Path Convergence and Approximation of Common Zeroes of a Finite Family of 𝑚-Accretive Mappings in Banach Spaces

Abstract

Let 𝐸 be a real Banach space which is uniformly smooth and uniformly convex. Let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of 𝐸, where 𝑄 is the sunny nonexpansive retraction. If 𝐸 admits weakly sequentially continuous duality mapping 𝑗, path convergence is proved for a nonexpansive mapping 𝑇∶𝐾→𝐾. As an application, we prove strong convergence theorem for common zeroes of a finite family of 𝑚-accretive mappings of 𝐾 to 𝐸. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from 𝐾 to 𝐸 under certain mild conditions.

1. Introduction

Let 𝐸 be a real Banach space with dual 𝐸^∗ and 𝐾 a nonempty, closed and convex subset of 𝐸. A mapping 𝑇∶𝐾→𝐾 is said to be nonexpansive if for all 𝑥,𝑦∈𝐾, we have ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖. (1.1) A point 𝑥∈𝐾 is called a fixed point of 𝑇 if 𝑇𝑥=𝑥. The fixed points set of 𝑇 is the set 𝐹(𝑇)∶={𝑥∈𝐾∶𝑇𝑥=𝑥}.

Construction of fixed points of nonexpansive mappings is an important subject in nonlinear mapping theory and its applications; in particular, in image recovery and signal processing (see, e.g., [1–3]). Many authors have worked extensively on the approximation of fixed points of nonexpansive mappings. For example, the reader can consult the recent monographs of Berinde [4] and Chidume [5].

We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥∶={𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²}, (1.2) where ⟨⋅,⋅⟩ denotes the generalized duality pairing between members of 𝐸 and 𝐸^∗. It is well known that if 𝐸^∗ is strictly convex then 𝐽 is single valued (see, e.g., [5, 6]). In the sequel, we will denote the single-valued normalized duality mapping by 𝑗.

A mapping 𝐴∶𝐷(𝐴)⊆𝐸→𝐸 is called accretive if, for all 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥0. (1.3) By the results of Kato [7], (1.3) is equivalent to ‖𝑥−𝑦‖≤‖𝑥−𝑦+𝑠(𝐴𝑥−𝐴𝑦)‖,∀𝑠>0. (1.4) If 𝐸 is a Hilbert space, accretive mappings are also called monotone. A mapping 𝐴 is called m-accretive if it is accretive and 𝑅(𝐼+𝑟𝐴), range of (𝐼+𝑟𝐴), is 𝐸 for all 𝑟>0; and 𝐴 is said to satisfy the range condition if cl(𝐷(𝐴))⊆𝑅(𝐼+𝑟𝐴),forall𝑟>0, where cl(𝐷(𝐴)) denotes the closure of the domain of 𝐴. 𝐴 is said to be maximal accretive if it is accretive and the inclusion 𝐺(𝐴)⊂𝐺(𝐵), where 𝐺(𝐴) is a graph of 𝐴, with 𝐵 accretive, implies 𝐺(𝐴)=𝐺(𝐵). It is known (see e.g., [8]) that every maximal accretive mapping is 𝑚-accretive and the converse holds if 𝐸 is a Hilbert space. Interest in accretive mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., [9]) that many physically significant problems can be modelled by initial-value problems of the following form: 𝑢^′(𝑡)+𝐴𝑢(𝑡)=0,𝑢(0)=𝑢₀, (1.5) where 𝐴 is an accretive mapping in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or Schrödinger equations. One of the fundamental results in the theory of accretive mappings, due to Browder [10], states that if 𝐴 is locally Lipschitzian and accretive, then 𝐴 is 𝑚-accretive. This result was subsequently generalized by Martin [11] to the continuous accretive mappings. If in (1.5), 𝑢(𝑡) is independent of 𝑡, then (1.5) reduces to 𝐴𝑢=0, (1.6) whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research effects have been devoted, especially within the past 30 years or so, to iterative methods for approximating these equilibrium points.

Closely related to the class of accretive mappings is the class of pseudocontractive mappings. A mapping 𝑇 with domain 𝐷(𝑇) in 𝐸 and range 𝑅(𝑇) in 𝐸 is called pseudocontractive if 𝐴∶=𝐼−𝑇 is accretive. It is then clear that any zero of 𝐴 is a fixed point of 𝑇. Consequently, the study of approximating fixed points of pseudocontractive mappings, which correspond to equilibrium points of the system (1.5), became a flourishing area of research for numerous mathematicians (see, e.g., [12–14] and the references therein).

It is not difficult to deduce from (1.4) that the mapping 𝐴 is accretive if and only if (𝐼+𝑟𝐴)⁻¹,forall𝑟>0, is nonexpansive on the range of (𝐼+𝑟𝐴). Thus, in particular, 𝐽_𝐴∶=(𝐼+𝐴)⁻¹ is nonexpansive and single valued on the range of (𝐼+𝐴). Furthermore, 𝐹(𝐽_𝐴)∶=𝑁(𝐴)∶={𝑥∈𝐷(𝐴)∶𝐴𝑥=0}. It is well known that every nonexpansive mapping is pseudocontractive and the converse does not, however, hold.

