Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Abstract

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

1. Introduction

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to 𝑅, where 𝑅 is the set of real numbers. The equilibrium problem for 𝐹∶𝐶×𝐶→𝑅 is to find 𝑥∈𝐶 such that

𝐹(𝑥,𝑦)≥0,∀𝑦∈𝐶. (1.1)

The set of solutions of (1.1) is denoted by 𝐸𝑃(𝐹). Given a mapping 𝑇∶𝐶→𝐻, let 𝐹(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. Then, 𝑧∈𝐸𝑃(𝐹) if and only if ⟨𝑇𝑧,𝑦−𝑧⟩≥0 for all 𝑦∈𝐶. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); for more details, see [1, 2].

Recall that a self-mapping 𝑆 of a closed convex subset 𝐶 of 𝐻 is nonexpansive [3] if there holds that ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶. (1.2) We denote the set of fixed points of 𝑆 by 𝐹(𝑆). There are some methods for approximation of fixed points of a nonexpansive mapping. In 2000, Moudafi [4] introduced the viscosity approximation method for nonexpansive mappings (see [5] for further developments in both Hilbert and Banach spaces). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1, 2, 6, 7]. Recently, Combettes and Hirstoaga [6] introduced an iterative scheme of finding the best approximation to the initial data when 𝐸𝑃(𝐹) is nonempty and proved a strong convergence theorem. S. Takahashi and W. Takahashi [7] introduced a Mann iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem.

On the other hand, Ishikawa [8] introduced the following iterative process defined recursively by

𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑆𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑥_𝑛,∀𝑛∈𝑁, (1.3) where the initial guess 𝑥₀ is taking in 𝐶 arbitrarily, {𝛼_𝑛} and {𝛽_𝑛} are sequences in the interval [0,1].

In this paper, motivated by the ideas in [4–8], we introduce an Ishikawa iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Starting with an arbitrary 𝑥₁∈𝐻, define sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} by

𝐹(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑢_𝑛,∀𝑛∈𝑁, (1.4) where {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞).

We will prove in Section 3 that if the sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛} of parameters satisfy appropriate conditions, then the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} generated by (1.4) converge strongly to 𝑧∈𝐹(𝑆)∩𝐸𝑃(𝐹). The results in this paper extend and generalize S. Takahashi and W. Takahashi's results [7].

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩, and norm ‖⋅‖ and let 𝐶 be a nonempty closed convex subset of 𝐻. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥 and 𝑥_𝑛⇀𝑥 means that {𝑥_𝑛} converges weakly to 𝑥. In a real Hilbert space 𝐻, we have

‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖² (2.1) for all 𝑥,𝑦∈𝐻 and 𝜆∈𝑅; see [9].

For any 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶(𝑥), such that ‖𝑥−𝑃_𝐶(𝑥)‖≤‖𝑥−𝑦‖ for all 𝑦∈𝐶. Such a 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is also known that 𝑦=𝑃_𝐶(𝑥) is equivalent to ⟨𝑥−𝑦,𝑦−𝑧⟩≥0 for all 𝑧∈𝐶.

For solving the equilibrium problem, let us assume that the bifunction 𝐹 satisfies 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 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦); (2.2)(A4)for each 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

We recall some lemmas needed later.

Lemma 2.1 (see [2]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let 𝐹 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that 1𝐹(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶. (2.3)

Lemma 2.2 (see [5]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐹 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟1(𝑥)={𝑧∈𝐶∶𝐹(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶} (2.4) for all 𝑥∈𝐻. Then, the following statements hold: (1)𝑇_𝑟 is single-valued;(2)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝑇_𝑟(𝑥)−𝑇_𝑟(𝑦)‖²≤⟨𝑇_𝑟(𝑥)−𝑇_𝑟(𝑦),𝑥−𝑦⟩; (2.5)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹);(4)𝐸𝑃(𝐹) is closed and convex.

