Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Pseudocontractive Mappings

Abstract

It is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa iteration method to a common fixed point of finite family of Lipschitz pseudocontractive mappings. There is no compactness assumption imposed either on C or on T. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.

1. Introduction

Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. The mapping 𝑇∶𝐶→𝐻 is called Lipschitz if there exists 𝐿≥0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.1)

If 𝐿=1, then 𝑇 is called nonexpansive, and if 𝐿<1, then 𝑇 is called a contraction. It follows from (1.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.

A mapping 𝑇∶𝐶→𝐻 is called 𝛼-strictly pseudocontractive [1] if for all 𝑥,𝑦∈𝐶 there exists 𝛼∈[0,1) such that 𝑥−𝑦,𝑇𝑥−𝑇𝑦≤𝑥−𝑦²−𝛼(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦². (1.2)

A mapping 𝑇 is called pseudocontractive if 𝑥−𝑦,𝑇𝑥−𝑇𝑦≤𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (1.3)

We note that (1.2) and (1.3) can be equivalently written as 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝛼(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶.𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶, (1.4)(1.5)respectively.

We observe from (1.4) and (1.5) that every nonexpansive mapping is 𝛼-strict pseudocontractive mapping and every 𝛼-strict pseudocontractive mapping is pseudocontractive mapping, and hence class of pseudocontractive mappings is a more general class of mappings. Furthermore, pseudocontractive mappings are related with the important class of nonlinear monotone mappings, where a mapping 𝐴 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝐻 is called monotone if the inequality 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥0, (1.6) holds for every 𝑥,𝑦∈𝐷(𝐴). We note that 𝑇 is pseudocontractive if and only if 𝐴∶=𝐼−𝑇 is monotone, and hence a fixed point of 𝑇, 𝐹(𝑇)∶={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥} is a zero of 𝐴, 𝑁(𝐴)∶={𝑥∈𝐷(𝐴)∶𝐴𝑥=0}. It is now well known (see, e.g., [2]) that if 𝐴 is monotone, then the solutions of the equation 𝐴𝑥=0 correspond to the equilibrium points of some evolution systems. Consequently, many researchers have made efforts to obtain iterative methods for approximating fixed points of 𝑇, when 𝑇 is pseudocontractive (see, e.g., [3–10] and the references contained therein).

Let 𝐶 be a closed subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a contraction. Then the Picard iteration method given by 𝑥₀∈𝐶,𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛≥1, (1.7) converges to the unique fixed point of 𝑇. However, this Picard iteration method may not always converge to a fixed point of 𝑇, when 𝑇 is nonexpansive mapping. We can take, for example, 𝑇 to be the anticlockwise rotation of the unit disk in ℝ² (with the Euclidean norm) about the origin of coordinate of an angle, say, 𝜃.

The scheme that has been used to approximate fixed points of nonexpansive mappings is the Mann iteration method [5] given by 𝑥₀∈𝐶,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.8) where {𝛼_𝑛} is a real sequence in the interval (0,1) satisfying certain conditions. But it is worth mentioning that the Mann iteration process does not always converge strongly to a fixed point of nonexpansive mapping 𝑇. One has to impose compactness assumption on 𝐶 (e.g., 𝐶 is compact) or on 𝑇 (e.g., 𝑇 is semicompact) to get strong convergence of Mann iteration method to a fixed point of nonexpansive self-map 𝑇 (see, e.g., [11, 12]).

We also note that efforts to approximate a fixed point of a Lipschitz pseudocontractive mapping defined even on a compact convex subset of a Hilbert space by Mann iteration method proved abortive. One can see an example of a Lipschitz pseudocontractive self-map of a compact convex subset of a Hilbert space with a unique fixed point for which no Mann sequence converges by Chidume and Mutangadura [13]. This leads now to our next concern.

Can we construct an iterative sequence for approximating fixed point of the Lipschitz pseudocontractive mappings?

In 1974, Ishikawa [14] introduced an iteration process which converges to a fixed point of Lipschitz pseudocontractive self-map 𝑇 of 𝐶, when 𝐶 is compact. In fact, he proved the following theorem.

Theorem I. If 𝐶 is a compact convex subset of a Hilbert space 𝐻,𝑇∶𝐶↦𝐶 is a Lipschitz pseudocontractive mapping and 𝑥₀ is any point of 𝐶, then the sequence {𝑥_𝑛}_𝑛≥0 converges strongly to a fixed point of 𝑇, where {𝑥_𝑛} is defined iteratively for each integer 𝑛≥0 by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛, (1.9) here {𝛼_𝑛},{𝛽_𝑛} are sequences of positive numbers satisfying the conditions (i)0≤𝛼_𝑛≤𝛽_𝑛<1,(ii)lim_𝑛→∞𝛽_𝑛=0,(iii)_𝑛≥0𝛼_𝑛𝛽_𝑛=∞. (1.10)

We observe that Theorem I imposes compactness assumption on 𝐶, and it is still an open problem whether or not scheme (1.9), known as the Ishikawa iterative method, can be used to approximate fixed points of Lipschitz pseudocontractive mappings without compactness assumption on 𝐶 or on 𝑇.

In order to obtain a strong convergence theorem for pseudocontractive mappings without the compactness assumption, Zhou [15] established the hybrid Ishikawa algorithm for Lipschitz pseudocontractive mappings as follows: 𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑧_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑧_𝑛−𝑧²≤𝑥_𝑛−𝑧²−𝛼_𝑛𝛽_𝑛1−2𝛼_𝑛−𝐿²𝛼_𝑛²𝑥_𝑛−𝑇𝑥_𝑛²,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1. (1.11)

He proved that the sequence {𝑥_𝑛} defined by (1.11) converges strongly to 𝑃_𝐹(𝑇)𝑥₀, where 𝑃_𝐶 is the metric projection from 𝐻 into 𝐶.

