘𝐴1}∘𝐽_𝜆^1𝑘_𝐴1𝑥_𝑘+𝑒_𝑘≤𝜙𝑝,𝑥_𝑘𝑥_𝑘+𝑒_𝑘+𝑒_𝑘.(3.4) This shows that ℱ⊂𝐶_𝑘+1. By induction, we can conclude that ℱ⊂𝐶_𝑛 for all 𝑛≥1.
Step  2. lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₁) exists.
From 𝑥_𝑛=Π_𝐶𝑛(𝑥₁) and 𝑥_𝑛+1=Π_𝐶𝑛+1(𝑥₁)∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₁≤𝜙𝑥_𝑛+1,𝑥₁,∀𝑛≥1.(3.5) From Lemma 2.3, for any 𝑝∈ℱ⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝐶𝑛𝑥₁,𝑥₁≤𝜙𝑝,𝑥₁−𝜙𝑝,𝑥_𝑛≤𝜙𝑝,𝑥₁.(3.6) Combining (3.5) and (3.6), we conclude that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₁) exists.
Step  3. lim_𝑛→∞‖𝐽(𝑇_𝑖𝑦_𝑛)−𝐽(𝑥_𝑛+𝑒_𝑛)‖=0.
Since 𝑥_𝑚=Π_𝐶𝑚(𝑥₁)∈𝐶_𝑚⊂𝐶_𝑛 for 𝑚>𝑛≥1, by Lemma 2.3, it follows that 𝜙𝑥_𝑚,𝑥_𝑛=𝜙𝑥_𝑚,Π_𝐶𝑛𝑥₁≤=𝜙𝜙𝑥𝑥_𝑚𝑚,𝑥₁−𝜙Π_𝐶𝑛𝑥₁,𝑥₁,𝑥₁−𝜙𝑥_𝑛,𝑥₁.(3.7) Letting 𝑚,𝑛→∞, we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0. By Lemma 2.1, it follows that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. Therefore, {𝑥_𝑛} is a Cauchy sequence. By the completeness of the space 𝐸 and the closedness of 𝐶, we can assume that 𝑥_𝑛→𝑞∈𝐶 as 𝑛→∞. In particular, we obtain that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.8) Since 𝑒_𝑛→0, we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛‖‖=0.(3.9) Since 𝑥_𝑛+1=Π_𝐶𝑛+1(𝑥₁)∈𝐶_𝑛+1, for each 𝑖≥1, 𝜙𝑥_𝑛+1,𝑇_𝑖𝑢_𝑛≤=𝜙𝑥𝑥_𝑛+1𝑛+1,𝑥_𝑛+𝑒_𝑛,𝐽𝑥_𝑛+1−𝐽𝑥_𝑛+𝑒_𝑛+𝑥_𝑛+1−𝑥_𝑛+𝑒_𝑛,𝐽𝑥_𝑛+1.(3.10) Since 𝐸 is uniformly smooth, 𝐽 is uniformly norm-to-norm continuous on bounded sets. It follows from (3.9) and by the boundedness of {𝑥_𝑛} that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑇_𝑖𝑢_𝑛=0(3.11) for all 𝑖=1,2,…. So from Lemma 2.1, we have lilmim_{𝑛→∞𝑛→∞}‖‖𝑥_𝑛+1−𝑇_𝑖𝑢_𝑛‖‖=0,‖‖𝑇_𝑖𝑢_𝑛−𝑥_𝑛‖‖=0,(3.12) and, since 𝑒_𝑛→0, therefore lim_𝑛→∞‖‖𝑇_𝑖𝑢_𝑛−𝑥_𝑛+𝑒_𝑛‖‖=0,(3.13) for all 𝑖=1,2,…. Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, lim_𝑛→∞‖‖𝐽𝑇_𝑖𝑢_𝑛−𝐽𝑥_𝑛+𝑒_𝑛‖‖=0(3.14) for all 𝑖=1,2,….
Step  4. lim_𝑛→∞‖𝑇_𝑖𝑢_𝑛−𝑢_𝑛‖=0 for all 𝑖=1,2,….
Denote that Θ^𝑖𝑛=𝐽_𝜆^𝑖𝑛_𝐴𝑖∘𝐽_𝜆^𝑛𝑖−1_𝐴𝑖−1∘…∘𝐽_𝜆^1𝑛_𝐴1 for each 𝑖∈{1,2,…,𝑁} and Θ^0𝑛=𝐼 for each 𝑛≥1. We note that 𝑦_𝑛=Θ^𝑁𝑛(𝑥_𝑛+𝑒_𝑛) for each 𝑛≥1.
To this end, we will show that lim_𝑛→∞‖‖𝐽Θ^𝑖𝑛𝑥_𝑛+𝑒_𝑛−𝐽Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛‖‖=0(3.15) for all 𝑖=1,2,…,𝑁.