Very recently, Yao et al. [15] proved path convergence for a nonexpansive mapping in a real Hilbert space. In particular, they proved the following theorem.

Theorem 1.1 (Yao et al. [15]). Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping with 𝐹(𝑇)≠∅. For 𝑡∈(0,1), let the net {𝑥_𝑡} be generated by 𝑥_𝑡=𝑇𝑃_𝐾[(1−𝑡)𝑥_𝑡], then as 𝑡→0, the net {𝑥_𝑡} converges strongly to a fixed point of 𝑇.

Furthermore, they applied Theorem 1.1 to prove the following theorem.

Theorem 1.2 (Yao et al. [15]). Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping such that 𝐹(𝑇)≠∅. Let {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 be two real sequences in (0,1). For an arbitrary 𝑥₁∈𝐾, let the sequence {𝑥_𝑛}^∞_𝑛=1 be generated iteratively by 𝑦_𝑛=𝑃_𝐾[(1−𝛼_𝑛)𝑥_𝑛𝑥],_𝑛+1=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛,𝑛≥1. (1.7) Suppose that the following conditions are satisfied:(a)lim𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,
then the sequence {𝑥_𝑛}^∞_𝑛=1 generated by (1.7) converges strongly to a fixed point of 𝑇.

Motivated by the results of Yao et al. [15], we proved path convergence for a nonexpansive mapping in a uniformly smooth real Banach space which is also uniformly convex and E admits weakly sequentially continuous duality mapping j. As an application, a strong convergence is proved for common zeroes of a finite family of m-accretive mappings of K to E. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from K to E under certain mild conditions.

2. Preliminaries

Let 𝐸 be a real Banach space and let 𝑆∶={𝑥∈𝐸∶‖𝑥‖=1}. 𝐸 is said to have a Gâteaux differentiable norm (and 𝐸 is called smooth) if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (2.1)

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. Further, 𝐸 is said to be uniformly smooth if the limit exists uniformly for (𝑥,𝑦)∈𝑆×𝑆. The modulus of smoothness of 𝐸 is defined by 𝜌_𝐸(𝜏)∶=sup{‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖=1,‖𝑦‖=𝜏}. (2.2)

𝐸 is equivalently said to be smooth if 𝜌_𝐸(𝜏)>0, for any 𝜏>0.

Let dim 𝐸≥2. The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)∶=inf{1−‖𝑥+𝑦2‖∶‖𝑥‖=‖𝑦‖=1;𝜖=‖𝑥−𝑦‖}. (2.3)

𝐸 is uniformly convex if for any 𝜖∈(0,2], there exists a 𝛿=𝛿(𝜖)>0 such that if 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1,‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜖, then ‖(1/2)(𝑥+𝑦)‖≤1−𝛿. Equivalently, 𝐸 is uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2]. 𝐸 is called strictly convex if for all 𝑥,𝑦∈𝐸,𝑥≠𝑦,‖𝑥‖=‖𝑦‖=1, we have ‖𝜆𝑥+(1−𝜆)𝑦‖<1,forall𝜆∈(0,1). It is known that every uniformly convex Banach space is reflexive.

Let 𝐾⊂𝐸 be closed and convex and 𝑄 be a mapping of 𝐸 onto 𝐾. Then 𝑄 is said to be sunny if 𝑄(𝑄(𝑥)+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥) for all 𝑥∈𝐸 and 𝑡≥0. A mapping 𝑄 of 𝐸 into 𝐸 is said to be a retraction if 𝑄²=𝑄. If a mapping 𝑄 is a retraction, then 𝑄(𝑧)=(𝑧) for every 𝑧∈𝑅(𝑄), where 𝑅(𝑄) is the range of 𝑄. A subset 𝐾 of 𝐸 is said to be a sunny nonexpansive retract of 𝐸 if there exists a sunny nonexpansive retraction of 𝐸 onto 𝐾 and it is said to be a nonexpansive retract of 𝐸 if there exists a nonexpansive retraction of 𝐸 onto 𝐾. If 𝐸=𝐻, the metric projection 𝑃_𝐾 is a sunny nonexpansive retraction from 𝐻 to any closed and convex subset of 𝐻. But this is not true in a general Banach spaces. We note that if 𝐸 is smooth and 𝑄 is retraction of 𝐾 onto 𝐹(𝑇), then 𝑄 is sunny and nonexpansive if and only if for each 𝑥∈𝐾 and 𝑧∈𝐹(𝑇) we have ⟨𝑄𝑥−𝑥,𝐽(𝑄𝑥−𝑧)⟩≤0, (see [16–18] for more details).

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at 𝑝 if whenever {𝑥_𝑛}^∞_𝑛=1 is a sequence in 𝐷(𝑇) such that 𝑥_𝑛⇀𝑥∈𝐷(𝑇) and 𝑇𝑥_𝑛→𝑝 then 𝑇𝑥=𝑝.

Suppose that 𝐽 is single valued. Then, 𝐽 is said to be weakly sequentially continuous if for each {𝑥_𝑛}^∞_𝑛=1⊂𝐸 which converges weakly to 𝑥 implies 𝐽(𝑥_𝑛) converges in weak∗ to 𝐽(𝑥).