Lemma 2.3 (see [10]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤(1−𝑐_𝑛)𝑎_𝑛+𝑏_𝑛,∀𝑛∈𝑁, (2.6) where {𝑏_𝑛} is a sequence of real numbers and {𝑐_𝑛} is a sequence in (0,1) such that(i)∑^∞_𝑛=1𝑐_𝑛=∞,(ii)limsup_𝑛→∞(𝑏_𝑛/𝑐_𝑛)≤0 or ∑^∞_𝑛=1|𝑏_𝑛|<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

3. Strong Convergence Theorem

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹 be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4) and let 𝑆 be a nonexpansive mapping of 𝐶 into 𝐻 such that 𝐹(𝑆)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐻 into itself and let {𝑥_𝑛}, {𝑢_𝑛}, and {𝑦_𝑛} be sequences generated by 𝑥₁∈𝐻 and (1.4). If {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞) satisfy the following conditions: lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=1𝛼_𝑛=∞,^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,liminf_𝑛→∞𝑟_𝑛>0,^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞, (3.1) then, {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} converge strongly to 𝑧∈𝐹(𝑆)∩𝐸𝑃(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐹)}𝑓(𝑧).

Proof. Let 𝑄=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐹)}. Then 𝑄𝑓 is a contraction of 𝐻 into itself. In fact, there exists 𝑎∈[0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑎‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. So, we have that ‖𝑄𝑓(𝑥)−𝑄𝑓(𝑦)‖≤‖𝑓(𝑥)−𝑓(𝑦)‖≤𝑎‖𝑥−𝑦‖ (3.2) for all 𝑥,𝑦∈𝐻. Since 𝐻 is complete, there exists a unique element 𝑧∈𝐻 such that 𝑧=𝑄𝑓(𝑧). Such a 𝑧∈𝐻 is an element of 𝐶.
Let 𝑣∈𝐹(𝑆)∩𝐸𝑃(𝐹). Then from 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have
‖𝑢_𝑛−𝑣‖=‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑣‖≤‖𝑥_𝑛−𝑣‖ (3.3) for all 𝑛∈𝑁. Put 𝑀=max{‖𝑥₁−𝑣‖,(1/(1−𝑎))‖𝑓(𝑣)−𝑣‖}. It is obvious that ‖𝑥₁−𝑣‖≤𝑀. Suppose ‖𝑥_𝑛−𝑣‖≤𝑀. Then, we have ‖𝑥_𝑛+1−𝑣‖≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖+(1−𝛼_𝑛)‖𝑆𝑦_𝑛−𝑣‖≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑓(𝑣)‖+𝛼_𝑛‖𝑓(𝑣)−𝑣‖+(1−𝛼_𝑛)‖𝑆𝑦_𝑛−𝑣‖≤𝑎𝛼_𝑛‖𝑥_𝑛−𝑣‖+𝛼_𝑛‖𝑓(𝑣)−𝑣‖+(1−𝛼_𝑛)‖𝑦_𝑛−𝑣‖. (3.4) On the other hand ‖𝑦_𝑛−𝑣‖≤𝛽_𝑛‖𝑥_𝑛−𝑣‖+(1−𝛽_𝑛)‖𝑆𝑢_𝑛−𝑣‖≤𝛽_𝑛‖𝑥_𝑛−𝑣‖+(1−𝛽_𝑛)‖𝑢_𝑛−𝑣‖≤𝛽_𝑛‖𝑥_𝑛−𝑣‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑣‖=‖𝑥_𝑛−𝑣‖. (3.5) Putting (3.5) into (3.4), we have ‖𝑥_𝑛+1−𝑣‖≤𝑎𝛼_𝑛‖𝑥_𝑛−𝑣‖+𝛼_𝑛‖𝑓(𝑣)−𝑣‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑣‖=[1−𝛼_𝑛(1−𝑎)]‖𝑥_𝑛−𝑣‖+𝛼_𝑛(1−𝑎)‖𝑓(𝑣)−𝑣‖1−𝑎≤[1−𝛼_𝑛(1−𝑎)]𝑀+𝛼_𝑛(1−𝑎)𝑀=𝑀. (3.6) So, we have that ‖𝑥_𝑛+1−𝑣‖≤𝑀 for any 𝑛∈𝑁. And hence {𝑥_𝑛} is bounded. We also obtain that {𝑢_𝑛},{𝑦_𝑛},{𝑆𝑢_𝑛},{𝑆𝑦_𝑛}, and {𝑓(𝑥_𝑛)} are bounded. Next, we show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. In fact, ‖𝑦_𝑛−𝑦_𝑛−1‖=‖𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑢_𝑛−[𝛽_𝑛−1𝑥_𝑛−1+(1−𝛽_𝑛−1)𝑆𝑢_𝑛−1]‖=‖𝛽_𝑛(𝑥_𝑛−𝑥_𝑛−1)+(𝛽_𝑛−𝛽_𝑛−1)𝑥_𝑛−1+(1−𝛽_𝑛)(𝑆𝑢_𝑛−𝑆𝑢_𝑛−1)+(𝛽_𝑛−1−𝛽_𝑛)𝑆𝑢_𝑛−1‖≤|𝛽_𝑛−𝛽_𝑛−1|‖𝑥_𝑛−1‖+𝛽_𝑛‖𝑥_𝑛−𝑥_𝑛−1‖+(1−𝛽_𝑛)‖𝑢_𝑛−𝑢_𝑛−1‖+|𝛽_𝑛−𝛽_𝑛−1|‖𝑆𝑢_𝑛−1‖, (3.7) and hence ‖𝑥_𝑛+1−𝑥_𝑛‖=‖𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑦_𝑛−𝛼_𝑛−1𝑓(𝑥_𝑛−1)−(1−𝛼_𝑛−1)𝑆𝑦_𝑛−1‖=‖𝛼_𝑛𝑓(𝑥_𝑛)−𝛼_𝑛𝑓(𝑥_𝑛−1)+𝛼_𝑛𝑓(𝑥_𝑛−1)−𝛼_𝑛−1𝑓(𝑥_𝑛−1)+(1−𝛼_𝑛)𝑆𝑦_𝑛−(1−𝛼_𝑛)𝑆𝑦_𝑛−1+(1−𝛼_𝑛)𝑆𝑦_𝑛−1−(1−𝛼_𝑛−1)𝑆𝑦_𝑛−1‖≤𝛼_𝑛𝑎‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|‖𝑓(𝑥_𝑛−1)‖+(1−𝛼_𝑛)‖𝑦_𝑛−𝑦_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|‖𝑆𝑦_𝑛−1‖≤𝛼_𝑛𝑎‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|‖𝑓(𝑥_𝑛−1)‖+(1−𝛼_𝑛)×[|𝛽_𝑛−𝛽_𝑛−1|‖𝑥_𝑛−1‖+𝛽_𝑛‖𝑥_𝑛−𝑥_𝑛−1‖+(1−𝛽_𝑛)‖𝑢_𝑛−𝑢_𝑛−1‖+|𝛽_𝑛−𝛽_𝑛−1|‖𝑆𝑢_𝑛−1‖]+|𝛼_𝑛−𝛼_𝑛−1|‖𝑆𝑦_𝑛−1‖=[𝛽_𝑛−𝛼_𝑛(𝛽_𝑛−𝑎)]‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|[‖𝑓(𝑥_𝑛−1)‖+‖𝑆𝑦_𝑛−1‖]+(1−𝛼_𝑛)|𝛽_𝑛−𝛽_𝑛−1|[‖𝑥_𝑛−1‖+‖𝑆𝑢_𝑛−1‖]+(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑢_𝑛−𝑢_𝑛−1‖≤[𝛽_𝑛−𝛼_𝑛(𝛽_𝑛−𝑎)]‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|𝐾₁+(1−𝛼_𝑛)|𝛽_𝑛−𝛽_𝑛−1|𝐾₂+(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑢_𝑛−𝑢_𝑛−1‖, (3.8) where 𝐾₁=sup{‖𝑓(𝑥_𝑛)‖+‖𝑆𝑦_𝑛‖∶𝑛∈𝑁} and 𝐾₂=sup{‖𝑥_𝑛‖+‖𝑆𝑢_𝑛‖∶𝑛∈𝑁}.
On the other hand, from 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑢_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1, we have
𝐹(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,𝐹(𝑢_𝑛+11,𝑦)+𝑟_𝑛+1⟨𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1⟩≥0,∀𝑦∈𝐶. (3.9)(3.10)
Putting 𝑦=𝑢_𝑛+1 in (3.9) and 𝑦=𝑢_𝑛 in (3.10), we have