Recently, several authors (see, e.g., [16–18]) also used the hybrid Mann and hybrid Ishikawa algorithm methods to obtain strong convergence to a fixed point of Lipschitz pseudocontractive mappings. But it is worth mentioning that the hybrid schemes are not easy to compute. They involve computation of 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1.

Another iteration scheme was introduced and studied by Chidume and Zegeye [19] with which they approximated fixed point of Lipschitz pseudocontractive mapping in a more general real Banach space.

Let 𝐾 be a convex nonempty subset of real Banach space 𝐸, and let 𝑇∶𝐾→𝐾 be a mapping. From arbitrary 𝑥₁∈𝐾, define {𝑥_𝑛}_𝑛≥1 by 𝑥_𝑛+1=1−𝜆_𝑛𝑥_𝑛+𝜆_𝑛𝑇𝑥_𝑛−𝜆_𝑛𝜃_𝑛𝑥_𝑛−𝑥₁,𝑛∈ℕ, (1.12) where {𝜆_𝑛}_𝑛≥1 and {𝜃_𝑛}_𝑛≥1 are real sequences in (0,1) satisfying the following conditions: (𝑖)lim_𝑛→∞𝜃_𝑛=0; (𝑖𝑖)𝜆_𝑛=𝑜(𝜃_𝑛); (𝑖𝑖𝑖)∑^∞_𝑛=1𝜆_𝑛𝜃_𝑛=∞; (𝑖𝑣)lim_𝑛→∞((𝜃_𝑛−1/𝜃_𝑛−1)/𝜆_𝑛𝜃_𝑛)=0, 𝜆_𝑛(1+𝜃_𝑛)<1. Examples of real sequences which satisfy these conditions are 𝜆_𝑛=1/(𝑛+1)^𝑎 and 𝜃_𝑛=1/(𝑛+1)^𝑏, where 0<𝑏<𝑎 and 𝑎+𝑏<1. They proved the following theorem.

Theorem CZ. Let 𝐶 be a nonempty closed convex subset of a reflexive real Banach space 𝐸 with a uniformly Gâteaux differentiable norm. Let 𝑇∶𝐶→𝐶 be a Lipschitz pseudocontractive mapping with Lipschitz constant 𝐿>0 and 𝐹(𝑇)≠∅. Suppose every closed convex and bounded subset of 𝐾 has the fixed point property for nonexpansive self-mappings. Let a sequence {𝑥_𝑛}_𝑛≥1 be generated iteratively by (1.12). Then {𝑥_𝑛}_𝑛≥1 converges strongly to a fixed point of 𝑇.

Theorem CZ solves the open problem of approximating fixed point of Lipschitz pseudocontractive mappings that has been in the air for many years. However, it is still an open problem whether or not this scheme can be used to approximate a common fixed point of a family of Lipschitz pseudocontractive mappings. Moreover, we observe that the conditions on the real sequences {𝜃_𝑛} and {𝜆_𝑛} excluded the natural choice, 𝜃_𝑛=1/(𝑛+1) and 𝜆_𝑛=1/(𝑛+1).

Our concern now is the following: can we construct an iterative sequence for a common fixed point of a family of Lipschitz pseudocontractive mappings?

For a sequence {𝛼_𝑛} of real numbers in [0,1] and an arbitrary 𝑢∈𝐶, let the sequence {𝑥_𝑛} in 𝐶 be iteratively defined by 𝑥₀∈𝐶∶𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥1. (1.13) The recursion formula (1.13) known as Halpern scheme was first introduced in 1967 by Halpern [20] in the framework of Hilbert spaces. He proved that {𝑥_𝑛} convergs strongly to a fixed point of nonexpansive self-mapping 𝑇 of 𝐶.

Recently, considerable research efforts have been devoted to developing iterative methods for approximating a common fixed point of a family of several nonlinear mappings (see, e.g., [4, 21, 22]). In 1996, Bauschke [3] introduced the following Halpern-type iterative process for approximating a common fixed point for a finite family of 𝑁 nonexpansive self-mappings. In fact, he proved the following theorem.

Theorem B. Let C be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝑇₁,𝑇₂,…,𝑇_𝑁 be a finite family of nonexpansive mappings of 𝐶 into itself with 𝐹∶=𝐹(𝑇₁𝑇_𝑁⋯𝑇₂)=⋯=𝐹(𝑇_𝑁−2⋯𝑇₁𝑇_𝑁)≠∅. Let {𝛼_𝑛} be a real sequence in [0,1] which satisfies certain mild conditions. Given points, 𝑥₀∈𝐶, let {𝑥_𝑛} be generated by 𝑥_𝑛+1=𝛼_𝑛+1𝑢+1−𝛼_𝑛+1𝑇_𝑛+1𝑥_𝑛,𝑛≥0, (1.14) where 𝑇_𝑛=𝑇_{𝑛(mod𝑁)}. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑢, where 𝑃_𝐹𝑢∶𝐻→𝐹 is the metric projection.

But it is worth mentioning that it is still an open problem whether or not this scheme can be used to approximate a common fixed points of Lipschitz pseudocontractive mappings?

In 2008, Zhou [22] studied weak convergence of an implicit scheme to a common fixed point of finite family of pseudocontractive mappings. More precisely, he proved the following theorem.