We need the following lemmas in the sequel.

Lemma 2.1 (Browder [19], Goebel and Kirk [20]). Let 𝐸 be a real uniformly convex Banach space and let 𝐾 be a nonempty, closed, and convex subset of 𝐸 and 𝑇∶𝐾→𝐾 is a nonexpansive mapping such that 𝐹(𝑇)≠∅, then, 𝐼−𝑇 is demiclosed at zero.

Lemma 2.2 (Suzuki [21]). Let {𝑥_𝑛}^∞_𝑛=1 and {𝑦_𝑛}^∞_𝑛=1 be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛}^∞_𝑛=1 be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥1 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0, then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.3 (Chidume [5], Reich [22]). Let 𝐸 be a uniformly real smooth Banach space, then there exists a nondecreasing continuous function 𝛽∶[0,∞)→[0,∞) with lim_𝑡→0⁺𝛽(𝑠)=0 and 𝛽(𝑐𝑠)≤𝑐𝛽(𝑠) for 𝑐≥1 such that for all 𝑥,𝑦∈𝐸, the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥)⟩+max{‖𝑥‖,1}‖𝑦‖𝛽(‖𝑦‖). (2.4)

Lemma 2.4 (Xu [23]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers which satisfies the following relation: 𝑎_𝑛+1≤(1−𝛼_𝑛)𝑎_𝑛+𝛼_𝑛𝜎_𝑛,𝑛≥1, (2.5) where {𝑎_𝑛}^∞_𝑛=1⊂[0,1] and {𝜎_𝑛}^∞_𝑛=1 is a sequence in ℝ satisfying the following: (i)∑𝛼_𝑛=∞,(ii)limsup𝜎_𝑛≤0, then, 𝑎_𝑛→0 as 𝑛→∞.

Lemma 2.5 (Cioranescu [8]). Let 𝐴 be a continuous accretive mapping defined on a real Banach space 𝐸 with 𝐷(𝐴)=𝐸, then 𝐴 is 𝑚-accretive.

Lemma 2.6 (Zegeye and Shahzad [24]). Let 𝐾 be a nonempty, closed, and convex subset of a real strictly convex Banach space 𝐸. For each 𝑟=1,2,…,𝑁 let 𝐴_𝑟∶𝐾→𝐸 be an 𝑚-accretive mapping such that ⋂^𝑁_𝑟=1𝑁(𝐴_𝑖)≠∅. Let 𝑎₀,𝑎₁,𝑎₂,…,𝑎_𝑁 be real numbers in (0,1) such that ∑^𝑁_𝑖=0𝑎_𝑖=1, and let 𝑆_𝑁∶=𝑎₀𝐼+𝑎₁𝐽_𝐴1+𝑎₂𝐽_𝐴2+⋯+𝑎_𝑁𝐽_𝐴𝑁, with 𝐽_𝐴𝑟∶=(𝐼+𝐴_𝑟)⁻¹, then 𝑆_𝑁 is nonexpansive and 𝐹(𝑆_𝑁)=⋂^𝑁_𝑟=1𝑁(𝐴_𝑟).

3. Path Convergence Theorem

Let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of a uniformly smooth Banach space 𝐸 which is also uniformly convex where 𝑄_𝐾 is the sunny nonexpansive retraction of 𝐸 onto 𝐾. Let 𝑇∶𝐾→𝐾 be nonexpansive. For each 𝑡∈(0,1), we define the mapping 𝑇_𝑡∶𝐾→𝐾 by 𝑇_𝑡𝑥∶=𝑇𝑄_𝐾[(1−𝑡)𝑥]. (3.1) We will show that 𝑇_𝑡 is a contraction.

From (3.1), we have‖𝑇_𝑡𝑥−𝑇_𝑡𝑦‖≤‖𝑄_𝐾(1−𝑡)𝑥−𝑄_𝐾(1−𝑡)𝑦‖≤(1−𝑡)‖𝑥−𝑦‖, (3.2) which implies that 𝑇_𝑡 is a contraction. Therefore, by the Banach contraction mapping principle, there exists a unique fixed point 𝑧_𝑡 of 𝑇_𝑡 in 𝐾. That is,𝑧_𝑡=𝑇𝑄_𝐾[(1−𝑡)𝑧_𝑡]. (3.3)

Next, we prove that {𝑧_𝑡} is bounded. Let 𝑥^∗∈𝐹(𝑇), then using (3.3), we have ‖𝑧_𝑡−𝑥^∗‖=‖𝑇𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑇Q_𝐾𝑥^∗‖≤‖𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑄_𝐾𝑥^∗‖≤‖(1−𝑡)𝑧_𝑡−𝑡𝑥^∗+𝑡𝑥^∗−𝑥^∗‖=‖(1−𝑡)(𝑧_𝑡−𝑥^∗)−𝑡𝑥^∗‖≤(1−𝑡)‖𝑧_𝑡−𝑥^∗‖+𝑡‖𝑥^∗‖. (3.4)

Thus, ‖𝑧_𝑡−𝑥^∗‖≤‖𝑥^∗‖. This implies that {𝑧_𝑡} is bounded.