𝐹(𝑢_𝑛,𝑢_𝑛+11)+𝑟_𝑛⟨𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,𝐹(𝑢_𝑛+1,𝑢_𝑛1)+𝑟_𝑛+1⟨𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1⟩≥0. (3.11) So, from the monotonicity of 𝐹, we get ⟨𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛−𝑢_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1⟩≥0, (3.12) and hence ⟨𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1(𝑢_𝑛+1−𝑥_𝑛+1)⟩≥0. (3.13) Without loss of generality, let us assume that there exists a real number 𝑏 such that 𝑟_𝑛>𝑏>0 for all 𝑛∈𝑁. Then, we have ‖𝑢_𝑛+1−𝑢_𝑛‖²≤⟨𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛𝑟+(1−^𝑛𝑟_𝑛+1)(𝑢_𝑛+1−𝑥_𝑛+1)⟩≤‖𝑢_𝑛+1−𝑢_𝑛‖{‖𝑥_𝑛+1−𝑥_𝑛𝑟‖+|1−^𝑛𝑟_𝑛+1|‖𝑢_𝑛+1−𝑥_𝑛+1‖}, (3.14) and hence ‖𝑢_𝑛+1−𝑢_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑟_𝑛+1|𝑟_𝑛+1−𝑟_𝑛|‖𝑢_𝑛+1−𝑥_𝑛+1‖≤‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑏|𝑟_𝑛+1−𝑟_𝑛|𝐿, (3.15) where 𝐿=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈𝑁}. So from (3.8), we have ‖𝑥_𝑛+1−𝑥_𝑛‖≤[𝛽_𝑛−𝛼_𝑛(𝛽_𝑛−𝑎)]‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|𝐾₁+(1−𝛼_𝑛)|𝛽_𝑛−𝛽_𝑛−1|𝐾₂+(1−𝛼_𝑛)(1−𝛽_𝑛)[‖𝑥_𝑛−𝑥_𝑛−11‖+𝑏|𝑟_𝑛−𝑟_𝑛−1|𝐿]=(1−𝛼_𝑛(1−𝑎))‖𝑥_𝑛−𝑥_𝑛−1‖+|𝛼_𝑛−𝛼_𝑛−1|𝐾₁+(1−𝛼_𝑛)|𝛽_𝑛−𝛽_𝑛−1|𝐾₂+(1−𝛼_𝑛)(1−𝛽_𝑛)1𝑏|𝑟_𝑛−𝑟_𝑛−1|𝐿. (3.16) Using Lemma 2.1 in [10], we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.17) From (3.15) and |𝑟_𝑛+1−𝑟_𝑛|→0, we have lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. (3.18) It follows from (3.7) that lim_𝑛→∞‖𝑦_𝑛+1−𝑦_𝑛‖=0. (3.19) Since 𝑥_𝑛=𝛼_𝑛−1𝑓(𝑥_𝑛−1)+(1−𝛼_𝑛−1)𝑆𝑦_𝑛−1, we have ‖𝑥_𝑛−𝑆𝑦_𝑛‖≤‖𝑥_𝑛−𝑆𝑦_𝑛−1‖+‖𝑆𝑦_𝑛−1−𝑆𝑦_𝑛‖≤𝛼_𝑛−1‖𝑓(𝑥_𝑛−1)−𝑆𝑦_𝑛−1‖+‖𝑦_𝑛−1−𝑦_𝑛‖. (3.20) From 𝛼_𝑛→0, we have ‖𝑥_𝑛−𝑆𝑦_𝑛‖→0. For 𝑣∈𝐹(𝑆)∩𝐸𝑃(𝐹), we have ‖𝑢_𝑛−𝑣‖²=‖𝑇_𝑟𝑛𝑥_𝑛−𝑇_r𝑛𝑣‖²≤⟨𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛𝑣,𝑥_𝑛−𝑣⟩=⟨𝑢_𝑛−𝑣,𝑥_𝑛=1−𝑣⟩2(‖𝑢_𝑛−𝑣‖²+‖𝑥_𝑛−𝑣‖²−‖𝑥_𝑛−𝑢_𝑛‖²), (3.21) and hence ‖𝑢_𝑛−𝑣‖²≤‖𝑥_𝑛−𝑣‖²−‖𝑥_𝑛−𝑢_𝑛‖². (3.22) Therefore, from the convexity of ‖⋅‖², we have ‖𝑦_𝑛−𝑣‖²≤𝛽_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛽_𝑛)‖𝑆𝑢_𝑛−𝑣‖²≤𝛽_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛽_𝑛)‖𝑢_𝑛−𝑣‖²≤𝛽_𝑛‖𝑥_𝑛−𝑣‖²+(1−𝛽_𝑛)[‖𝑥_𝑛−𝑣‖²−‖𝑥_𝑛−𝑢_𝑛‖²]=‖𝑥_𝑛−𝑣‖²−(1−𝛽_𝑛)‖𝑥_𝑛−𝑢_𝑛‖², (3.23) and hence ‖𝑥_𝑛+1−𝑣‖²=‖𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑦_𝑛−𝑣‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+(1−𝛼_𝑛)‖𝑆𝑦_𝑛−𝑣‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+(1−𝛼_𝑛)‖𝑦_𝑛−𝑣‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+(1−𝛼_𝑛)[‖𝑥_𝑛−𝑣‖²−(1−𝛽_𝑛)‖𝑥_𝑛−𝑢_𝑛‖²]≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+‖𝑥_𝑛−𝑣‖²−(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑢_𝑛‖². (3.24) So, we