Viscosity Approximation Methods for Two Accretive Operators in Banach Spaces

Abstract

We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of Ceng et al. (2009) and some others.

1. Introduction and Preliminaries

Let 𝐸 be a real Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇∶𝐶→𝐶 a mapping. Recall that 𝑇 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶. A point 𝑥∈𝐶 is a fixed point of 𝑇 provided that 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶,𝑇𝑥=𝑥}. Throughout this paper, we assume that 𝑇 is a nonexpansive mapping such that 𝐹(𝑇)≠∅. Recall that a self-mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶. Let Σ_𝐶={𝑓∶𝐶→𝐶∣𝑓 is a contraction with constant 𝛼}. The normalized duality mapping 𝐽 from 𝐸 into 2^𝐸∗ is given by 𝐽(𝑥)={𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²}, 𝑥∈𝐸, where 𝐸^∗ denotes the dual space of 𝐸 and ⟨⋅,⋅⟩ denotes the generalized duality pairing.

A Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1, for all 𝑥≠𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1. It is said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, for any two sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝐸 such that ‖𝑥‖=‖𝑦‖=1 and lim_𝑛→∞(‖𝑥_𝑛+𝑦_𝑛‖/2)=1.

The norm of 𝐸 is said to be Gâteaux differentiable if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1) exists for each 𝑥,𝑦 in its unit sphere 𝑈={𝑥∈𝐸,‖𝑥‖=1}. Such an 𝐸 is called a smooth Banach space. The norm is said to be uniformly Gâteaux differentiable if, for each 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. It is well known that 𝐸 is smooth if and only if the duality mapping 𝐽 is single valued and that, if 𝐸 has a uniformly Gâteaux differentiable norm, 𝐽 is uniformly norm to weak* continuous on each bounded subset of 𝐸 (cf. [1]).

Let 𝐷 be a subset of 𝐶. Then 𝑄∶𝐶→𝐷 is called a retraction from 𝐶 onto 𝐷 if 𝑄(𝑥)=𝑥 for all 𝑥∈𝐷. A retraction 𝑄∶𝐶→𝐷 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥 for all 𝑥∈𝐶 and 𝑡≥0 whenever 𝑄𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶. A subset 𝐷 of 𝐶 is said to be a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction of 𝐶 onto 𝐷. In a smooth Banach space 𝐸, it is known that 𝑄∶𝐶→𝐷 is a sunny nonexpansive retraction if and only if the following condition holds (cf. [2, page 48]): ⟨𝑥−𝑄𝑥,𝐽(𝑧−𝑄𝑥)⟩≤0,𝑥∈𝐶,𝑧∈𝐷. (2)

Recall that an operator 𝐴 with 𝐷(𝐴) and 𝑅(𝐴) in 𝐸 is said to be accretive if, for each 𝑥_𝑖∈𝐷(𝐴) and 𝑦_𝑖∈𝐴𝑥_𝑖, 𝑖=1,2, there exists a 𝑗∈𝐽(𝑥₂−𝑥₁) such that ⟨𝑦₂−𝑦₁,𝑗⟩≥0. An accretive operator 𝐴 is 𝑚-accretive if 𝑅(𝐼+𝜆𝐴)=𝐸, for all 𝜆>0. Denote by 𝐴⁻¹0 the set of zeros of 𝐴; that is, 𝐴⁻¹0={𝑥∈𝐷(𝐴),𝐴𝑥=0}.

Denote by 𝐽_𝑟  (𝑟>0) the resolvent of 𝐴; that is, 𝐽_𝑟=(𝐼+𝑟𝐴)⁻¹. It is well known that 𝐹(𝐽_𝑟)=𝐴⁻¹0, for all 𝑟>0. And if 𝐷(𝐴) is convex, then 𝐽_𝑟 is a nonexpansive mapping from 𝐸 to 𝐷(𝐴). If 𝐸 is a Hilbert space, then 𝐴 is a maximal monotone operator if and only if 𝐴 is an 𝑚-accretive operator.

Recently, the approximation of zeros of accretive operators has been studied extensively (see, e.g., [3–9]). Specially, Ceng et al. [10] studied the following composite iterative scheme in uniformly smooth Banach spaces: 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑦_𝑛+1−𝛽_𝑛𝐽_𝑟𝑛𝑦_𝑛, (3) where 𝑢∈𝐷(𝐴) is an arbitrary (but fixed) element. They proved that {𝑥_𝑛} generated by (3) converges strongly to a zero of 𝑚-accretive operator 𝐴 under certain appropriate conditions.

Very recently, Chen et al. [11] considered the following viscosity iterative scheme in a reflexive Banach space having a weakly sequentially continuous duality mapping: 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛, (4) where {𝛼_𝑛},{𝛽_𝑛}⊂(0,1). Under some conditions, they showed that {𝑥_𝑛} generated by (4) converges strongly to a zero of 𝑚-accretive operator 𝐴.

In this paper, motivated by [10–14], we will consider the following so-called composite viscosity iterative scheme for finding a common zero of two 𝑚-accretive operators: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛, (5) where 𝐴 and 𝐵 are 𝑚-accretive operators, 𝑆_𝑟𝑛=(1−𝜆)𝐽^𝐴_𝑟+𝜆𝐽^𝐵_𝑟 such that 𝐹(𝑆_𝑟)=𝐴⁻¹0∩𝐵⁻¹0≠∅, 𝐶=𝐷(𝐴)=𝐷(𝐵), 𝑓∈Σ_𝐶, and {𝛼_𝑛},{𝛽_𝑛}⊂(0,1). Under some conditions, we will prove that {𝑥_𝑛} generated by (5) converges strongly to a common zero of 𝐴 and 𝐵 in a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm, which improve the corresponding results in [10–13].