We next show that ‖𝑧_𝑡−𝑇𝑧_𝑡‖→0 as 𝑡→0, as follows:‖𝑧_𝑡−𝑇𝑧_𝑡‖=‖𝑇𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑇𝑄_𝐾𝑧_𝑡‖≤‖𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑄_𝐾𝑧_𝑡‖≤‖(1−𝑡)𝑧_𝑡−𝑧_𝑡‖≤𝑡‖𝑧_𝑡‖→0,(since𝑡→0). (3.5)

Next, we show that {𝑧_𝑡} is relatively norm compact as 𝑡→0. Let {𝑡_𝑛} be a sequence in (0,1) such that 𝑡_𝑛→0 as 𝑛→∞. Put 𝑧_𝑛∶=𝑧_𝑡𝑛. From (3.5), we obtain that‖𝑧_𝑛−𝑇𝑧_𝑛‖→0. (3.6)

Remark 3.1. Let 𝑥^∗∈𝐹(𝑇) and 𝑟₁>0 be sufficiently large such that 𝑧_𝑡∈𝐵_𝑟1(𝑥^∗)∩𝐾 for each 𝑡∈(0,1), where 𝐵_𝑟1(𝑥^∗)∶={𝑧∈𝐸∶‖𝑧−𝑥^∗‖≤𝑟₁}. For the next theorem, we define 𝐴∶=max{1,2𝑟₁} and assume that the function 𝛽 from Lemma 2.3 satisfies the following condition: 𝛽(𝑠)≤𝑠/𝐴.

Theorem 3.2. Let 𝐸 be a real Banach space which is uniformly smooth and uniformly convex and let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of 𝐸, where 𝑄_𝐾 is the sunny nonexpansive retraction of 𝐸 onto 𝐾. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping with 𝐹(𝑇)≠∅. For each 𝑡∈(0,1), let {𝑧_𝑡} be generated by (3.3), then as 𝑡→0,{𝑧_𝑡} converges strongly to a fixed point of 𝑇 if 𝐸 admits weak sequential continuous duality mapping 𝑗.

Proof. From (3.3), we get for 𝑢∈𝐹(𝑇), ‖𝑧_𝑡−𝑢‖²=‖𝑇𝑄_𝐾[(1−𝑡)𝑧_𝑡]−𝑇𝑄_𝐾𝑢‖²≤‖(1−𝑡)𝑧_𝑡−𝑢‖²=‖𝑧_𝑡−𝑢−𝑡𝑧_𝑡‖²≤‖𝑧_𝑡−𝑢‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑢)⟩+max{‖𝑧_𝑡−𝑢‖,1}𝑡‖𝑧_𝑡‖𝛽(𝑡‖𝑧_𝑡‖)≤‖𝑧_𝑡−𝑢‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑢)⟩+max{2𝑟₁,1}𝑡‖𝑧_𝑡‖𝛽(𝑡‖𝑧_𝑡‖)≤‖𝑧_𝑡−𝑢‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑢)⟩+𝑡²‖𝑧_𝑡‖²=‖𝑧_𝑡−𝑢‖²−2𝑡⟨𝑧_𝑡−𝑢,𝑗(𝑧_𝑡−𝑢)⟩−2𝑡⟨𝑢,𝑗(𝑧_𝑡−𝑢)⟩+𝑡²‖𝑧_𝑡‖²≤‖𝑧_𝑡−𝑢‖²−2𝑡‖𝑧_𝑡−𝑢‖²−2𝑡⟨𝑢,𝑗(𝑧_𝑡−𝑢)⟩+𝑡²‖𝑧_𝑡‖². (3.7) This implies that ‖𝑧_𝑡−𝑢‖²≤⟨𝑢,𝑗(𝑢−𝑧_𝑡𝑡)⟩+2‖𝑧_𝑡‖². (3.8) In particular, ‖𝑧_𝑛−𝑢‖²≤⟨𝑢,𝑗(𝑢−𝑧_𝑛𝑡)⟩+^𝑛2‖𝑧_𝑛‖². (3.9) Since {𝑧_𝑛} is bounded, without loss of generality, we can assume that {𝑧_𝑛} converges weakly to 𝑧^∗. Using the demiclosedness property of (𝐼−𝑇) at zero and the fact that ‖𝑧_𝑛−𝑇𝑧_𝑛‖→0 as 𝑛→∞, we obtain that 𝑧^∗∈𝐹(𝑇). Therefore, we can substitute 𝑧^∗ for 𝑢 in (3.9) to obtain ‖𝑧_𝑛−𝑧^∗‖²≤⟨𝑧^∗,𝑗(𝑧^∗−𝑧_𝑛𝑡)⟩+^𝑛2‖𝑧_𝑛‖². (3.10) Using the fact that 𝑗 is weakly sequentially continuous, we have from the last inequality that {𝑧_𝑛} converges strongly to 𝑧^∗. We now show that {𝑧_𝑡} actually converges to 𝑧^∗. Suppose that {𝑧_𝑡𝑚} converges strongly to 𝑥^∗. Put 𝑧_𝑚∶=𝑧_𝑡𝑚, then since ‖𝑧_𝑚−𝑇𝑧_𝑚‖→0 as 𝑚→∞ and 𝐼−𝑇 is demiclosed at zero, we have that 𝑥^∗∈𝐹(𝑇).
Claim 3 (𝑧^∗=𝑥^∗). Suppose in contradiction that 𝑥^∗≠𝑧^∗. Using (3.3), we obtain using similar argument as above that ‖𝑧_𝑚−𝑧^∗‖²≤⟨𝑧^∗,𝑗(𝑧^∗−𝑧_𝑚𝑡)⟩+^𝑚2‖𝑧_𝑚‖². (3.11) Thus, ‖𝑥^∗−𝑧^∗‖²≤⟨𝑧^∗,𝑗(𝑧^∗−𝑥^∗)⟩. (3.12) Interchanging 𝑥^∗ and 𝑧^∗, we obtain ‖𝑧^∗−𝑥^∗‖²≤⟨𝑥^∗,𝑗(𝑥^∗−𝑧^∗)⟩. (3.13) Adding (3.12) and (3.13) yields 2‖𝑥^∗−𝑧^∗‖²≤‖𝑥^∗−𝑧^∗‖² (3.14) and implies that 𝑥^∗=𝑧^∗. This completes the proof.