have (1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑢_𝑛‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+‖𝑥_𝑛−𝑣‖²−‖𝑥_𝑛+1−𝑣‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+‖𝑥_𝑛−𝑥_𝑛+1‖(‖𝑥_𝑛−𝑣‖+‖𝑥_𝑛+1−𝑣‖). (3.25) Without loss of generality, let us assume that there exists two real numbers 𝛽^∗ and 𝛽 such that 1>𝛽≥𝛽_𝑛≥𝛽^∗>0 for all 𝑛∈𝑁. Hence, (1−𝛼_𝑛)(1−𝛽)‖𝑥_𝑛−𝑢_𝑛‖²≤(1−𝛼_𝑛)(1−𝛽_𝑛)‖𝑥_𝑛−𝑢_𝑛‖²≤𝛼_𝑛‖𝑓(𝑥_𝑛)−𝑣‖²+‖𝑥_𝑛−𝑥_𝑛+1‖(‖𝑥_𝑛−𝑣‖+‖𝑥_𝑛+1−𝑣‖). (3.26) It follows that ‖𝑥_𝑛−𝑢_𝑛‖→0. We also have ‖𝑆𝑢_𝑛−𝑥_𝑛‖≤‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑆𝑢_𝑛−𝑆𝑦_𝑛‖≤‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑢_𝑛−𝑦_𝑛‖≤‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑦_𝑛‖=‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆𝑢_𝑛‖. (3.27) It follows that 𝛽^∗‖𝑆𝑢_𝑛−𝑥_𝑛‖≤𝛽_𝑛‖𝑆𝑢_𝑛−𝑥_𝑛‖≤‖𝑆𝑦_𝑛−𝑥_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖. (3.28) Hence, ‖𝑆𝑢_𝑛−𝑥_𝑛‖→0. Since ‖𝑆𝑢_𝑛−𝑢_𝑛‖≤‖𝑆𝑢_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑢_𝑛‖, (3.29) we also have lim_𝑛→∞‖𝑆𝑢_𝑛−𝑢_𝑛‖=0. Next, we show that limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧⟩≤0, (3.30) where 𝑧=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐹)}𝑓(𝑧). To show this inequality, we choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that lim_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛𝑖−𝑧⟩=limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧⟩. (3.31) Since {𝑢_𝑛𝑖} is bounded, there exists a subsequence {𝑢_𝑛𝑖𝑗} of {𝑢_𝑛𝑖} which converges weakly to 𝑤. Without loss of generality, we can assume that {𝑢_𝑛𝑖}⇀𝑤. From ‖𝑆𝑢_𝑛−𝑢_𝑛‖→0, we obtain 𝑆𝑢_𝑛𝑖⇀𝑤. Let us show 𝑤∈𝐸𝑃(𝐹). By 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, we have 𝐹(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶. (3.32)
From (A2), we also have