For any 𝑝∈ℱ, by (3.4), we see that 𝜙𝑝,Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛≤≤𝜙𝜙𝑝𝑝,,ΘΘ^𝑛𝑁−2_𝑛𝑁−3𝑥_𝑛+𝑒_𝑛≤𝜙⋮𝑝,𝑥_𝑛𝑥_𝑛+𝑒_𝑛+𝑒_𝑛.(3.16) Since 𝑝∈ℱ, by Lemma 2.5 and (3.16), it follows that 𝜙𝑦_𝑛,Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛≤≤𝜙𝜙𝑝𝑝,,Θ𝑥^𝑛𝑁−1_𝑛𝑥_𝑛+𝑒_𝑛−𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛+𝑒_𝑛−𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛+𝑒_𝑛−𝜙𝑝,𝑢_𝑛=‖‖𝑥_𝑛+𝑒_𝑛−𝜙𝑝,𝑇_𝑖𝑢_𝑛+𝑒_𝑛‖‖²−‖‖𝑇_𝑖𝑢_𝑛‖‖²−2𝑝,𝐽𝑥_𝑛+𝑒_𝑛−𝐽𝑇_𝑖𝑢_𝑛.(3.17) From (3.13) and (3.14), we get that lim_𝑛→∞𝜙(𝑦_𝑛,Θ^𝑛𝑁−1(𝑥_𝑛+𝑒_𝑛))=0. So we obtain that lim_𝑛→∞‖‖𝑦_𝑛−Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛‖‖=0.(3.18) Again, since 𝑝∈ℱ, 𝜙Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛,Θ^𝑛𝑁−2𝑥_𝑛+𝑒_𝑛≤≤𝜙𝜙p𝑝,,Θ𝑥^𝑛𝑁−2_𝑛𝑥_𝑛+𝑒_𝑛−𝜙𝑝,Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛≤𝜙𝑝,𝑥_𝑛+𝑒_𝑛−𝜙𝑝,Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛+𝑒_𝑛−𝜙𝑝,𝑇_𝑖𝑢_𝑛.(3.19) From (3.13) and (3.14), we get that lim_𝑛→∞𝜙Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛,Θ^𝑛𝑁−2𝑥_𝑛+𝑒_𝑛=0.(3.20) It also follows that lim_𝑛→∞‖‖Θ^𝑛𝑁−1𝑥_𝑛+𝑒_𝑛−Θ^𝑛𝑁−2𝑥_𝑛+𝑒_𝑛‖‖=0.(3.21) Continuing in this process, we can show that lim_𝑛→∞‖‖Θ^𝑛𝑁−2𝑥_𝑛+𝑒_𝑛−Θ^𝑛𝑁−3𝑥_𝑛+𝑒_𝑛‖‖=⋯=lim_𝑛→∞‖‖Θ^1𝑛𝑥_𝑛+𝑒_𝑛−𝑥_𝑛+𝑒_𝑛‖‖=0.(3.22) So, we now conclude that lim_𝑛→∞‖‖Θ^𝑖𝑛𝑥_𝑛+𝑒_𝑛−Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛‖‖=0(3.23) for each 𝑖=1,2,…,𝑁. By the uniform norm-to-norm continuity of 𝐽, we also have lim_𝑛→∞‖‖𝐽Θ^𝑖𝑛𝑥_𝑛+𝑒_𝑛−𝐽Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛‖‖=0(3.24) for each 𝑖=1,2,…,𝑁. Using (3.23), it is easily seen that lim_𝑛→∞‖‖𝑦_𝑛−𝑥_𝑛+𝑒_𝑛‖‖=0.(3.25) From 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛, by Lemma 2.8, it follows that 𝜙𝑢_𝑛,𝑦_𝑛≤=𝜙𝜙𝑝𝑇,_𝑟𝑛𝑦𝑦_𝑛𝑛,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑇_𝑟𝑛𝑦_𝑛≤𝜙𝑝,𝑥_𝑛+𝑒_𝑛−𝜙𝑝,𝑢_𝑛+𝑒_𝑛−𝜙𝑝,𝑇_𝑖𝑢_𝑛.(3.26) This implies that lim_𝑛→∞𝜙(𝑢_𝑛,𝑦_𝑛)=0 and hence lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.27) Combining (3.13), (3.25), and (3.27), we obtain that lim_𝑛→∞‖‖𝑇_𝑖𝑢_𝑛−𝑢_𝑛‖‖=0(3.28) for all 𝑖≥1.
Step  5. 𝑞∈⋂^∞𝑖=1𝐹(𝑇_𝑖).