Corollary 3.3. Let 𝐸∶=𝑙_𝑝,1<𝑝<∞ and let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of 𝐸, where 𝑄_𝐾 is the sunny nonexpansive retraction of 𝐸 onto 𝐾. Let 𝑇∶𝐾→𝐾 be a nonexpansive mapping with 𝐹(T)≠∅. For each 𝑡∈(0,1), let {𝑧_𝑡} be generated by (3.3) then as 𝑡→0,{𝑧_𝑡} converges strongly to a fixed point of 𝑇.

4. Iterative Methods and Convergence Theorems

Theorem 4.1. Let 𝐸 be a real Banach space which is uniformly smooth and uniformly convex, and let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of 𝐸, where 𝑄_𝐾 is the sunny nonexpansive retraction of 𝐸 onto 𝐾. For each 𝑟=1,2,…,𝑁, let 𝐴_𝑟∶𝐾→𝐸 be an 𝑚-accretive mapping such that ⋂^𝑁_𝑟=1𝑁(𝐴_𝑟)≠∅. Let {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 be two real sequences in (0,1). For an arbitrary 𝑥₁∈𝐾, let the sequence {𝑥_𝑛}^∞_𝑛=1 be generated iteratively by 𝑦_𝑛=𝑄_𝐾[(1−𝛼_𝑛)𝑥_𝑛𝑥],_𝑛+1=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑆_𝑁𝑦_𝑛,𝑛≥1, (4.1) where 𝑆_𝑁∶=𝑎₀𝐼+𝑎₁𝐽_𝐴1+𝑎₂𝐽_𝐴2+⋯+𝑎_𝑁𝐽_𝐴𝑁, with 𝐽_𝐴𝑟∶=(𝐼+𝐴_𝑟)⁻¹,𝑟=1,2,…,𝑁 for 0<𝑎_𝑖<1,𝑖=0,1,2,…,𝑁,∑^𝑁_𝑖=0𝑎_𝑖=1. Suppose that the following conditions are satisfied:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_n=1𝛼_𝑛=∞,(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, then the sequence {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common zero of {𝐴_𝑟}^𝑁_𝑟=1 if 𝐸 admits weakly sequentially continuous duality mapping 𝑗.