Theorem Z. Let 𝐸 be a real uniformly convex Banach space with a Frêchet differentiable norm. Let 𝐶 be a closed convex subset of 𝐸, and let {𝑇_𝑖}^𝑟_𝑖=1 be a finite family of Lipschitzian pseudocontractive self-mappings of 𝐶 such that 𝐹∶=∩^𝑟_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝑥_𝑛} be defined by 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑛≥1, (1.15) where 𝑇_𝑛=𝑇_{𝑛(mod𝑟)}. If {𝛼_𝑛} is chosen so that 𝛼_𝑛∈(0,1) with limsup_𝑛→∞𝛼_𝑛<1, then {𝑥_𝑛} converges weakly to a common fixed point of the family {𝑇_𝑖}^𝑟_𝑖=1.

Here, we remark that the scheme in Theorem Z is implicit, and the convergence is weak convergence.

More recently, Zegeye et al. [23] proved the following strong convergence of Ishikawa iterative process for a common fixed point of finite family of Lipschitz pseudocontractive mappings.

Theorem ZSA (see [23]). Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶,𝑖=1,2,…,𝑁, be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constants 𝐿_𝑖, for 𝑖=1,2,…,𝑁, respectively. Assume that the interior of 𝐹∶=∩^𝑛_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated from an arbitrary 𝑥₀∈𝐸 by 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇_𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇_𝑛𝑦_𝑛,𝑛≥1, (1.16) where 𝑇_𝑛∶=𝑇_{𝑛(mod𝑁)} and {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) satisfying certain appropriate conditions. Then, {𝑥_𝑛} converges strongly to a common fixed point of {𝑇₁,𝑇₂,…,𝑇_𝑁}.

From Theorem ZSA, we observe that the assumption that the interior of 𝐹(𝑇)  is nonempty is severe restriction.

Motivated by Halpern [20] and Zegeye et al. [23], it is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa algorithm (3.3) to a common fixed point of a finite family of Lipschitz pseudocontractive mappings. No compactness assumption is imposed either on one of the mappings or on 𝐶. The assumption that interior of 𝐹(𝑇)   is nonempty is dispensed with. Moreover, computation of closed and convex set 𝐶_𝑛 for each 𝑛≥1 is not required. The results obtained in this paper improve and extend the results of Theorems I and ZSA, Zhou [15], Yao et al. [17], and Tang et al. [16].

2. Preliminaries

In what follows we will make use of the following lemmas.

Lemma 2.1. Let 𝐻 be a real Hilbert space. Then for any given 𝑥,𝑦∈𝐸, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦. (2.1)

Lemma 2.2 (see [24]). Let 𝐶 be a convex subset of a real Hilbert space 𝐻. Let 𝑥∈𝐻. Then 𝑥₀=𝑃_𝐶𝑥 if and only if 𝑧−𝑥₀,𝑥−𝑥₀≤0,∀𝑧∈𝐶. (2.2)

Lemma 2.3 (see [25]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛽_𝑛𝑎_𝑛+𝛽_𝑛𝛿_𝑛,𝑛≥𝑛₀, (2.3) where {𝛽_𝑛}⊂(0,1) and {𝛿_𝑛}⊂𝑅 satisfying the following conditions: lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.4 (see [18]). Let 𝐻 be a real Hilbert space, let 𝐶 be a closed convex subset of 𝐻, and let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping; then(i)𝐹(𝑇) is closed convex subset of 𝐶;  (ii)(𝐼−𝑇) is demiclosed at zero; that is, if {𝑥_𝑛} is a sequence in 𝐶 such that x_𝑛⇀𝑥 and 𝑇𝑥_𝑛−𝑥_𝑛→0, as 𝑛→∞, then 𝑥=𝑇(𝑥).

Lemma 2.5 (see [26]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈𝑁. Then there exists a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞,and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈𝑁: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1. (2.4) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+1}.

Lemma 2.6 (see [27]). Let 𝐻 be a real Hilbert space. Then for all 𝑥_𝑖∈𝐻 and 𝛼_𝑖∈[0,1] for 𝑖=1,2,…,𝑛 such that 𝛼₁+𝛼₂+⋯+𝛼_𝑛=1 the following equality holds: 𝛼₀𝑥₀+𝛼₁𝑥₁+⋯+𝛼_𝑛𝑥_𝑛²=^𝑛_𝑖=0𝛼_𝑖𝑥_𝑖²−_{0≤𝑖,𝑗≤𝑛}𝛼_𝑖𝛼_𝑗𝑥_𝑖−𝑥_𝑗². (2.5)

3. Main Result

We now prove the following lemma and theorems.

Lemma 3.1. Let 𝐶 be a nonempty convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶,𝑖=1,2,…,𝑁, be a finite family of Lipschitz pseudocontractive mappings with constants 𝐿_𝑖, respectively. Let 𝑆=𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑁𝑇_𝑁, where 𝜃₁+𝜃₂+⋯+𝜃_𝑁=1. Then 𝑆 is Lipschitz pseudocontractive mapping on 𝐶.

Proof. Let 𝑥,𝑦∈𝐶. Then 𝑆𝑥−𝑆𝑦,𝑥−𝑦=𝜃₁𝑇₁𝑥−𝑇₁𝑦,𝑥−𝑦+𝜃₂𝑇₂𝑥−𝑇₂𝑦,𝑥−𝑦+⋯+𝜃_𝑁𝑇_𝑁𝑥−𝑇_𝑁𝑦,𝑥−𝑦≤𝜃₁𝑥−𝑦²+𝜃₂𝑥−𝑦²+⋯+𝜃_𝑁𝑥−𝑦²=𝑥−𝑦². (3.1)
Hence 𝑆 is pseudocontractive. Moreover, since 𝑆𝑥−𝑆𝑦=𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑁𝑇_𝑁𝑥−𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑁𝑇_𝑁𝑦≤𝜃₁𝑇₁𝑥−𝑇₁𝑦+𝜃₂𝑇₂𝑥−𝑇₂𝑦+⋯+𝜃_𝑁𝑇_𝑁𝑥−𝑇_𝑁𝑦≤𝐿𝑥−𝑦, (3.2) where 𝐿∶=max{𝐿_𝑖∶𝑖=1,2,…,𝑁}, we get that 𝑆 is 𝐿-Lipschitz. The proof is complete.