Lemma 1 (see [10]). In a Banach space 𝐸, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑥,𝑦∈𝐸, (6) where 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).

Lemma 2 (see [10, 13]). Let {𝛼_𝑛} be a sequence of nonnegative real numbers satisfying the condition 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝜎_𝑛𝛾_𝑛,∀𝑛≥0, (7) where {𝛾_𝑛}⊂(0,1) and {𝜎_𝑛} such that(i)lim_𝑛→∞𝛾_𝑛=0 and ∑^∞_𝑛=1𝛾_𝑛=∞;(ii)either limsup_𝑛→∞𝜎_𝑛≤0 or ∑^∞_𝑛=1|𝛾_𝑛𝜎_𝑛|<∞.Then lim_𝑛→∞𝛼_𝑛=0.

Lemma 3 (the resolvent identity [10]). For 𝜆>0, 𝜇>0 and 𝑥∈𝐸, 𝐽_𝜆𝑥=𝐽_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽_𝜆𝑥. (8)

Lemma 4 (see [3, Theorem 4.1, page 287]). Let 𝐸 be a uniformly smooth Banach space, 𝐶 be a closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅, and 𝑓∈Σ_𝐶. Then {𝑧_𝑡} defined by the following 𝑧_𝑡=𝑡𝑓𝑧_𝑡+(1−𝑡)𝑇𝑧_𝑡,𝑧_𝑡∈𝐶, (9) converges strongly to a point in Fix(𝑇). If, moreover, one defines 𝑄∶Σ_𝐶→𝐹(𝑇) by 𝑄𝑓∶=lim_𝑡→0𝑧_𝑡,𝑓∈Σ_𝐶, (10) then 𝑄(𝑓) solves the variational inequality 𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑝≤0,𝑓∈Σ_𝐶,𝑝∈𝐹(𝑇). (11)

Recall that a mapping 𝑔∶𝐶→𝐶 is said to be weakly contractive [15, 16] if 𝑔(𝑥)−𝑔𝑦≤𝑥−𝑦−𝜓𝑥−𝑦,∀𝑥,𝑦∈𝐶, (12) where 𝜓∶[0,+∞)→[0,+∞) is a continuous and strictly increasing function such that 𝜓 is positive on (0,+∞) and 𝜓(0)=0. As a special case, if 𝜓(𝑡)=(1−𝑘)𝑡 for 𝑡∈[0,+∞), where 𝑘∈(0,1), then the weakly contractive mapping 𝑔 is a contraction with constant 𝑘. Rhodes [17] obtained the following result for weakly contractive mapping (see also [16]).

Lemma 5 (see [17, Theorem 2]). Let (𝑋,𝑑) be a complete metric space and 𝑔 a weakly contractive mapping on 𝑋. Then 𝑔 has a unique fixed point 𝑝 in 𝑋.

Lemma 6. Let {𝑠_𝑛} and {𝛾_𝑛} be two sequences of nonnegative real numbers and {𝜆_𝑛} a sequence of positive numbers satisfying the conditions(i)∑^∞_𝑛=1𝜆_𝑛=+∞,(ii)lim_𝑛→∞(𝛾_𝑛/𝜆_𝑛)=0.
Let the recursive inequality 𝑠_𝑛+1≤𝑠_𝑛−𝜆_𝑛𝜓𝑠_𝑛+𝜆_𝑛𝛾_𝑛,𝑛=0,1,2,…, (13) be given where 𝜓(𝑡) is a continuous and strict increasing function on [0,+∞) with 𝜓(0)=0. Then lim_𝑛→∞𝑠_𝑛=0.

2. Main Results

Throughout this section, we assume the following:

(i) 𝐸 is a strictly convex Banach space which has a uniformly Gâteaux differentiable norm, and 𝐶 is a nonempty closed convex subset of 𝐸.

(ii) Take 𝑆_𝑟=(1−𝜆)𝐽^𝐴_𝑟+𝜆𝐽^𝐵_𝑟, 0<𝜆<1. Obviously 𝑆_𝑟 is nonexpansive mapping and 𝐹(𝑆_𝑟)=𝐴⁻¹0∩𝐵⁻¹0, if 𝐸 is a strictly convex Banach space. Indeed, it is easy to see that 𝐹(𝑆_𝑟)⊃𝐴⁻¹0∩𝐵⁻¹0. Let 𝑞∈𝐹(𝑆_𝑟), 𝑝∈𝐴⁻¹0∩𝐵⁻¹0; then 𝑞−𝑝≤(1−𝜆)𝐽^𝐴_𝑟𝑞−𝑝+𝜆𝐽^𝐵_𝑟≤(𝐽𝑞−𝑝1−𝜆)^𝐴_𝑟𝑞−𝑝+𝜆𝑞−𝑝≤𝑞−𝑝. (14) From the above formula, we obtain (1−𝜆)‖𝐽^𝐴_𝑟𝑞−𝑝‖+𝜆‖𝑞−𝑝‖=‖𝑞−𝑝‖, so ‖𝐽^𝐴_𝑟𝑞−𝑝‖=‖𝑞−𝑝‖. Similarly, ‖𝐽^𝐵_𝑟𝑞−𝑝‖=‖𝑞−𝑝‖. But 𝑞−𝑝=(1−𝜆)𝐽^𝐴_𝑟𝑞−𝑝+𝜆𝐽^𝐵_𝑟𝑞−𝑝. (15) Then the strict convexity of 𝐸 implies that 𝑞−𝑝=𝐽^𝐴_𝑟𝑞−𝑝=𝐽^𝐵_𝑟𝑞−𝑝, that is, 𝑞=𝐽^𝐴_𝑟𝑞=𝐽^𝐵_𝑟𝑞, or 𝑞∈𝐴⁻¹0∩𝐵⁻¹0.