Proof. By Lemma 2.6, 𝑆_𝑁 is nonexpansive and 𝐹(𝑆_𝑁)=⋂^𝑁_𝑟=1𝑁(𝐴_𝑟). Now, we first show that the sequence {𝑥_𝑛}^∞_𝑛=1 is bounded. Let 𝑥^∗∈⋂^𝑁_𝑟=1𝑁(𝐴_𝑟)=𝐹(𝑆_𝑁), we have from (4.1) that ‖𝑥_𝑛+1−𝑥^∗‖=‖(1−𝛽_𝑛)(𝑥_𝑛−𝑥^∗)+𝛽_𝑛(𝑆_𝑁𝑦_𝑛−𝑥^∗)‖≤(1−𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖+𝛽_𝑛‖𝑆_𝑁𝑦_𝑛−𝑥^∗‖≤(1−𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖+𝛽_𝑛[(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖+𝛼_𝑛‖𝑥^∗‖]=(1−𝛼_𝑛𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖+𝛼_𝑛𝛽_𝑛‖𝑥^∗‖≤max{‖𝑥_𝑛−𝑥^∗‖,‖𝑥^∗⋮‖}≤max{‖𝑥₁−𝑥^∗‖,‖𝑥^∗‖}. (4.2) Hence, {𝑥_𝑛}^∞_𝑛=1 is bounded and {𝑆_𝑁𝑥_𝑛} is also bounded. Set 𝑢_𝑛=𝑆_𝑁𝑦_𝑛,𝑛≥1. It follows that ‖𝑢_𝑛+1−𝑢_𝑛‖=‖𝑆_𝑁𝑦_𝑛+1−𝑆_𝑁𝑦_𝑛‖≤‖𝑦_𝑛+1−𝑦_𝑛‖≤‖(1−𝛼_𝑛+1)𝑥_𝑛+1−(1−𝛼_𝑛)𝑥_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛‖+𝛼_𝑛+1‖𝑥_𝑛+1‖+𝛼_𝑛‖𝑥_𝑛‖. (4.3) Hence, limsup_𝑛→∞(‖𝑢_𝑛+1−𝑢_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. This together with Lemma 2.2 implies that lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. Thus, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=lim_𝑛→∞𝛽_𝑛‖𝑥_𝑛−𝑢_𝑛‖=0,‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑆_𝑁𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛽_𝑛‖𝑆_𝑁𝑦_𝑛−𝑆_𝑁𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛽_𝑛‖𝑦_𝑛−𝑥_𝑛‖=‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛽_𝑛‖𝑄_𝐾[(1−𝛼_𝑛)𝑥_𝑛]−𝑄_𝐾𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛽_𝑛‖(1−𝛼_𝑛)𝑥_𝑛−𝑥_𝑛‖=‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛼_𝑛𝛽_𝑛‖𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+(1−𝛽_𝑛)‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛‖+𝛼_𝑛‖𝑥_𝑛‖,since𝛽_𝑛∈(0,1), (4.4) that is, ‖𝑥_𝑛−𝑆_𝑁𝑥_𝑛1‖≤𝛽_𝑛{‖𝑥_𝑛−𝑥_𝑛+1‖+𝛼_𝑛‖𝑥_𝑛‖}→0. (4.5) Let {𝑧_𝑡} be defined by (3.3) for 𝑇=𝑆_𝑁, then from Theorem 3.2, 𝑧_𝑡→𝑥^∗∈𝐹(𝑆_𝑁)=⋂^𝑁_𝑟=1𝑁(𝐴_𝑟) as 𝑡→0 (This is guaranteed because 𝐸 admits weakly sequentially continuous duality mapping). Next, we show that limsup_𝑛→∞⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩≤0. (4.6) Now, since {𝑥_𝑛}^∞_𝑛=1 and {𝑧_𝑡} are bounded, there exist 𝑟₁>0,𝑟₂>0 such that 𝑧_𝑡∈𝐵_𝑟1(𝑥^∗)∩𝐾 for each 𝑡∈(0,1) and 𝑥_𝑛∈𝐵_𝑟2(𝑥^∗)∩𝐾 for any 𝑛≥1. Let 𝜌=𝑟₁+𝑟₂, 𝐴=max{2𝜌,1}, and 𝛽(𝑠)≤𝑠/𝐴. Hence, by Lemma 2.3, we have ‖𝑧_𝑡−𝑥_𝑛‖²=‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛+𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖²≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+2⟨𝑆_𝑁𝑥_𝑛−𝑥_𝑛,𝑗(𝑧_𝑡−𝑆_𝑁𝑥_𝑛)⟩+max{‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖,1}‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖𝛽(‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖)≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+2⟨𝑆_𝑁𝑥_𝑛−𝑥_𝑛,𝑗(𝑧_𝑡−𝑆_𝑁𝑥_𝑛)⟩+max{‖𝑧_𝑡−𝑥^∗‖+‖𝑥_𝑛−𝑥^∗‖,1}‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖𝛽(‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖)≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+2⟨𝑆_𝑁𝑥_𝑛−𝑥_𝑛,𝑗(𝑧_𝑡−𝑆_𝑁𝑥_𝑛)⟩+max{2𝜌,1}‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖𝛽(‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖)≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+2⟨𝑆_𝑁𝑥_𝑛−𝑥_𝑛,𝑗(𝑧_𝑡−𝑆_𝑁𝑥_𝑛)⟩+‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖²≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+2‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖+‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖²≤‖𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+𝑀‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖≤‖𝑆_𝑁𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑆_𝑁𝑥_𝑛‖²+𝑀‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖≤‖𝑄_𝐾(1−𝑡)𝑧_𝑡−𝑥_𝑛‖²+𝑀‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖=‖(1−𝑡)𝑧_𝑡−𝑥_𝑛‖²+𝑀‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖=‖𝑧_𝑡−𝑥_𝑛−𝑡𝑧_𝑡‖²+𝑀‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖≤‖𝑧_𝑡−𝑥_𝑛‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩+max{‖𝑧_𝑡−𝑥_𝑛‖,1}𝑡‖𝑧_𝑡‖𝛽(𝑡‖𝑧_𝑡‖)+𝑀‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑧_𝑡−𝑥_𝑛‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩+max{‖𝑧_𝑡−𝑥^∗‖+‖𝑥_𝑛−𝑥^∗‖,1}𝑡‖𝑧_𝑡‖𝛽(𝑡‖𝑧_𝑡‖)+𝑀‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑧_𝑡−𝑥_𝑛‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩+max{2𝜌,1}𝑡‖𝑧_𝑡‖𝛽(𝑡‖𝑧_𝑡‖)+𝑀‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑧_𝑡−𝑥_𝑛‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩+𝑡²‖𝑧_𝑡‖²+𝑀‖𝑇𝑥_𝑛−𝑥_𝑛‖≤‖𝑧_𝑡−𝑥_𝑛‖²−2𝑡⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩+𝑡²𝑀₁+𝑀‖𝑇𝑥_𝑛−𝑥_𝑛‖ (4.7)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) for some 𝑀>0 and 𝑀₁>0. Thus, ⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩≤𝑀₁𝑡/2+(𝑀/2𝑡)‖𝑆_𝑁𝑥_𝑛−𝑥_𝑛‖. Therefore, lim_𝑡→0limsup_𝑛→∞⟨𝑧_𝑡,𝑗(𝑧_𝑡−𝑥_𝑛)⟩≤0. (4.8) Moreover, ⟨−𝑧_𝑡,𝑗(𝑥_𝑛−𝑧_𝑡)⟩=⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑥^∗)⟩+⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑧_𝑡)⟩−⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑥^∗)⟩+⟨𝑥^∗−𝑧_𝑡,𝑗(𝑥_𝑛−𝑧_𝑡)⟩=⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑥^∗)⟩+⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑧_𝑡)−𝑗(𝑥_𝑛−𝑥^∗)⟩+⟨𝑥^∗−𝑧_𝑡,𝑗(𝑥_𝑛−𝑧_𝑡)⟩. (4.9) Since {𝑥_𝑛}^∞_𝑛=1 is bounded, we have that ⟨𝑥^∗−𝑧_𝑡,𝑗(𝑥_𝑛−𝑧_𝑡)⟩→0 as 𝑡→0 and since 𝑗 is norm-to-weak∗ uniformly continuous on bounded sets, we have ⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑧_𝑡)−𝑗(𝑥_𝑛−𝑥^∗)⟩→0 as 𝑡→0. Using (4.8) and (4.9), we obtain limsup_𝑛→∞⟨−𝑥^∗,𝑗(𝑥_𝑛−𝑥^∗)⟩≤0. (4.10) From (4.1), we have ‖𝑦_𝑛−𝑥^∗‖=‖𝑆_𝑁𝑄_𝐾(1−𝛼_𝑛)𝑥_𝑛−𝑥^∗‖²≤‖(1−𝛼_𝑛)𝑥_𝑛−𝑥^∗‖²=‖𝑥_𝑛−𝑥^∗−𝛼_𝑛𝑥^∗‖²≤‖𝑥_𝑛−𝑥^∗‖²−2𝛼_𝑛⟨𝑥_𝑛,𝑗(𝑥_𝑛−𝑥^∗)⟩+max{1,2𝜌}𝛼_𝑛‖𝑥_𝑛‖𝛽(𝛼_𝑛‖𝑥_𝑛‖)≤‖𝑥_𝑛−𝑥^∗‖²−2𝛼_𝑛⟨𝑥_𝑛,𝑗(𝑥_𝑛−𝑥^∗)⟩+𝛼²_𝑛‖𝑥_𝑛‖²=‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥_𝑛−𝑥^∗+𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖²=‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩−2𝛼_𝑛⟨𝑥^∗−𝑥_𝑛,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖²=‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖²−2𝛼_𝑛‖𝑥_𝑛−𝑥^∗‖²=(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²−𝛼_𝑛‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖²≤(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖². (4.11) Also, from (4.1), we obtain ‖𝑥_𝑛+1−𝑥^∗‖²≤(1−𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛽_𝑛‖𝑦_𝑛−𝑥^∗‖²≤(1−𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛽_𝑛[(1−𝛼_𝑛)‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛‖𝑥_𝑛‖²]≤(1−𝛼_𝑛𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖²+2𝛼_𝑛𝛽_𝑛⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛)⟩+𝛼²_𝑛𝛽_𝑛𝑀₂=(1−𝛼_𝑛𝛽_𝑛)‖𝑥_𝑛−𝑥^∗‖²+𝛼_𝑛𝛽_𝑛[2⟨𝑥^∗,𝑗(𝑥^∗−𝑥_𝑛𝛼)⟩+^𝑛𝛽_𝑛𝑀₂], (4.12) where 𝑀₂∶=sup_𝑛≥1‖𝑥_𝑛‖². Using Lemma 2.4, we get that {𝑥_𝑛}^∞_𝑛=1 converges strongly to 𝑥^∗∈𝐹(𝑆_𝑁)=⋂^𝑁_𝑟=1𝑁(𝐴_𝑟). This completes the proof.