Let {𝑇_𝑖∶𝑖=1,2,…,𝑁} be a finite family of pseudocontractive mappings. The family is said to satisfy condition (𝐻) if ⟨𝑇_𝑖𝑥−𝑥,𝑇_𝑗𝑥−𝑥⟩≥0, for 𝑖,𝑗∈{1,2,…,𝑁}.

Theorem 3.2. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶,𝑖=1,2,…,𝑁 be a finite family of Lipschitz pseudocontractive mappings with Lipschitz constants 𝐿_𝑖, respectively, satisfying condition (𝐻). Assume that 𝐹∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let a sequence {𝑥_𝑛} be a sequence generated from an arbitrary 𝑥₁=𝑤∈𝐶 by 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛, (3.3) where 𝑆_𝑛∶=𝜃_𝑛,1𝑇₁+𝜃_𝑛,2𝑇₂+⋯+𝜃_𝑛,𝑁𝑇_𝑁, for {𝜃_𝑛,𝑖}⊆[𝑎,𝑏]⊂(0,1) such that 𝜃_𝑛,1+𝜃_𝑛,2+⋯+𝜃_𝑛,𝑁=1, for all 𝑛≥1 and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}⊂(0,1) satisfying the following conditions: (𝑖) 0≤𝛼_𝑛≤𝑐<1, for all 𝑛≥1 such that lim_𝑛→∞𝛼_𝑛=0 and ∑𝛼_𝑛=∞; (𝑖𝑖) 0<𝛼≤𝛾_𝑛≤𝛽_𝑛≤𝛽<1/[(1+𝐿²)+1], for all 𝑛≥1, for 𝐿∶=max{𝐿_𝑖∶𝑖=1,2,…,𝑁}. Then, {𝑥_𝑛} converges strongly to a common fixed point of {𝑇₁,𝑇₂,…𝑇_𝑁} nearest to 𝑥₁=𝑤.

Proof. Let 𝑝=𝑃_𝐹𝑤. Then from (3.3), Lemma 2.6, (1.5), and Lemma 3.1 we have the following: 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑤+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛−𝑝²=𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑝²+1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²=𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛+𝑦_𝑛−𝑝²+𝑦_𝑛−𝑆_𝑛𝑦_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²=𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑦_𝑛−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑦_𝑛−𝑆_𝑛𝑦_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛼_𝑛1−𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛². (3.4)

In addition, we have that 𝑦_𝑛−𝑆_𝑛𝑦_𝑛²=1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛+𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²=1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²≤1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽_𝑛𝐿²𝑥_𝑛−𝑦_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²=1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽³_𝑛𝐿²𝑥_𝑛−𝑆_𝑛𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²=1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽_𝑛𝐿²𝛽²_𝑛+𝛽_𝑛−1𝑥_𝑛−𝑆_𝑛𝑥_𝑛²,||𝑦_𝑛−𝑝||²=1−𝛽_𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝²=1−𝛽_𝑛𝑥_𝑛−𝑝²+𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²≤1−𝛽_𝑛𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑆_𝑛𝑥_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²=𝑥_𝑛−𝑝²+𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛². (3.5)(3.6) Substituting (3.5) and (3.6) into (3.4) we obtain that 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑝²+𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛𝛾_𝑛1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽_𝑛𝐿²𝛽²_𝑛+𝛽_𝑛×𝑥−1_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²+1−𝛼_𝑛𝛾_𝑛𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛𝛾_𝑛𝛽_𝑛𝐿²𝛽²_𝑛+𝛽_𝑛×𝑥−1_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛1−𝛽_𝑛𝛾_𝑛−1−𝛼_𝑛1−𝛾_𝑛𝛾_𝑛×𝑥_𝑛−𝑆_𝑛𝑦_𝑛²=𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾_𝑛𝛽_𝑛1−2𝛽_𝑛−𝐿²𝛽²_𝑛×𝑥_𝑛−𝑆_𝑛x_𝑛²+1−𝛼_𝑛𝛾_𝑛𝛾_𝑛−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²=𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝛾_𝑛+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝². (3.7) Since from (𝑖𝑖), we have that (𝛾_𝑛−𝛽_𝑛)≤0 and 1−2𝛽_𝑛−𝐿²𝛽²_𝑛≥1−2𝛽−𝐿²𝛽²>0 for all 𝑛≥1, (3.7) implies that 𝑥_𝑛+1−𝑝²≤𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾_𝑛𝛽_𝑛1−2𝛽−𝐿²𝛽²𝑥_𝑛−𝑆_𝑛𝑥_𝑛²≤𝛼_𝑛𝑤−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝². (3.8) Thus, by induction, 𝑥_𝑛+1−𝑝²≤max𝑥₁−𝑝²,𝑤−𝑝²,∀𝑛≥1, (3.9) which implies that {𝑥_𝑛} and hence {𝑦_𝑛} are bounded.