Since 𝑥_𝑛→𝑞 and 𝑒_𝑛→0, 𝑥_𝑛+𝑒_𝑛→𝑞. So from (3.25) and (3.27), we have 𝑢_𝑛→𝑞. Note that 𝑇_𝑖 (𝑖=1,2,…) are weak relatively nonexpansive. Using (3.28), we can conclude that 𝑞∈𝐹(𝑇_𝑖)=𝐹(𝑇_𝑖) for all 𝑖≥1. Hence 𝑞∈⋂^∞𝑖=1𝐹(𝑇_𝑖).
Step  6. 𝑞∈⋂^𝑁𝑖=1𝐴^𝑖−1(0^∗).
Noting that Θ^𝑖𝑛(𝑥_𝑛+𝑒_𝑛)=𝐽_𝜆^𝑖𝑛_𝐴𝑖Θ^𝑛𝑖−1(𝑥_𝑛+𝑒_𝑛) for each 𝑖=1,2,…,𝑁, we obtain that ‖‖𝐴_𝜆𝑖𝑛Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛‖‖=1𝜆^𝑖𝑛‖‖𝐽Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛−𝐽Θ^𝑖𝑛𝑥_𝑛+𝑒_𝑛‖‖.(3.29) From (3.24) and liminf_𝑛→∞𝜆^𝑖𝑛>0, we have lim_𝑛→∞‖‖𝐴_𝜆𝑖𝑛Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛‖‖=0.(3.30) We note that (Θ^𝑖𝑛(𝑥_𝑛+𝑒_𝑛),𝐴_𝜆^𝑖𝑛Θ^𝑛𝑖−1(𝑥_𝑛+𝑒_𝑛))∈𝐺(𝐴_𝑖) for each 𝑖=1,2,…,𝑁. If (𝑤,𝑤^∗)∈𝐺(𝐴_𝑖) for each 𝑖=1,2,…,𝑁, then it follows from the monotonicity of 𝐴_𝑖 that 𝑤^∗−𝐴_𝜆^𝑖𝑛Θ^𝑛𝑖−1𝑥_𝑛+𝑒_𝑛,𝑤−Θ^𝑖𝑛𝑥_𝑛+𝑒_𝑛≥0.(3.31) We see that Θ^𝑖𝑛(𝑥_𝑛+𝑒_𝑛)→𝑞 for each 𝑖=1,2,…,𝑁. Thus, from (3.30) and (3.31), we have ⟨𝑤^∗,𝑤−𝑞⟩≥0.(3.32) By the maximality of 𝐴_𝑖, it follows that 𝑞∈𝐴^𝑖−1(0^∗) for each 𝑖=1,2,…,𝑁. Therefore, 𝑞∈⋂^𝑁𝑖=1𝐴^𝑖−1(0^∗).
Step  7. 𝑞∈EP(𝐹).
From 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛, we have 𝐹𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.33) By (A2), we have ‖‖𝑦−𝑢_𝑛‖‖‖‖𝐽𝑢^𝑛−𝐽𝑦_𝑛‖‖𝑟_𝑛≥1𝑟_𝑛≥−𝐹⟨𝑦−𝑢_𝑛𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩,𝑦≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶.(3.34) Note that ‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖/𝑟_𝑛→0 since liminf_𝑛→∞𝑟_𝑛>0. From (A4) and 𝑢_𝑛→𝑞, we get 𝐹(𝑦,𝑞)≤0 for all 𝑦∈𝐶. For 0<𝑡<1 and 𝑦∈𝐶, define that 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑞. Then 𝑦_𝑡∈𝐶, which implies that 𝐹(𝑦_𝑡,𝑞)≤0. From (A1), we obtain that 0=𝐹(𝑦_𝑡,𝑦_𝑡)≤𝑡𝐹(𝑦_𝑡,𝑦)+(1−𝑡)𝐹(𝑦_𝑡,𝑞)≤𝑡𝐹(𝑦_𝑡,𝑦). Thus, 𝐹(𝑦_𝑡,𝑦)≥0. From (A3), we have 𝐹(𝑞,𝑦)≥0 for all 𝑦∈𝐶. Hence, 𝑞∈EP(𝐹). From Steps 5, 6, and 7, we now can conclude that 𝑞∈ℱ.
Step  8. 𝑞=Π_ℱ(𝑥₁).
From 𝑥_𝑛=Π_𝐶𝑛(𝑥₁), we have 𝐽𝑥₁−𝐽𝑥_𝑛,𝑥_𝑛−𝑧≥0,∀𝑧∈𝐶_𝑛.(3.35) Since ℱ⊂𝐶_𝑛, we also have 𝐽𝑥₁−𝐽𝑥_𝑛,𝑥_𝑛−𝑧≥0,∀𝑧∈ℱ.(3.36) Letting 𝑛→∞ in (3.36), we obtain that 𝐽𝑥₁−𝐽(𝑞),𝑞−𝑧≥0,∀𝑧∈ℱ.(3.37) This shows that 𝑞=Π_ℱ(𝑥₁) by Lemma 2.2. We thus complete the proof.