Theorem 7. Let 𝐸 be a strictly convex Banach space which has a uniformly Gâteaux differentiable norm, 𝐴,𝐵 two m-accretive maps in 𝐸 such that 𝐶=𝐷(𝐴)=𝐷(𝐵) is convex and 𝐴⁻¹0∩𝐵⁻¹0≠∅, and 𝑓∶𝐶→𝐶 a fixed contraction mapping with contract constant 𝛼. Suppose that 𝛼_𝑛⊂(0,1), 𝛽_𝑛⊂(0,1), and 𝑟_𝑛>0 satisfy the following conditions:(i)∑^∞_𝑛=1𝛼_𝑛=∞, 𝛼_𝑛→0, as 𝑛→∞;(ii)𝛽_𝑛→0, and 𝑟_𝑛→𝑟>𝜀>0 as 𝑛→∞;(iii)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞.
Let {𝑥_𝑛} be the composite viscosity process defined by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛. (16) Then {𝑥_𝑛} converges strongly to 𝑝∈𝐴⁻¹0∩𝐵⁻¹0, where 𝑝 is the unique solution of the following variational inequality: 𝐼−𝑓𝑝,𝐽𝑝−𝑞≤0,𝑓∈Σ_𝐶,𝑞∈𝐴⁻¹0∩𝐵⁻¹0. (17)