Furthermore, from (3.3), Lemma 2.1, and following the methods used in (3.7) we get that 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑤−𝑝+1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛−𝑝²≤1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1=−𝑝1−𝛼_𝑛𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑝²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1≤−𝑝1−𝛼_𝑛𝛾_𝑛𝑦_𝑛−𝑝²+𝑦_𝑛−𝑆_𝑛𝑦_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1=−𝑝1−𝛼_𝑛𝛾_𝑛𝑦_𝑛−𝑝²+1−𝛼_𝑛𝛾_𝑛𝑦_𝑛−𝑆_𝑛𝑦_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²≤1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑝²+𝛽²_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛𝛾_𝑛×1−𝛽_𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛²+𝛽_𝑛𝐿²𝛽²_𝑛+𝛽_𝑛−1𝑥_𝑛−𝑆_𝑛𝑥_𝑛²+1−𝛼_𝑛1−𝛾_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛1−𝛾_𝑛1−𝛼_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1≤−𝑝1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−−𝑝1−𝛼_𝑛𝛾_𝑛𝛽_𝑛1−2𝛽_𝑛−𝛽²_𝑛𝐿²𝑆_𝑛𝑦_𝑛−𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−()𝛼−𝑝1−𝑐²1−2𝛽−𝛽²𝐿²𝑆_𝑛𝑥_𝑛−𝑥_𝑛²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−𝑝. (3.10)

On the other hand, using Lemma 2.6 and condition (𝐻), we get that ||𝑥_𝑛−𝑆_𝑛𝑥_𝑛||²=𝑥_𝑛−𝜃_𝑛,1𝑇₁+𝜃_𝑛,2𝑇₂+⋯+𝜃_𝑛,𝑁𝑇_𝑁𝑥_𝑛²=𝜃_𝑛,1𝑥_𝑛−𝑇₁𝑥_𝑛+𝜃_𝑛,2𝑥_𝑛−𝑇₂𝑥_𝑛+⋯+𝜃_𝑛,𝑁𝑥_𝑛−𝑇_𝑁𝑥_𝑛²=𝜃_𝑛,1𝑥_𝑛−𝑇₁𝑥_𝑛²+𝜃_𝑛,2𝑥_𝑛−𝑇₂𝑥_𝑛²+⋯+𝜃_𝑛,𝑁𝑥_𝑛−𝑇_𝑁𝑥_𝑛²−_{1≤𝑖,𝑗≤𝑁}𝜃_𝑛,𝑖𝜃_𝑛,𝑗𝑇_𝑖𝑥_𝑛−𝑇_𝑗𝑥_𝑛²−_{1≤𝑖,𝑗≤𝑁,𝑖≠𝑗}𝜃_𝑛,𝑖𝜃_𝑛,𝑗𝑇_𝑖𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑇_𝑗𝑥_𝑛²=𝜃_𝑛,11−𝜃_𝑛,2−𝜃_𝑛,3−⋯−𝜃_𝑛,𝑁𝑥_𝑛−𝑇₁𝑥_𝑛²+𝜃_𝑛,21−𝜃_𝑛,1−𝜃_𝑛,3−𝜃_𝑛,4−⋯−𝜃_𝑛,𝑁𝑥_𝑛−𝑇₂𝑥_𝑛²…+𝜃_𝑛,𝑁1−𝜃_𝑛,1−𝜃_𝑛,2−𝜃_𝑛,4−⋯−𝜃_𝑛,𝑁𝑥_𝑛−𝑇_𝑁𝑥_𝑛²≥𝜃_𝑛,1𝑥_𝑛−𝑇₁𝑥_𝑛²+𝜃_𝑛,2𝑥_𝑛−𝑇₂𝑥_𝑛²+⋯+𝜃_𝑛,𝑁𝑥_𝑛−𝑇_𝑁𝑥_𝑛². (3.11) Thus, substituting (3.11) into (3.10) we obtain that𝑥_𝑛+1−𝑝²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−()𝛼−𝑝1−𝑐²1−2𝛽−𝛽²𝐿²×𝜃_𝑛,11−𝜃_𝑛,2−𝜃_𝑛,3−⋯−𝜃_𝑛,𝑁𝑥_𝑛−𝑇₁𝑥_𝑛²+𝜃_𝑛,21−𝜃_𝑛,1−𝜃_𝑛,3−𝜃_𝑛,4−⋯−𝜃_𝑛,𝑁𝑥_𝑛−𝑇₂𝑥_𝑛²…+𝜃_𝑛,𝑁1−𝜃_𝑛,1−𝜃_𝑛,2−𝜃_𝑛,4−⋯−𝜃_{𝑛,𝑁−1}𝑥_𝑛−𝑇_𝑁𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−𝑝. (3.12)(3.13) Now, we consider the following two cases.

Case  1. Suppose that there exists 𝑛₀∈𝑁 such that {||𝑥_𝑛−𝑝||} is nonincreasing. Then, we get that {||𝑥_𝑛−𝑝||)} is convergent. Thus, from (3.12) and the fact that 𝛼_𝑛→0, as 𝑛→∞, we have that 𝑥_𝑛−𝑇_𝑖𝑥_𝑛→0,as𝑛→∞, (3.14) for each 𝑖=1,2,…,𝑁. Let 𝑧_𝑛=𝛾_𝑛𝑆_𝑛𝑦_𝑛+(1−𝛾_𝑛)𝑥_𝑛. Then from (3.3) we obtain that 𝑥_𝑛+1−𝑧_𝑛=𝛼_𝑛𝑤−𝑧_𝑛→0,as𝑛→∞. (3.15)