As a direct consequence of Theorem 3.1, we can also apply to a system of convex minimization problems.

Theorem 3.2. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑓_𝑖∶𝐸→(−∞,∞] (𝑖=1,2,…,𝑁) be proper lower semicontinuous convex functions, let 𝐹∶𝐶×𝐶→ℝ be a bifunction, and let 𝑇_𝑖∶𝐶→𝐶 (𝑖=1,2,…) be weak relatively nonexpansive mappings such that ℱ∶=[⋂^𝑁𝑖=1(𝜕𝑓^𝑖−1)(0^∗)]∩[⋂^∞𝑖=1𝐹(𝑇_𝑖)]∩𝐸𝑃(𝐹)≠∅. Let {𝑒_𝑛}^∞𝑛=1⊂𝐸 be the sequence such that lim_𝑛→∞𝑒_𝑛=0. Define the sequence {𝑥_𝑛}^∞𝑛=1 in 𝐶 as follows: 𝑧𝑥₁∈𝐶₁=𝐶,_1𝑛=argmin_𝑦∈𝐸𝑓¹(𝑦)+12𝜆^1𝑛‖𝑦‖²+1𝜆^1𝑛𝑧⋮𝑦,𝐽𝑥_𝑛+𝑒_𝑛_𝑛𝑁−1,=argmin_𝑦∈𝐸𝑓^𝑁−1(𝑦)+12𝜆^𝑛𝑁−1‖𝑦‖²+1𝜆^𝑛𝑁−1𝑦𝑦,𝐽𝑧^𝑛𝑁−2_𝑛,=argmin_𝑦∈𝐸𝑓^𝑁(𝑦)+12𝜆^𝑁𝑛‖𝑦‖²+1𝜆^𝑁𝑛𝑢𝑦,𝐽𝑧^𝑛𝑁−1_𝑛,𝐶_𝑛+1=𝑇_𝑟𝑛𝑦_𝑛,=𝑧∈𝑥𝐶_𝑛𝑛+1∶sup_𝑖≥1𝜙𝑧,𝑇_𝑖𝑢_𝑛≤𝜙𝑧,𝑥_𝑛+𝑒_𝑛,=Π_𝐶𝑛+1𝑥₁,∀𝑛≥1.(3.38) If liminf_𝑛→∞𝜆^𝑖𝑛>0 for each 𝑖=1,2,…,𝑁 and liminf_𝑛→∞𝑟_𝑛>0, then the sequence {𝑥_𝑛} converges strongly to 𝑞=Π_ℱ(𝑥₁).