Proof. First, by using Lemma 4, we know that there exists the unique solution 𝑝 of a variational inequality 𝐼−𝑓𝑝,𝐽𝑝−𝑞≤0,𝑓∈Σ_𝐶,𝑞∈𝐴⁻¹0∩𝐵⁻¹0, (18) where 𝑝=lim_𝑡→0𝑧_𝑡 and 𝑧_𝑡 is defined by 𝑧_𝑡=𝑡𝑓(𝑧_𝑡)+(1−𝑡)𝑆_𝑟(𝑧_𝑡) for each 𝑟>0 and 0<𝑡<1.
Next, we will divide our discussion into the following steps.
Step  1. We will show that {𝑥_𝑛} is bounded.
In fact, take 𝑝∈𝐴⁻¹0∩𝐵⁻¹0. Then 𝑦_𝑛−𝑝=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛≤−𝑝1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝. (19) Therefore, 𝑥_𝑛+1−𝑝=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛≤−𝑝1−𝛽_𝑛(1−𝛼)𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑝≤()𝛽−𝑝1−1−𝛼_𝑛+𝛼_𝑛−𝛼_𝑛𝛽_𝑛(1−𝛼)𝑥_𝑛+𝛽−𝑝_𝑛+𝛼_𝑛−𝛼_𝑛𝛽_𝑛(𝑓𝑝11−𝛼)−𝑝≤max𝑓𝑝,𝑥1−𝛼−𝑝_𝑛−𝑝. (20) Using the induction method, we have 𝑥_𝑛1−𝑝≤max𝑓𝑝,𝑥1−𝛼−𝑝₀−𝑝,𝑛≥0, (21) which implies that {𝑥_𝑛}, {𝑓(𝑥_𝑛)}, {𝑦_𝑛}, and {𝑓(𝑦_𝑛)} are all bounded. Since ‖𝑆_𝑟𝑛𝑥_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, then {𝑆_𝑟𝑛𝑥_𝑛} is bounded. Following the conditions of (i) and (ii), we obtain that 𝑥_𝑛+1−𝑦_𝑛=𝛽_𝑛𝑓𝑦_𝑛−𝑦_𝑛𝑦⟶0,as𝑛⟶∞,_𝑛−𝑆_𝑟𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑆_𝑟𝑛𝑥_𝑛⟶0,as𝑛⟶∞. (22)(23)
Step  2. We show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0.
For this, we estimate 𝑦_𝑛+1−𝑦_𝑛 first. From (16), we know that 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑦_𝑛−1=𝛼_𝑛−1𝑓𝑥_𝑛−1+1−𝛼_𝑛−1𝑆_𝑟𝑛−1𝑥_𝑛−1. (24) Then simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝑆_𝑟𝑛−1𝑥_𝑛−1. (25) It follows from (25) that 𝑦_𝑛−𝑦_𝑛−1≤1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝛼𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝑆_𝑟𝑛−1𝑥_𝑛−1. (26) In view of Lemma 3, we have 𝐽^𝐴_𝑟𝑛𝑥_𝑛=𝐽^𝐴_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛. (27) If 𝑟_𝑛−1≤𝑟_𝑛, then 𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝐽^𝐴_𝑟𝑛−1𝑥_𝑛−1=𝐽^𝐴_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝐽^𝐴_𝑟𝑛−1𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝜀𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1. (28) Similarly, 𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝐽^𝐵_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝜀𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1. (29) Thus, let 𝑀=sup{(1/𝜀)‖𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1‖,(1/𝜀)‖𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1‖}; we have 𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛−1𝑥_𝑛−1≤(1−𝜆)𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝐽^𝐴_𝑟𝑛−1𝑥_𝑛−1+𝜆𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝐽^𝐵_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀. (30) Substituting (30) into (26) we get 𝑦_𝑛−𝑦_𝑛−1≤1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀+𝛼_𝑛𝛼𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝑆_𝑟𝑛−1𝑥_𝑛−1≤1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝑟_𝑛−𝑟_𝑛−1𝑀+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝑆_𝑟𝑛−1𝑥_𝑛−1≤1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝑟_𝑛−𝑟_𝑛−1, (31) where 𝑀₁ is a constant such that 𝑀₁>max𝑀,𝑓𝑥_𝑛−1−𝑆_𝑟𝑛−1𝑥_𝑛−1. (32) On the other hand, we have 𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛,𝑥_𝑛=𝛽_𝑛−1𝑓𝑦_𝑛−1+1−𝛽_𝑛−1𝑦_𝑛−1. (33) Simple calculations show that 𝑥_𝑛+1−𝑥_𝑛=1−𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛽_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑦_𝑛−1−𝑦_𝑛−1. (34) It follows that 𝑥_𝑛+1−𝑥_𝑛≤1−𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛽_𝑛𝛼𝑦_𝑛−𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑦_𝑛−1−𝑦_𝑛−1=1−𝛽_𝑛(1−𝛼)𝑦_𝑛−𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑦_𝑛−1−𝑦_𝑛−1. (35) Substituting (31) into (35) we get 𝑥_𝑛+1−𝑥_𝑛≤1−𝛽_𝑛(1−𝛼)1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝑟_𝑛−𝑟_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑓𝑦_𝑛−1−𝑦_𝑛−1≤1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑥_𝑛−1+𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑟_𝑛−𝑟_𝑛−1, (36) where 𝑀₂ is a constant such that 𝑀₂>max𝑓𝑦_𝑛−1−𝑦_𝑛−1,𝑀₁. (37) From conditions (i)–(iii), we have that ^∞_𝑛=1𝛼_𝑛=∞,𝛼_𝑛⟶0,as𝑛⟶∞,_∞𝑛=1𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑟_𝑛−𝑟_𝑛−1<+∞. (38) Hence, noticing (36) and applying Lemma 2, we obtain ‖𝑥_𝑛+1−𝑥_𝑛‖→0. Then by (22) we obtain 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑦_𝑛⟶0,(𝑛⟶∞). (39)
Step  3. We prove that ‖𝑥_𝑛−𝑆_𝑟𝑥_𝑛‖→0, ‖𝑦_𝑛−𝑆_𝑟𝑦_𝑛‖→0.