1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹(𝑦,𝑢_𝑛), (3.33) and hence, ⟨𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖⟩≥𝐹(𝑦,𝑢_𝑛𝑖). (3.34) Since (𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑤, from (A4), we have 𝑓(𝑦,𝑤)≤0,∀𝑦∈𝐶. (3.35) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. Since 𝑦∈𝐶 and 𝑤∈𝐶, we obtain 𝑦_𝑡∈𝐶 and hence 𝐹(𝑦_𝑡,𝑤)≤0. So we have 0=𝐹(𝑦_𝑡,𝑦_𝑡)≤𝑡𝐹(𝑦_𝑡,𝑦)+(1−𝑡)𝐹(𝑦_𝑡,𝑤)≤𝑡𝐹(𝑦_𝑡,𝑦). (3.36) Dividing by 𝑡, we get 𝐹(𝑦_𝑡,𝑦)≥0. (3.37) Letting 𝑡→0 and from (A3), we get 𝐹(𝑤,𝑦)≥0 (3.38) for all 𝑦∈𝐶 and hence 𝑤∈𝐸𝑃(𝐹). We shall show that 𝑤∈𝐹(𝑆). Assume 𝑤∉𝐹(𝑆). Since 𝑢_𝑛𝑖⇀𝑤 and 𝑤≠𝑆𝑤, from the Opial theorem [11] we have liminf_𝑖→∞‖𝑢_𝑛𝑖−𝑤‖<liminf_𝑖→∞‖𝑢_𝑛𝑖−𝑆𝑤‖≤liminf_𝑖→∞{‖𝑢_𝑛𝑖−𝑆𝑢_𝑛𝑖‖+‖𝑆𝑢_𝑛𝑖−𝑆𝑤‖}≤liminf_𝑖→∞‖𝑢_𝑛𝑖−𝑤‖. (3.39) This is a contradiction. So, we get 𝑤∈𝐹(𝑆). Therefore, 𝑤∈𝐹(𝑆)∩𝐸𝑃(𝐹). Since 𝑧=𝑃_{𝐹(𝑆)∩𝐸𝑃(𝐹)}𝑓(𝑧), we have limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧⟩=lim_𝑖→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛𝑖−𝑧⟩=lim_𝑖→∞⟨𝑓(𝑧)−𝑧,𝑢_𝑛𝑖−𝑧⟩=⟨𝑓(𝑧)−𝑧,𝑤−𝑧⟩≤0. (3.40) From 𝑥_𝑛+1−𝑧=𝛼_𝑛(𝑓(𝑥_𝑛)−𝑧)+(1−𝛼_𝑛)(𝑆𝑦_𝑛−𝑧), we have (1−𝛼_𝑛)²‖𝑆𝑦_𝑛−𝑧‖²≥‖𝑥_𝑛+1−𝑧‖²−2𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑧,𝑥_𝑛+1−𝑧⟩,‖𝑦_𝑛−𝑧‖²=‖𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)S𝑢_𝑛−𝑧‖²≤𝛽_𝑛‖𝑥_𝑛−𝑧‖²+(1−𝛽_𝑛)‖𝑆𝑢_𝑛−𝑧‖²≤𝛽_𝑛‖𝑥_𝑛−𝑧‖²+(1−𝛽_𝑛)‖𝑢_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖². (3.41)(3.42) It follows that ‖𝑥_𝑛+1−𝑧‖²≤(1−𝛼_𝑛)²‖𝑆𝑦_𝑛−𝑧‖²+2𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑧,𝑥_𝑛+1−𝑧⟩≤(1−𝛼_𝑛)²‖𝑦_𝑛−𝑧‖²+2𝛼_𝑛⟨𝑓(𝑥_𝑛)−𝑓(𝑧),𝑥_𝑛+1−𝑧⟩+2𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩≤(1−𝛼_𝑛)²‖𝑥_𝑛−𝑧‖²+2𝛼_𝑛𝑎‖𝑥_𝑛−𝑧‖‖𝑥_𝑛+1−𝑧‖+2𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩≤(1−𝛼_𝑛)²‖𝑥_𝑛−𝑧‖²+𝛼_𝑛𝑎{‖𝑥_𝑛−𝑧‖²+‖𝑥_𝑛+1−𝑧‖²}+2𝛼_𝑛⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩. (3.43) Hence ‖𝑥_𝑛+1−𝑧‖²≤(1−𝛼_𝑛)²+𝛼_𝑛𝑎1−𝛼_𝑛𝑎‖𝑥_𝑛−𝑧‖²+2𝛼_𝑛1−𝛼_𝑛𝑎⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩. (3.44) From 𝛼_𝑛→0, we know that there exists a positive integer 𝑛₀, such that 1>1−𝛼_𝑛𝑎>1/2 for all 𝑛≥𝑛₀. Then (1−𝛼_𝑛)²+𝛼_𝑛𝑎1−𝛼_𝑛𝑎=1−2𝛼_𝑛+𝛼_𝑛𝑎1−𝛼_𝑛𝑎+𝛼²_𝑛1−𝛼_𝑛𝑎=1−2(1−𝑎)𝛼_𝑛1−𝛼_𝑛𝑎+𝛼²_𝑛1−𝛼_𝑛𝑎≤1−2(1−𝑎)𝛼_𝑛+2𝛼²_𝑛,∀𝑛≥𝑛₀. (3.45) Putting above inequality into (3.44), we get ‖𝑥_𝑛+1−𝑧‖²≤(1−2(1−𝑎)𝛼_𝑛)‖𝑥_𝑛−𝑧‖²+2𝑀𝛼²_𝑛+2𝛼_𝑛1−𝛼_𝑛𝑎𝜎_𝑛,∀𝑛≥𝑛₀, (3.46) where 𝑀=sup{‖𝑥_𝑛−𝑧‖²∶𝑛∈𝑁}, and 𝜎_𝑛=⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩.
It follows from Lemma 2.3 that
𝑥_𝑛→𝑧∈𝐹(𝑆)∩𝐸𝑃(𝐹). (3.47) It follows from ‖𝑥_𝑛−𝑢_𝑛‖→0 and (3.42) that 𝑢_𝑛→𝑧 and 𝑦_𝑛→𝑧.