Furthermore, from (3.3) and (3.14) we get that 𝑦_𝑛−𝑥_𝑛=𝛽_𝑛𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝜃_𝑛,1𝑇₁𝑥_𝑛−𝑥_𝑛+𝜃_𝑛,2𝑇₂x_𝑛−𝑥_𝑛+⋯+𝜃_𝑛,𝑁𝑇_𝑁𝑥_𝑛−𝑥_𝑛→0, (3.16) as 𝑛→∞, and hence (3.16) and the fact that 𝑆_𝑛 is 𝐿-Lipschitz imply that 𝑧_𝑛−𝑥_𝑛=𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑥_𝑛=𝛾_𝑛𝑆_𝑛𝑦_𝑛−𝑆_𝑛𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝑥_𝑛≤𝛾_𝑛𝐿𝑦_𝑛−𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑥_𝑛−𝑥_𝑛→0,as𝑛→∞. (3.17) Now, (3.15) and (3.17) imply that 𝑥_𝑛+1−𝑥_𝑛→0,as𝑛→∞. (3.18)

Moreover, since {𝑥_𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑥_𝑛𝑖+1} of {𝑥_𝑛} such that 𝑥_𝑛𝑖+1⇀𝑧 and limsup_𝑛→∞⟨𝑤−𝑝,𝑥_𝑛+1−𝑝⟩=lim_𝑖→∞⟨𝑤−𝑝,𝑥_𝑛𝑖+1−𝑝⟩. This implies from (3.18) that 𝑥_𝑛𝑖⇀𝑧. Then, from (3.14) and Lemma 2.4 we have that 𝑧∈𝐹(𝑇_𝑖), for each 𝑖=1,2,…,𝑁. Hence, 𝑧∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖). Therefore, by Lemma 2.2, we immediately obtain that limsup_𝑛→∞𝑤−𝑝,𝑥_𝑛+1−𝑝=lim_𝑖→∞𝑤−𝑝,𝑥_𝑛𝑖+1=−𝑝𝑤−𝑝,𝑧−𝑝≤0. (3.19) Then, since from (3.13) we have that 𝑥_𝑛+1−𝑝²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−𝑝. (3.20)

It follows from (3.20), (3.19), and Lemma 2.3 that ||𝑥_𝑛−𝑝||→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑝.

Case  2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑥_𝑛𝑖−𝑝<𝑥_𝑛𝑖+1−𝑝, (3.21) for all 𝑖∈𝑁. Then, by Lemma 2.5, there exists a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞, ||𝑥_𝑚𝑘−𝑝||≤||𝑥_𝑚𝑘+1−𝑝|| and ||𝑥_𝑘−𝑝||≤||𝑥_𝑚𝑘+1−𝑝|| for all 𝑘∈𝑁. Now, from (3.12) and the fact that 𝛼_𝑛→0, we get that 𝑥_𝑚𝑘−𝑇_𝑖𝑥_𝑚𝑘→0, as 𝑘→∞, for each 𝑖=1,2,…,𝑁. Thus, as in Case 1, we obtain that 𝑥_𝑚𝑘+1−𝑥_𝑚𝑘→0 and thatlimsup_𝑘→∞𝑤−𝑝,𝑥_𝑚𝑘+1−𝑝≤0. (3.22)

Now, from (3.13) we have that 𝑥_𝑚𝑘+1−𝑝²≤1−𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑝²+2𝛼_𝑚𝑘𝑤−𝑝,𝑥_𝑚𝑘+1−𝑝, (3.23) and hence, since ‖𝑥_𝑚𝑘−𝑝‖²≤‖𝑥_𝑚𝑘+1−𝑝‖², (3.23) implies that 𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑝²≤𝑥_𝑚𝑘−𝑝²−𝑥_𝑚𝑘+1−𝑝²+2𝛼_𝑚𝑘𝑤−𝑝,𝑥_𝑚𝑘+1−𝑝≤2𝛼_𝑚𝑘𝑤−𝑝,𝑥_𝑚𝑘+1−𝑝. (3.24) But noting that 𝛼_𝑚𝑘>0, we obtain that 𝑥_𝑚𝑘−𝑝²≤2𝑤−𝑝,𝑥_𝑚𝑘+1−𝑝. (3.25)

Then, from (3.22) we get that ||𝑥_𝑚𝑘−𝑝||→0, as 𝑘→∞. This together with (3.23) gives that ||𝑥_𝑚𝑘+1−𝑝||→0, as 𝑘→∞. But ||𝑥_𝑘−𝑝||≤||𝑥_𝑚𝑘+1−𝑝||, for all 𝑘∈𝑁; thus we obtain that 𝑥_𝑘→𝑝. Therefore, from the previous two cases, we can conclude that {𝑥_𝑛} converges strongly to an element of 𝐹, and the proof is complete.

If, in Theorem 3.2, we consider single Lipschitz pseudocontractive mapping, then the assumption of condition (𝐻) is not required. In fact, we have the following corollary.

Corollary 3.3. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a Lipschitz pseudocontractive mapping with Lipschitz constants 𝐿. Assume that 𝐹(𝑇) is nonempty. Let a sequence {𝑥_𝑛} be a sequence generated from an arbitrary 𝑥₁=𝑤∈𝐶 by 𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝛾_𝑛𝑇𝑦_𝑛+1−𝛾_𝑛𝑥_𝑛, (3.26) where {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) satisfying the following conditions: (𝑖)0<𝛼_𝑛≤𝑐<1, for all 𝑛≥1 such that lim_𝑛→∞𝛼_𝑛=0 and ∑𝛼_𝑛=∞; (𝑖𝑖)  0<𝛼≤𝛾_𝑛≤𝛽_𝑛≤𝛽<1/[(1+𝐿²)+1], for all 𝑛≥1. Then, {𝑥_𝑛} converges strongly to a fixed point of 𝑇 nearest to 𝑥₁=𝑤.