In fact, since 𝑥_𝑛−𝑆_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛+𝑦_𝑛−𝑆_𝑟𝑛𝑥_𝑛, (40) from (22) and (23), we have ‖𝑥_𝑛−𝑆_𝑟𝑛𝑥_𝑛‖→0. Then 𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑥_𝑛≤(1−𝜆)𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝐽^𝐴_𝑟𝑥_𝑛+𝜆𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝐽^𝐵_𝑟𝑥_𝑛≤(1−𝜆)𝐽^𝐴_𝑟𝑟𝑟_𝑛𝑥_𝑛+𝑟1−𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝐽^𝐴_𝑟𝑥_𝑛+𝜆𝐽^𝐵_𝑟𝑟𝑟_𝑛𝑥_𝑛+𝑟1−𝑟_𝑛𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝐽^𝐵_𝑟𝑥_𝑛≤(𝑟1−𝜆)𝑟_𝑛𝑥_𝑛+𝑟1−𝑟_𝑛𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛𝑟+𝜆𝑟_𝑛𝑥_𝑛+𝑟1−𝑟_𝑛𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑥_𝑛≤𝑟1−𝑟_𝑛max𝐽^𝐴_𝑟𝑛𝑥_𝑛−𝑥_𝑛,𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑥_𝑛⟶0. (41) Hence, we have 𝑥_𝑛−𝑆_𝑟𝑥_𝑛≤𝑥_𝑛−𝑆_𝑟𝑛𝑥_𝑛+𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑥_𝑛𝑦⟶0,𝑛⟶∞,_𝑛−𝑆_𝑟𝑦_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑥_𝑛−𝑆_𝑟𝑥_𝑛+𝑆_𝑟𝑦_𝑛−𝑆_𝑟𝑥_𝑛≤2𝑥_𝑛−𝑦_𝑛+𝑥_𝑛−𝑆_𝑟𝑥_𝑛⟶0,𝑛⟶∞. (42)
Step  4. We show that limsup_𝑛→∞⟨(𝐼−𝑓)𝑝,𝐽(𝑦_𝑛−𝑝)⟩≤0, limsup_𝑛→∞⟨(𝐼−𝑓)𝑝,𝐽(𝑥_𝑛+1−𝑝)⟩≤0.
To prove this, let {𝑦_𝑛𝑗} be a subsequence of {𝑦_𝑛} such that limsup_𝑛→∞𝐼−𝑓𝑝,𝐽𝑦_𝑛−𝑝=lim_𝑗→∞𝐼−𝑓𝑝,𝐽𝑦_𝑛𝑗−𝑝. (43)
By Lemma 4, lim_𝑡→0⁺𝑧_𝑡=𝑝∈𝐹(𝑆_𝑟), where 𝑧_𝑡=𝑡𝑓(𝑧_𝑡)+(1−𝑡)𝑆_𝑟(𝑧_𝑡). Then 𝑧_𝑡𝑛−𝑦_𝑛𝑗=𝑡_𝑛𝑓𝑧_𝑡𝑛−𝑦_𝑛𝑗+1−𝑡_𝑛𝑆_𝑟𝑧_𝑡𝑛−𝑦_𝑛𝑗. (44) For each integer 𝑛≥0, let 𝑡_𝑛∈(0,1) such that 𝑡_𝑛𝑆⟶0,^𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗𝑡_𝑛⟶0,𝑛⟶∞. (45) Using Lemma 1, we get 𝑧_𝑡𝑛−𝑦_𝑛𝑗²≤1−𝑡_𝑛²𝑆_𝑟𝑧_𝑡𝑛−𝑦_𝑛𝑗²+2𝑡_𝑛𝑓𝑧_𝑡𝑛−𝑦_𝑛𝑗,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗≤1−2𝑡_𝑛+𝑡²_𝑛𝑆_𝑟𝑧_𝑡𝑛−𝑆_𝑟𝑦_𝑛𝑗+𝑦_𝑛𝑗−𝑆_𝑟𝑦_𝑛𝑗²+2𝑡_𝑛𝑓𝑧_𝑡𝑛−𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗+2𝑡_𝑛𝑧_𝑡𝑛−𝑦_𝑛𝑗²≤1+𝑡²_𝑛𝑧_𝑡𝑛−𝑦_𝑛𝑗²+1+𝑡_𝑛²𝑆_𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗×2𝑧_𝑡𝑛−𝑦_𝑛𝑗+𝑆_𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗+2𝑡_𝑛𝑓𝑧_𝑡𝑛−𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗, (46) and hence 𝑧_𝑡𝑛−𝑓𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗≤𝑡_𝑛2𝑧_𝑡𝑛−𝑦_𝑛𝑗²+1+𝑡_𝑛²𝑆_𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗2𝑡_𝑛×2𝑧_𝑡𝑛−𝑦_𝑛𝑗+𝑆_𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗. (47) Since {𝑥_𝑛}, {𝑧_𝑡𝑛}, and {𝑆_𝑟𝑦_𝑛} are bounded, then ‖𝑆_𝑟𝑦_𝑛𝑗−𝑦_𝑛𝑗‖/2𝑡_𝑛→0. Therefore, limsup_𝑗→∞𝑧_𝑡𝑛−𝑓𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗≤0. (48) We also know that 𝑝−𝑓𝑧_𝑡𝑛,𝐽𝑦_𝑛𝑗=𝑧−𝑝𝑝−𝑓_𝑡𝑛,𝐽𝑦_𝑛𝑗−𝑝−𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗+𝑝−𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗+𝑧_𝑡𝑛−𝑓𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗. (49) Notice that 𝑧_𝑡𝑛→𝑝, 𝑝∈𝐹(𝑆_𝑟), 𝑛→∞, and 𝐽 is norm to weak* uniformly continuous on bounded subset of 𝐸; then we obtain 𝑝−𝑧_𝑡𝑛,𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗𝑧⟶0,𝑛⟶∞,𝑝−𝑓_𝑡𝑛,𝐽𝑦_𝑛𝑗−𝑝−𝐽𝑧_𝑡𝑛−𝑦_𝑛𝑗⟶0,𝑛⟶∞. (50) From (48), (49), and the two results mentioned above, we have limsup_𝑛→∞𝐼−𝑓𝑝,𝐽𝑦_𝑛−𝑝≤0. (51) Using (22) and the property of 𝐽, we obtain the result that limsup_𝑛→∞𝐼−𝑓𝑝,𝐽𝑦_𝑛−𝑝=limsup_𝑛→∞𝐼−𝑓𝑝,𝐽𝑥_𝑛+1−𝑝≤0. (52)
Step  5. lim_𝑛→∞‖𝑥_𝑛−𝑝‖=0.
Using (16), we have 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+1−𝛼_𝑛𝑆_𝑟𝑥_𝑛−𝑝²=1−𝛼_𝑛𝑆_𝑟𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤1−𝛼_𝑛(1−𝛼)𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝. (53) Applying Lemma 1, we obtain 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑓𝑦_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝²=1−𝛽_𝑛𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑦_𝑛−𝑓𝑝+𝛽_𝑛𝑓𝑝−𝑝²≤1−𝛽_𝑛(1−𝛼)𝑦_𝑛−𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛼_𝑛(1−𝛼)1−𝛽_𝑛(1−𝛼)𝑥_𝑛−𝑝²+2𝛼_𝑛1−𝛽_𝑛(1−𝛼)𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1≤𝛼−𝑝1−_𝑛+𝛽_𝑛(1−𝛼)𝑥_𝑛−𝑝²+𝛼_𝑛𝛽_𝑛(1−𝛼)²𝑥_𝑛−𝑝²−2𝛼_𝑛𝛽_𝑛(1−𝛼)𝑓𝑝−𝑝,𝐽𝑦_𝑛𝛼−𝑝+2_𝑛+𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝−𝐽𝑥_𝑛+1≤𝛼−𝑝1−_𝑛+𝛽_𝑛(1−𝛼)𝑥_𝑛−𝑝²+𝛼_𝑛+𝛽_𝑛𝛼_𝑛𝛽_𝑛𝛼_𝑛+𝛽_𝑛(1−𝛼)²𝐿²+2𝛼_𝑛+𝛽_𝑛𝛼_𝑛𝛽_𝑛𝛼_𝑛+𝛽_𝑛(1−𝛼)𝐿²+2𝛼_𝑛+𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1+𝛼−𝑝_𝑛+𝛽_𝑛2𝛼_𝑛𝛼_𝑛+𝛽_𝑛×𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝−𝐽𝑥_𝑛+1−𝑝, (54) where 𝐿=sup_𝑛≥0{‖𝑥_𝑛−𝑝‖,‖𝑦_𝑛−𝑝‖,‖𝑓(𝑝)−𝑝‖}. Put 𝛾_𝑛=𝛼_𝑛+𝛽_𝑛(),𝜎1−𝛼_𝑛=𝛼_𝑛𝛽_𝑛𝛼_𝑛+𝛽_𝑛(1−𝛼)𝐿²+2𝛼_𝑛𝛽_𝑛𝛼_𝑛+𝛽_𝑛𝐿²+2𝑓𝑝𝑥1−𝛼−𝑝,𝐽_𝑛+1+𝛼−𝑝_𝑛𝛼_𝑛+𝛽_𝑛2×𝑓𝑝𝑦1−𝛼−𝑝,𝐽_𝑛−𝑝−𝐽𝑥_𝑛+1−𝑝. (55) From the conditions (i) and (ii), the result of Step 4, and the facts that 𝛼_𝑛𝛽_𝑛/(𝛼_𝑛+𝛽_𝑛)→0 and ⟨𝑓(𝑝)−𝑝,𝐽(𝑦_𝑛−𝑝)−𝐽(𝑥_𝑛+1−𝑝)⟩→0, we know that 𝛾_𝑛→0, ∑^∞_𝑛=1𝛾_𝑛=+∞ and limsup_𝑛→∞𝜎_𝑛≤0. In view of Lemma 2, (54) reduces to 𝑥_𝑛+1−𝑝≤1−𝛾_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝜎_𝑛; (56) then we know that lim_𝑛→∞𝑥_𝑛−𝑝=0. (57) This completes the proof.