If in Theorem 4.1, we consider 𝐾=𝐸, the condition that each 𝐴_𝑟,𝑟=1,2,…,𝑁 is 𝑚-accretive may be replaced with continuity of each 𝐴_𝑟. Thus, we have this theorem.

Theorem 4.2. Let 𝐸 be a real Banach space which is uniformly smooth and also uniformly convex. For each 𝑟=1,2,…,𝑁, let 𝐴_𝑟∶𝐸→𝐸 be a continuous accretive mapping such that ⋂^𝑁_𝑟=1𝑁(𝐴_𝑟)≠∅. Let {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 be two real sequences in (0,1). For an arbitrary 𝑥₁∈𝐸, let the sequence {𝑥_𝑛}^∞_𝑛=1 be generated iteratively by 𝑦_𝑛=(1−𝛼_𝑛)𝑥_𝑛,𝑥_𝑛+1=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑆_𝑁𝑦_𝑛,𝑛≥1, (4.13) where 𝑆_𝑁∶=𝑎₀𝐼+𝑎₁𝐽_𝐴1+𝑎₂𝐽_𝐴2+⋯+𝑎_𝑁𝐽_𝐴𝑁, with 𝐽_𝐴𝑟∶=(𝐼+𝐴_𝑟)⁻¹,𝑟=1,2,…,𝑁 for 0<𝑎_𝑖<1,𝑖=0,1,2,…,𝑁,∑^𝑁_𝑖=0𝑎_𝑖=1. Suppose that the following conditions are satisfied:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,then the sequence {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common zero of {𝐴_𝑟}^𝑁_𝑟=1 if 𝐸 admits weakly sequentially continuous duality mapping 𝑗.