Proof. Putting 𝑆_𝑛∶=𝑇 in (3.3) the scheme reduces to scheme (3.26), and following the method of proof of Theorem 3.2 we get that (see, (3.10)) 𝑥_𝑛+1−𝑝²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−−𝑝1−𝛼_𝑛𝛾_𝑛𝛽_𝑛1−2𝛽−𝛽²𝐿²𝑇𝑥_𝑛−𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−()𝛾−𝑝1−𝑐_𝑛𝛽_𝑛1−2𝛽−𝛽²𝐿²𝑇𝑥_𝑛−𝑥_𝑛²≤1−𝛼_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑤−𝑝,𝑥_𝑛+1−𝑝. (3.27)
Now, considering cases as in the proof of Theorem 3.2 we obtain the required result.
We now state and prove a convergence theorem for a common zero of finite family of monotone mappings.

Corollary 3.4. Let 𝐻 be a real Hilbert space. Let 𝐴_𝑖∶𝐻→𝐻,𝑖=1,2,…,𝑁 be a finite family of Lipschitz monotone mappings with Lipschitz constants 𝐿_𝑖, respectively, satisfying ⟨𝐴_𝑖𝑥,𝐴_𝑗𝑥⟩≥0, for all 𝑖,𝑗∈{1,2,…,𝑁}.
Assume that 𝐹∶=⋂^𝑁_𝑖=1𝑁(𝐴_𝑖) is nonempty. Let a sequence {𝑥_𝑛} be generated from an arbitrary 𝑥₁∈𝐻 by 𝑦_𝑛=𝑥_𝑛−𝛽_𝑛𝐴_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑥_𝑛−𝛾_𝑛𝐴_𝑛𝑦_𝑛, (3.28) where 𝐴_𝑛∶=𝜃_𝑛,1𝐴₁+𝜃_𝑛,2𝐴₂+⋯+𝜃_𝑛,𝑁𝐴_𝑁, for {𝜃_𝑛,𝑖}⊆[𝑎,𝑏]⊂(0,1) such that 𝜃_𝑛,1+𝜃_𝑛,2+⋯+𝜃_𝑛,𝑁=1, for all 𝑛≥1 and {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}⊂(0,1) satisfying the following conditions: (𝑖)  0<𝛼_𝑛≤𝑐<1, for all 𝑛≥0 such that lim_𝑛→∞𝛼_𝑛=0 and ∑𝛼_𝑛=∞; (𝑖𝑖) 0<𝛼≤𝛾_𝑛≤𝛽_𝑛≤𝛽<1/[(1+𝐿²)+1], for all 𝑛≥1, for 𝐿∶=max{(1+𝐿_𝑖)∶𝑖=1,2,…,𝑁}. Then, {𝑥_𝑛} converges strongly to a common zero point of {𝐴₁,𝐴₂,…𝐴_𝑁} nearest to 𝑥₁=𝑤.

Proof . Let 𝑇_𝑖𝑥∶=(𝐼−𝐴_𝑖)𝑥, for 𝑖=1,2,…,𝑁. Then we get that every 𝑇_𝑖 for all 𝑖∈{1,2,…,𝑁} is Lipschitz pseudocontractive mapping with Lipschitz constants 𝐿^′_𝑖∶=(1+𝐿_𝑖) and ∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=∩^𝑁_𝑖=1(𝐴_𝑖)≠∅. Moreover, when 𝐴_𝑛 is replaced with (𝐼−𝑇_𝑛), for each 𝑖∈{1,2,…,𝑁}, we get that scheme (3.28) reduces to scheme (3.3), and hence the conclusion follows from Theorem 3.2.
If, in Corollary 3.4 we consider a single Lipschitz monotone mapping, then we obtain the following corollary.

Corollary 3.5. Let 𝐻 be a real Hilbert space. Let 𝐴∶𝐻→𝐻 be Lipschitz monotone mappings with Lipschitz constant 𝐿. Assume that 𝑁(𝐴) is nonempty. Let a sequence {𝑥_𝑛} be generated from an arbitrary 𝑥₁∈𝐻 by 𝑦_𝑛=𝑥_𝑛−𝛽_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑤+1−𝛼_𝑛𝑥_𝑛−𝛾_𝑛𝐴𝑦_𝑛, (3.29) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}⊂(0,1) satisfying the following conditions: (𝑖) 0<𝛼_𝑛≤𝑐<1, for all 𝑛≥1 such that lim_𝑛→∞𝛼_𝑛=0 and ∑𝛼_𝑛=∞; (𝑖𝑖) 0<𝛼≤𝛾_𝑛≤𝛽_𝑛≤𝛽<1/[(1+𝐿²)+1], forall𝑛≥1. Then, {𝑥_𝑛} converges strongly to a zero point of 𝐴 nearest to 𝑥₁=𝑤.

We now give examples of Lipschitz pseudocontractive mappings satisfying condition (𝐻). Let 𝑋∶=ℝ and 𝐶∶=[−24,3]⊂R. Let 𝑇₁,𝑇₂∶=𝐶→𝐶 be defined by 𝑇₁[),𝑥∶=𝑥,𝑥∈−24,0𝑥−𝑥³[],𝑇,𝑥∈0,3₂[),𝑥∶=𝑥,𝑥∈−24,23𝑥−𝑥²,𝑥∈[2,3]. (3.30)

Then we observe that 𝐹(𝑇₁)=[−24,0],and 𝐹(𝑇₂)=[−24,2], and hence common fixed point of 𝑇₁ and 𝑇₂ is [−24,0] which is nonempty. Now, we show that 𝑇₁ and 𝑇₂ are pseudocontractive mappings. But, since 𝐴₁𝑥∶=𝐼−𝑇₁[),𝑥𝑥=0,𝑥∈−24,0₃[],𝐴,𝑥∈0,3₂𝑥∶=𝐼−𝑇₂[),𝑥=0,𝑥∈−24,2−2𝑥+𝑥²,𝑥∈[2,3] (3.31) are monotone, we have that 𝑇₁ and 𝑇₂ are pseudocontractive mappings.