Remark 8. If we modify (16) as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛, (58)(59) or 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑆_𝑟𝑛𝑦_𝑛. (60) Then, imitating the proof of Theorem 7, we can also get the result of Theorem 7. Therefore, from the compare of iterative scheme, the conclusions of [10, 11] are special cases of Theorem 7.

Example 9. Next we study the following optimization problem: min_𝑥∈𝐶ℎ(𝑥),min_𝑥∈𝐶𝑘(𝑥), (61) where 𝐶 is an interior nonempty closed convex subset of a Hilbert space and ℎ,𝑘∶𝐶→𝑅 are two proper convex and lower semicontinuous functionals. To solve optimization problem (61), we will list the following well known results.

Proposition 10 (see [18]). Let 𝜑∶𝐶→𝑅 be a proper convex and lower semicontinuous functional. Then(i)𝜕𝜑∶𝐶→𝐸^∗ (𝜕 denotes the subdifferential in the sense of convex analysis) is a maximal monotone mapping;(ii)𝜕𝜑(𝑥₀)=min_𝑥∈𝐶𝜑(𝑥) if and only if 0∈𝜕𝜑(𝑥₀).

In Hilbert space 𝜕𝜑 is a 𝑚-accretive mapping. Thus 𝐴=𝜕ℎ, 𝐵=𝜕𝑘 are two 𝑚-accretive mappings. Solving optimization problem (61) is equivalent to finding a common zero of 𝐴 and 𝐵.

Let 𝐽^𝐴_𝑟=(𝐼+𝑟𝐴)⁻¹=(𝐼+𝑟𝜕ℎ)⁻¹,𝐽^𝐵_𝑟=(𝐼+𝑟𝐵)⁻¹=(𝐼+𝑟𝜕𝑘)⁻¹,𝑆_𝑟=(1−𝜆)𝐽^𝐴_𝑟+𝜆𝐽^𝐵_𝑟,𝛼_𝑛=𝛽_𝑛=1𝑛,𝑟_𝑛=𝑟𝑛,().𝑛+1𝑟>𝜀 (62) Then the conditions (i), (ii), and (iii) of Theorem 7 are satisfied. For arbitrary 𝑓∈Σ_𝐶 the sequence {𝑥_𝑛} generated by (16) converges strongly to a common zero of 𝐴 and 𝐵, which is also the solution of the optimization problem (60).