By Theorem 3.1, we can obtain the following new result.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑆 be a nonexpansive mapping of 𝐶 into 𝐻 such that 𝐹(𝑆)≠∅. Let 𝑓 be a contraction of 𝐻 into itself and let {𝑥_𝑛} and {𝑦_𝑛} be sequences generated by 𝑥₁∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝑆𝑦_𝑛,𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑆𝑃_𝐶𝑥_𝑛,∀𝑛∈𝑁. (3.48) If {𝛼_𝑛},{𝛽_𝑛}⊂[0,1] satisfy the following conditions: lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=1𝛼_𝑛=∞,^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, (3.49) then, {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑧∈𝐹(𝑆), where 𝑧=𝑃_𝐹(𝑆)𝑓(𝑧).

Proof. Put 𝐹(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝐶 and 𝑟_𝑛=1 for all 𝑛∈𝑁 in Theorem 3.1. Then, we get 𝑢_𝑛=𝑃_𝐶𝑥_𝑛. So from Theorem 3.1, the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑧∈𝐹(𝑆), where 𝑧=𝑃_𝐹(𝑆)𝑓(𝑧).

Remark 3.3. Theorem 3.1 and Corollary 3.2, respectively, extend and generalize Theorem 3.2 and Corollary 3.3 in [7] from the Mann iterative form to the Ishikawa iterative form.

Acknowledgments

The authors would like to express their thanks to the referees for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05).