Now, we show that 𝑇₁ and 𝑇₂ are Lipschitzian mappings. First, we show that 𝑇₁ is Lipschitzian with constant 𝐿=28. Let 𝐶₁=[−24,0), 𝐶₂=[0,3]. If 𝑥,𝑦∈𝐶₁, then we have that 𝑇₁𝑥−𝑇₁𝑦=𝑥−𝑦≤28𝑥−𝑦. (3.32) If 𝑥,𝑦∈𝐶₂,then we have that 𝑇₁𝑥−𝑇₁𝑦=𝑥−𝑥³−𝑦−𝑦³≤𝑥−𝑦+𝑥³−𝑦³=𝑥−𝑦+𝑥−𝑦𝑥²+𝑥𝑦+𝑦²=1+𝑥²+𝑥𝑦+𝑦²𝑥−𝑦≤28𝑥−𝑦. (3.33)

If 𝑥∈𝐶₁ and 𝑦∈𝐶_2, then we get that 𝑇₁𝑥−𝑇₁𝑦=𝑥−𝑦−𝑦³≤𝑥−𝑦+𝑦³=𝑥−𝑦+𝑦²𝑦≤𝑥−𝑦+𝑦²𝑦−𝑥=𝑦².+1𝑥−𝑦≤10𝑥−𝑦≤28𝑥−𝑦 (3.34) Therefore, from (3.32), (3.33), and (3.34), we obtain that 𝑇₁ is Lipschitz.

Next, we show that 𝑇₂ is Lipschitz with constant 𝐿=9.

Let 𝐷₁=[−24,2), 𝐷₂=[2,3]. If 𝑥,𝑦∈𝐷₁, then we have that 𝑇₂𝑥−𝑇₂𝑦=𝑥−𝑦≤9𝑥−𝑦. (3.35) If 𝑥,𝑦∈𝐷₂,then we get that 𝑇₂𝑥−𝑇₂𝑦=3𝑥−𝑥²−3𝑦−𝑦²≤3𝑥−𝑦+𝑥²−𝑦²+≤.=3𝑥−𝑦𝑥−𝑦𝑥+𝑦3+𝑥+𝑦𝑥−𝑦≤9𝑥−𝑦 (3.36) If 𝑥∈𝐷₁ and 𝑦∈𝐷₂ then we have that 𝑇₂𝑥−𝑇₂𝑦=𝑥−3𝑦−𝑦²≤𝑥−𝑦+𝑦²−2𝑦 (3.37) and for 𝑥∈[0,2) (3.37) implies that 𝑇₂𝑥−𝑇₂𝑦≤𝑥−𝑦+𝑦²−2𝑦−𝑥²≤+≤.−2𝑥𝑥−𝑦+2𝑥−𝑦𝑥−𝑦𝑥+𝑦3+𝑥+𝑦𝑥−𝑦≤9𝑥−𝑦 (3.38) For 𝑥∈[−24,0) inequality (3.37) gives that 𝑇₂𝑥−𝑇₂𝑦≤+𝑦,≤+=.𝑥−𝑦𝑦−2𝑥−𝑦𝑦−2𝑦−𝑥1+𝑦−2𝑥−𝑦≤2𝑥−𝑦≤9𝑥−𝑦 (3.39)

Therefore, from (3.35), (3.36), and (3.39) we obtain that 𝑇₂ is Lipschitz. Furthermore, we show that 𝑇₁ and 𝑇₂ satisfy condition (𝐻). If 𝑥∈𝐷₁,then we have that ⟨𝑇₁𝑥−𝑥,𝑇₂𝑥−𝑥⟩=0, and if 𝑥∈𝐷₂ we get that ⟨𝑇₁𝑥−𝑥,𝑇₂𝑥−𝑥⟩=⟨𝑥−𝑥³−𝑥,(3𝑥−𝑥²)−𝑥⟩=⟨−𝑥³,2𝑥−𝑥²⟩=−𝑥³(2𝑥−𝑥²)≥0. Therefore, 𝑇₁ and 𝑇₁ satisfy property (𝐻).

Remark 3.6. Theorem 3.2 provides convergence sequence to a common fixed point of finite family of Lipschitzian pseudocontractive mappings whereas Corollary 3.4 provides convergence sequence to a common zero of finite family of monotone mappings in Hilbert spaces. No compactness assumption is imposed either on 𝑇 or 𝐶. This provides affirmative answer to the question raised.

Remark 3.7. Theorem 3.2 improves Theorem I, Theorem 3.1 of Zhou [15], Theorem 3.1 of Yao et al. [17], and Theorem 3.1 of Tang et al. [16] in the sense that either our convergence does not require compactness of 𝑇 or computation of 𝐶_𝑛+1 from 𝐶_𝑛 for each 𝑛≥1.

Remark 3.8. Theorem 3.2 improves Theorems I and ZSA in the sense that our convergence is for a fixed point of a finite family of Lipschitz pseudocontractive mappings. The condition that interior of 𝐹(𝑇) is nonempty is dispensed with.