Theorem 11. Let 𝐴,𝐵 be two accretive maps in 𝐸 with 𝐴⁻¹0∩𝐵⁻¹0≠∅ and satisfying the following range conditions: 𝐷(𝐴)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵), 𝐷(𝐵)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵) which are convex. Let 𝑓, {𝛼_𝑛}, {𝛽_𝑛}, {𝑟_𝑛}, and {𝑥_𝑛}, {𝑦_𝑛} be the same as those in Theorem 7. Let {𝑥_𝑛} be a sequence generated by (16). Then {𝑥_𝑛} converges strongly to 𝑝∈𝐴⁻¹0∩𝐵⁻¹0, where 𝑝 is the unique solution of the following variational inequality: 𝐼−𝑓𝑝,𝐽𝑝−𝑞≤0,𝑓∈Σ_𝐶,𝑞∈𝐴⁻¹0∩𝐵⁻¹0. (63)

Theorem 12. Let 𝐴,𝐵 be two accretive maps in 𝐸 with 𝐹=𝐴⁻¹0∩𝐵⁻¹0≠∅ and satisfying the following range conditions: 𝐷(𝐴)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵), 𝐷(𝐵)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵) which are convex. Let 𝑓, {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛} be the same as those in Theorem 7. Let 𝑔∶𝐶→𝐶 be a weakly contractive mapping with the function 𝜓. Let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝛼_𝑛𝑔𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛. (64) Then {𝑥_𝑛} converges strongly to 𝑝=𝑄(𝑔(𝑝))∈𝐴⁻¹0∩𝐵⁻¹0, where 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐹.

Proof. Since 𝐸 is a uniformly smooth Banach space, then there is a sunny nonexpansive retraction 𝑄 from 𝐶 onto 𝐹. Then 𝑄∘𝑔 is a weakly contractive mapping of 𝐶 into itself. Indeed, for all 𝑥,𝑦∈𝐶, 𝑄𝑔(𝑥)𝑔𝑦≤𝑔(𝑥)𝑦≤.−𝑄−𝑔𝑥−𝑦−𝜓𝑥−𝑦 (65) Lemma 5 assures that there exists a unique element 𝑝∈𝐶 such that 𝑝=𝑄(𝑔(𝑝)). Such 𝑝∈𝐶 is an element of 𝐴⁻¹0∩𝐵⁻¹0. Now we define a iterative scheme as follows: 𝑧_𝑛=𝛼_𝑛𝑔𝑝+1−𝛼_𝑛𝑆_𝑟𝑛𝑤_𝑛,𝑤_𝑛+1=𝛽_𝑛𝑓𝑧_𝑛+1−𝛽_𝑛𝑧_𝑛. (66) Let 𝑤_𝑛 be the sequence generated by (66). Then Remark 8 (59) assures that 𝑤_𝑛 converges strongly to 𝑝=𝑄(𝑔(𝑝)) as 𝑛→∞. For any 𝑛, we have 𝑥_𝑛+1−𝑤_𝑛+1≤𝛽_𝑛𝑓𝑦_𝑛−𝑓𝑧_𝑛+1−𝛽_𝑛𝑦_𝑛−𝑧_𝑛≤𝑦_𝑛−𝑧_𝑛≤𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑝+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛𝑤_𝑛≤𝛼_𝑛𝑔𝑥_𝑛−𝑔𝑤_𝑛+𝛼_𝑛𝑔𝑤_𝑛𝑝+−𝑔1−𝛼_𝑛𝑥_𝑛−𝑤_𝑛≤𝑥_𝑛−𝑤_𝑛−𝛼_𝑛𝜓𝑥_𝑛−𝑤_𝑛+𝛼_𝑛𝑤_𝑛−𝑝−𝜓𝑤_𝑛≤𝑥−𝑝_𝑛−𝑤_𝑛−𝛼_𝑛𝜓𝑥_𝑛−𝑤_𝑛+𝛼_𝑛𝑤_𝑛−𝑝. (67) Thus, we obtain for 𝑠_𝑛=‖𝑥_𝑛−𝑤_𝑛‖ the following recursive inequality: 𝑠_𝑛+1=𝑠_𝑛−𝛼_𝑛𝜓𝑠_𝑛+𝛼_𝑛𝑤_𝑛−𝑝. (68) Since ‖𝑤_𝑛−𝑝‖→0, it follows from Lemma 6 that lim_𝑛→∞‖𝑥_𝑛−𝑤_𝑛‖=0. Hence lim_𝑛→∞𝑥_𝑛−𝑝≤lim_𝑛→∞𝑥_𝑛−𝑤_𝑛+𝑤_𝑛−𝑝=0. (69) This completes the proof.

In virtue of the weakly contractive mapping 𝑔 being a contraction, using Theorem 12 we may obtain the following.

Corollary 13. Let 𝐴,𝐵 be two accretive maps in 𝐸 with 𝐹=𝐴⁻¹0∩𝐵⁻¹0≠∅ and satisfying the following range conditions: 𝐷(𝐴)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵), 𝐷(𝐵)⊆𝐶⊂𝑅(𝐼+𝑟𝐴)∩𝑅(𝐼+𝑟𝐵) which are convex. Let 𝑓,𝑔∈Σ_𝐶, {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛} be the same as those in Theorem 7. Let {𝑥_𝑛} be a sequence generated by 𝑦_𝑛=𝛼_𝑛𝑔𝑥_𝑛+1−𝛼_𝑛𝑆_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑦_𝑛+1−𝛽_𝑛𝑦_𝑛. (70) Then {𝑥_𝑛} converges strongly to 𝑝=𝑄(𝑔(𝑝))∈𝐴⁻¹0∩𝐵⁻¹0, where 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐹.