Proof. Take 𝑄_𝐾=𝐼 in Theorem 4.1. By Lemma 2.5, we have that 𝐴_𝑟 is 𝑚-accretive for each 𝑟=1,2,…,𝑁. Then, the result follows from Theorem 4.1.

The following theorems give strong convergence to a common fixed point of a finite family of pseudocontractive mappings.

Theorem 4.3. Let 𝐸 be a real Banach space which is uniformly smooth and uniformly convex, and let 𝐾 be a nonempty, closed, and convex sunny nonexpansive retract of 𝐸, where 𝑄_𝐾 is the sunny nonexpansive retraction of 𝐸 onto 𝐾. For each 𝑟=1,2,…,𝑁, let 𝑇_𝑟∶𝐾→𝐸 be a pseudocontractive mapping such that (𝐼−𝑇_𝑟) is 𝑚-accretive on 𝐾 with ⋂^𝑁_𝑟=1𝐹(𝑇_𝑟)≠∅. Let {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 be two real sequences in (0,1) and 𝐽_𝑇𝑟∶=(2𝐼−𝑇_𝑟)⁻¹ for each 𝑟=1,2,…,𝑁. For an arbitrary 𝑥₁∈𝐾 let sequence {𝑥_𝑛}^∞_𝑛=1 be generated iteratively by 𝑦_𝑛=𝑄_𝐾[(1−𝛼_𝑛)𝑥_𝑛𝑥],_𝑛+1=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑆_𝑁𝑦_𝑛,𝑛≥1, (4.14) where 𝑆_𝑁∶=𝑎₀𝐼+𝑎₁𝐽_𝑇1+𝑎₂𝐽_𝑇2+⋯+𝑎_𝑁𝐽_𝑇𝑁 for 0<𝑎_𝑖<1,𝑖=0,1,2,…,𝑁,∑^𝑁_𝑖=0𝑎_𝑖=1. Suppose that the following conditions are satisfied:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,then the sequence {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common fixed point of {𝑇_𝑟}^𝑁_𝑟=1 if 𝐸 admits weakly sequentially continuous duality mapping 𝑗.

Proof. Let 𝐴_𝑟∶=(𝐼−𝑇_𝑟) for each 𝑟=1,2,…,𝑁. Then, clearly, 𝐹(𝑇_𝑟)=𝑁(𝐴_𝑟) and hence ⋂^𝑁_𝑟=1𝑁(𝐴_𝑟)=⋂^𝑁_𝑟=1𝐹(𝑇_𝑟)≠∅. Furthermore, each 𝐴_𝑟 for 𝑟=1,2,…,𝑁 is 𝑚-accretive. The result follows from Theorem 4.1.

Theorem 4.4. Let 𝐸 be a real Banach space which is uniformly smooth and uniformly convex. For each 𝑟=1,2,…,𝑁, let 𝑇_𝑟∶𝐾→𝐸 be a continuous pseudocontractive mapping on 𝐸 such that ⋂^𝑁_𝑟=1𝐹(𝑇_𝑟)≠∅. Let {𝛼_𝑛}^∞_𝑛=1 and {𝛽_𝑛}^∞_𝑛=1 be two real sequences in (0,1) and 𝐽_𝑇𝑟∶=(2𝐼−𝑇_𝑟)⁻¹ for each 𝑟=1,2,…,𝑁. For arbitrary 𝑥₁∈𝐾 let sequence {𝑥_𝑛}^∞_𝑛=1 be generated iteratively by 𝑦_𝑛=(1−𝛼_𝑛)𝑥_𝑛,𝑥_𝑛+1=(1−𝛽_𝑛)𝑥_𝑛+𝛽_𝑛𝑆_𝑁𝑦_𝑛,𝑛≥1, (4.15) where 𝑆_𝑁∶=𝑎₀𝐼+𝑎₁𝐽_𝑇1+𝑎₂𝐽_𝑇2+⋯+𝑎_𝑁𝐽_𝑇𝑁 for 0<𝑎_𝑖<1,𝑖=0,1,2,…,𝑁,∑^𝑁_𝑖=0𝑎_𝑖=1. Suppose that the following conditions are satisfied:(a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,then the sequence {𝑥_𝑛}^∞_𝑛=1 converges strongly to a common fixed point of {𝑇_𝑟}^𝑁_𝑟=1 if 𝐸 admits weakly sequentially continuous duality mapping 𝑗.

Proof. The proof follows from Theorem 4.2.

Acknowledgments

The authors are extremely grateful to Professor Simeon Reich and the anonymous referee for their valuable comments and useful suggestions which improve the presentation of this paper.