Proof. By Rockafellar's theorem [43, 44], 𝜕𝑓_𝑖 are maximal monotone operators for each 𝑖=1,2,…,𝑁. Let 𝜆^𝑖>0 for each 𝑖=1,2,…,𝑁. Then, 𝑧^𝑖=𝐽_𝜆^𝑖_𝜕𝑓𝑖(𝑥) if and only if 0∈𝜕𝑓_𝑖𝑧^𝑖+1𝜆^𝑖=𝜕𝑓_𝑖𝐽𝑧^𝑖−𝐽(𝑥)+1𝜆^𝑖‖⋅2‖²−𝐽(𝑥)𝑧^𝑖,(3.39) which is equivalent to 𝑧^𝑖=argmin_𝑦∈𝐸𝑓^𝑖(𝑦)+1𝜆^𝑖‖𝑦2‖²−⟨𝑦,𝐽(𝑥)⟩.(3.40) Using Theorem 3.1, we thus complete the proof.

If 𝐸=𝐻 is a real Hilbert space, we then obtain the following results.

Corollary 3.3. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝐴_𝑖∶𝐻→2^𝐻 (𝑖=1,2,…,𝑁) be maximal monotone operators, let 𝐹∶𝐶×𝐶→ℝ be a bifunction, and let 𝑇_𝑖∶𝐶→𝐶 (𝑖=1,2,…) be weak relatively nonexpansive mappings such that ℱ∶=[⋂^𝑁𝑖=1𝐴^𝑖−1(0)]∩[⋂^∞𝑖=1𝐹(𝑇_𝑖)]∩𝐸𝑃(𝐹)≠∅. Let {𝑒_𝑛}^∞𝑛=1⊂𝐻 be the sequence such that lim_𝑛→∞𝑒_𝑛=0. Define the sequence {𝑥_𝑛}^∞𝑛=1 in 𝐶 as follows: 𝑦𝑥₁∈𝐶₁=𝐶,_𝑛=𝐽_𝜆^𝑁𝑛_𝐴𝑁∘J𝑢_𝜆^𝑛𝑁−1_{𝐴𝑁−1𝑛}∘⋯∘𝐽_𝜆^1𝑛_𝐴1𝑥_𝑛+𝑒_𝑛,𝐶_𝑛+1=𝑇_𝑟𝑛𝑦_𝑛,=𝑧∈𝐶𝑥_𝑛∶_𝑛+1sup_𝑖≥1‖‖𝑧−𝑇_𝑖𝑢_𝑛‖‖≤‖‖𝑧−𝑥_𝑛+𝑒_𝑛‖‖,=𝑃_𝐶𝑛+1𝑥₁,∀𝑛≥1.(3.41) If liminf_𝑛→∞𝜆^𝑖𝑛>0 for each 𝑖=1,2,…,𝑁 and liminf_𝑛→∞𝑟_𝑛>0, then the sequence {𝑥_𝑛} converges strongly to 𝑞=𝑃_ℱ(𝑥₁).

Corollary 3.4. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑓_𝑖∶𝐻→(−∞,∞] (𝑖=1,2,…,𝑁) be proper lower semi-continuous convex functions, let 𝐹∶𝐶×𝐶→ℝ be a bifunction, and let 𝑇_𝑖∶𝐶→𝐶 (𝑖=1,2,…) be weak relatively nonexpansive mappings such that ℱ∶=[⋂^𝑁𝑖=1(𝜕𝑓^𝑖−1)(0)]∩[⋂^∞𝑖=1𝐹(𝑇_𝑖)]∩𝐸𝑃(𝐹)≠∅. Let {𝑒_𝑛}^∞𝑛=1⊂𝐻 be the sequence such that lim_𝑛→∞𝑒_𝑛=0. Define the sequence {𝑥_𝑛}^∞𝑛=1 in 𝐶 as follows: 𝑧𝑥₁∈𝐶₁=𝐶,_1𝑛=argmin_𝑦∈𝐻𝑓¹(𝑦)+12𝜆^1𝑛‖𝑦‖²+1𝜆^1𝑛𝑧_𝑛𝑁−1⋮⟨𝑦,𝑥_𝑛+𝑒_𝑛⟩,=argmin_𝑦∈𝐻𝑓^𝑁−1(𝑦)+12𝜆^𝑛𝑁−1‖𝑦‖²+1𝜆^𝑛𝑁−1𝑦_𝑛𝑦,𝑧^𝑛𝑁−2,=argmin_𝑦∈𝐻𝑓^𝑁(𝑦)+12𝜆^𝑁𝑛‖𝑦‖²+1𝜆^𝑁𝑛𝑢_𝑛𝑦,𝑧^𝑛𝑁−1,𝐶_𝑛+1=𝑇_𝑟𝑛𝑦_𝑛,=𝑧∈𝐶𝑥_𝑛∶_𝑛+1sup_𝑖≥1‖‖𝑧−𝑇_𝑖𝑢_𝑛‖‖≤‖‖𝑧−𝑥_𝑛+𝑒_𝑛‖‖,=𝑃_𝐶𝑛+1𝑥₁,∀𝑛≥1.(3.42) If liminf_𝑛→∞𝜆^𝑖𝑛>0 for each 𝑖=1,2,…,𝑁 and liminf_𝑛→∞𝑟_𝑛>0, then the sequence {𝑥_𝑛} converges strongly to 𝑞=𝑃_ℱ(𝑥₁).

Remark 3.5. Using the shrinking projection method, we can construct a hybrid-proximal point algorithm for solving a system of the zero-finding problems, the equilibrium problems, and the fixed point problems of weak relatively nonexpansive mappings.

Remark 3.6. Since every relatively nonexpansive mapping is weak relatively nonexpansive, our results also hold if 𝑇_𝑖∶𝐶→𝐶 (𝑖=1,2,…) are relatively nonexpansive mappings.

Acknowledgments

The authors thank the editor and the referee(s) for valuable suggestions. The first author was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. The second and the third authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.