Example 14. Next we give an essential example.
Let Ω be an bounded domain in a Euclidean space 𝑅^𝑁 with Lipschitz boundary Γ. Let 𝜙∶Γ×𝑅→𝑅 be a given function shch that for each 𝑥∈Γ(i)𝜙_𝑥=𝜙(𝑥,⋅)∶𝑅→𝑅 is a proper, convex, lower semicontinuous function with 𝜙_𝑥(0)=0.(ii)𝛽_𝑥=𝜕𝜙_𝑥 (subdifferential of 𝜙_𝑥) is the maximal monotone mapping on 𝑅 with 0∈𝛽_𝑥(0) and for each 𝑡∈𝑅, the function 𝑥∈Γ→(𝐼+𝜆𝛽_𝑥)⁻¹(𝑡)∈𝑅 is measurable for 𝜆>0.
Let 𝛼∶𝑅^𝑁→𝑅^𝑁 be a continuous, monotone function such that there exist constants 𝑘₁,𝑘₂ satisfying (i) |𝛼(𝜉)|≤𝑘₁|𝜉| and (ii) ⟨𝛼(𝜉),𝜉⟩≥𝑘₂|𝜉|² for each 𝜉∈𝑅^𝑁.

Definition 15 (see [19]). One first defines a mapping 𝐴^𝛼∶𝐻¹(Ω)→(𝐻¹(Ω))^∗ (𝐻¹(Ω) is a sobolev space) by 𝐴^𝛼𝑢,𝑣=_Ω𝛼grad𝑢,grad𝑣𝑑𝑥 (71) for 𝑢,𝑣∈𝐻¹(Ω). Clearly 𝐴^𝛼 is an everywhere defined, monotone, demicontinuous operator from 𝐻¹(Ω) into (𝐻¹(Ω))^∗. Second one defines an operator 𝐴^𝛼_𝑝∶𝐿^𝑝(Ω)→2^𝐿𝑝(Ω) for 1<𝑝<+∞ as follows.
(i) For 𝑝≥2 one defines the domain of 𝐴^𝛼_𝑝 by 𝐷𝐴^𝛼_𝑝=𝑢∈𝐿^𝑝(Ω)∶thereexistsan𝑓∈𝐿^𝑝(Ω)suchthat𝐴^𝛼𝑢+𝜕Φ(𝑢)∋𝑓. (72) Here Φ(𝑢)=∫_Γ𝜙_𝑥(𝑢|_Γ(𝑥))𝑑Γ(𝑥) is the proper, convex, l.s.c. function (see [19, Lemma 3.1]). For 𝑢∈𝐷(𝐴^𝛼_𝑝) we set 𝐴^𝛼_𝑝(𝑢)=𝑓∈𝐿^𝑝(Ω)∣𝐴^𝛼𝑢+𝜕Φ(𝑢)∋𝑓. (73)
(ii) For 1<𝑝<2, one defines 𝐴^𝛼_𝑝 as the 𝐿^𝑝-closure of 𝐴^𝛼₂ defined in (i) above.

For the operator 𝐴^𝛼_𝑝 one has following results.

Lemma 16 (see [19, Lemma 3.4]). 𝐴^𝛼_𝑝∶𝐿^𝑝(Ω)→2^𝐿𝑝(Ω) is 𝑚-accretive operator (1<𝑝<+∞).

Lemma 17 (see [19, Proposition 3.2]). Let 𝑓∈𝐿^𝑝(Ω),𝑢∈𝐿^𝑝(Ω) such that 𝑓∈𝐴^𝛼_𝑝𝑢. Then(i) div(𝛼(𝑔𝑟𝑎𝑑𝑢))=𝑓, a.e. on Ω and(ii) ⟨𝑛,𝛼(𝑔𝑟𝑎𝑑𝑢)⟩∈𝛽_𝑥(𝑢(𝑥)) for a.e. 𝑥∈Γ.

Lemma 18 (see [19, Proposition 3.3]). Let 𝛽_𝑥≡0 for 𝑥∈Γ. Then (𝐴^𝛼_𝑝)⁻¹0={𝑢∈𝐿^𝑝(Ω)∣𝑢=𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛}.
Clearly for different 𝛼₁,𝛼₂, 𝐴^𝛼1_𝑝,𝐴^𝛼2_𝑝 are two m-accretive operators. The above results show that ∅≠𝐴^𝛼1_𝑝⁻¹0∩𝐴^𝛼2_𝑝⁻¹0=𝑢∈𝐿^𝑝(Ω)∣𝑢=𝑎𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛. (74) For the sake of finding a common zero, Theorems 7, 11, and 12 provided three different iterative algorithms. Therefore the study of a common zero of two accretive operators makes sense.

Remark 19. The results presented in this paper substantially improve and extend the results of Ceng et al. [10] from the following aspects.(1)Theorems 7 and 12 extend the result on the iterative construction of the zero for a single accretive operator to the case of that for common zeros of two accretive operators. If we modify two accretive operators as finite accretive operators, then, imitating the proof of Theorem 7, we can also get the result of Theorem 7.(2)Our results include one or two different viscosity items. Remark 8 shows that the conclusions of Ceng et al. are special cases of this paper.(3)The viscosity item is changed from a contractive mapping 𝑓 to weakly contractive mapping 𝑔 in Theorem 12.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants no. 11101115, 11271106) and the Natural Science Foundation of Hebei Province (A2011201053).