# Weak and strong convergence theorems of implicit iteration process on Banach spaces

## Abstract

In this article, we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem of an implicit-shrinking iterative process for two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space.

## 1 Introduction

Let E be a Banach space, and let C be a nonempty closed convex subset of E. A mapping T: CE is nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for every x, y C. Let F(T): = {x C: x = Tx} denote the set of fixed points of T. Besides, a mapping T: CE is quasinonexpansive if $F\left(T\right)\ne \varnothing$ and ||Tx - y|| ≤ ||x - y|| for all x C and y F(T).

In 2008, Suzuki [1] introduced the following generalized nonexpansive mapping on Banach spaces. A mapping T: CE is said to satisfy condition (C) if for all x, y C,

$\frac{1}{2}||x-Tx||\le ||x-y||⇒||Tx-Ty||\le ||x-y||.$

In fact, every nonexpansive mapping satisfies condition (C), but the converse may be false [1, Example 1]. Besides, if T: CE satisfies condition (C) and $F\left(T\right)\ne \varnothing$, then T is a quasinonexpansive mapping. However, the converse may be false [1, Example 2].

Construction of approximating fixed points of nonlinear mappings is an important subject in the theory of nonlinear mappings and its applications in a number of applied areas.

Let C be a nonempty closed convex subset of a real Hilbert space H, and let T: CC be a mapping. In 1953, Mann [2] gave an iteration process:

${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.1)

where x0 is taken in C arbitrarily, and {α n } is a sequence in [0,1].

In 2001, Soltuz [3] introduced the following Mann-type implicit process for a nonexpansive mapping T: CC:

${x}_{n}={\alpha }_{n}{x}_{n-1}+\left(1-{\alpha }_{n}\right)T{x}_{n},\phantom{\rule{1em}{0ex}}n\in ℕ,$
(1.2)

where x0 is taken in C arbitrarily, and {t n } is a sequence in [0,1].

In 2001, Xu and Ori [4] have introduced an implicit iteration process for a finite family of nonexpansive mappings. Let T1, T2, ..., T N be N self-mappings of C and suppose that $F:={\cap }_{i=1}^{N}F\left({T}_{i}\right)\ne \varnothing$, the set of common fixed points of T i , i = 1, 2, ..., N. Let I: = {1, 2, ..., N}. Xu and Ori [4] gave an implicit iteration process for a finite family of nonexpansive mappings:

${x}_{n}={t}_{n}{x}_{n-1}+\left(1-{t}_{n}\right){T}_{n}{x}_{n},\phantom{\rule{1em}{0ex}}n\in ℕ,$
(1.3)

where x0 is taken in C arbitrarily, {t n } is a sequence in [0,1], and T k = Tk mod N. (Here the mod N function takes values in I.) And they proved the weak convergence of process (1.3) to a common fixed point in the setting of a Hilbert space.

In 2010, Khan et al. [5] presented an implicit iterative process for two nonexpansive mappings in Banach spaces. Let E be a Banach space, and let C be a nonempty closed convex subset of E, and let T, S: CC be two nonexpansive mappings. Khan et al. [5] considered the following implicit iterative process:

${x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}S{x}_{n}+{\gamma }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\in ℕ,$
(1.4)

where {α n }, {β n }, and {γ n } are sequences in [0,1] with α n + β n + γ n = 1.

Motivated by the above works in [5], we want to consider the following implicit iterative process. Let E be a Banach space, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a mapping which satisfy condition (C). We first consider the weak convergence theorems for the following implicit iterative process:

(1.5)

where {a n }, {b n }, {c n }, and {d n } are sequences in [0,1] with a n + b n + c n + d n = 1.

Next, we also consider weak convergence theorems for another implicit iterative process:

(1.6)

where {a n }, {b n }, {c n }, and {d n } are sequences in [0,1] with a n + b n + c n = 1.

In fact, for the above implicit iterative processes, most researchers always considered weak convergence theorems, and few researchers considered strong convergence theorem under suitable conditions. For example, one can see [57]. However, some conditions are not natural. For this reason, we consider the following shrinking-implicit iterative processes and study the strong convergence theorem. Let {x n } be defined by

(1.7)

where {a n }, {b n }, {c n }, and {d n } are sequences in (0, 1) with a n + b n + c n = 1.

In this article, we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfy condition (C). And we generalize Khan et al.'s result [5] as special case. Next, we consider strong convergence theorem of an implicit-shrinking iterative process for two non-expansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space.

## 2 Preliminaries

Throughout this article, let and be the sets of all positive integers and real numbers, respectively. Let E be a Banach space and let E* be the dual space of E. For a sequence {x n } of E and a point x E, the weak convergence of {x n } to x and the strong convergence of {x n } to x are denoted by x n x and x n x, respectively.

A Banach space E is said to satisfy Opial's condition if {x n } is a sequence in E with x n x, then

$\underset{n\to \infty }{limsup}||{x}_{n}-x||<\underset{n\to \infty }{limsup}||{x}_{n}-y||,\phantom{\rule{1em}{0ex}}\forall y\in E,y\ne x.$

Let E be a Banach space. Then, the duality mapping $J:E⊸{E}^{*}$ is defined by

$Jx:\left\{{x}^{*}\in {E}^{*}:⟨x,{x}^{*}⟩=||x|{|}^{2}=||{x}^{*}|{|}^{2}\right\},\phantom{\rule{1em}{0ex}}\forall x\in E.$

Let S(E) be the unit sphere centered at the origin of E. Then, the space E is said to be smooth if the limit

$\underset{t\to 0}{lim}\frac{||x+ty||-||x||}{t}$

exists for all x, y S(E). It is also said to be uniformly smooth if the limit exists uniformly in x, y S(E). A Banach space E is said to be strictly convex if $∥\frac{x+y}{2}∥<1$ whenever x, y S(E) and xy. It is said to be uniformly convex if for each ε (0, 2], there exists δ > 0 such that $∥\frac{x+y}{2}∥<1-\delta$ whenever x, y S(E) and ||x - y|| ≥ ε. Furthermore, we know that [8]

1. (i)

if E in smooth, then J is single-valued;

2. (ii)

if E is reflexive, then J is onto;

3. (iii)

if E is strictly convex, then J is one-to-one;

4. (iv)

if E is strictly convex, then J is strictly monotone;

5. (v)

if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

A Banach space E is said to have Kadec-Klee property if a sequence {x n } of E satisfying that x n x and ||x n || → ||x||, then x n x. It is known that if E uniformly convex, then E has the Kadec-Klee property [8].

Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this article, define the function ϕ: C × C by

$\varphi \left(x,y\right):=||x|{|}^{2}-2⟨x,Jy⟩+||y|{|}^{2},\phantom{\rule{1em}{0ex}}\forall x,y\in E.$

Observe that, in a Hilbert space H, ϕ(x, y) = ||x - y||2 for all x, y H. Furthermore, for each x, y, z, w E, we know that:

1. (1)

(||x|| - ||y||)2ϕ(x, y) ≤ (||x|| + ||y||)2;

2. (2)

ϕ(x, y) ≥ 0;

3. (3)

ϕ(x, y) = ϕ(x, z) + ϕ(z, y) + 2〈x - z, Jz - Jy〉;

4. (4)

2〈x - y, Jz - Jw〉 = ϕ(x, w) + ϕ(y, z) - ϕ(x, z) - ϕ(y, w);

5. (5)

if E is additionally assumed to be strictly convex, then

$\varphi \left(x,y\right)=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{only}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}x=y;$
6. (6)

ϕ(x, J -1Jy + (1 - λ)Jz)) ≤ λϕ(x, y) + (1 - λ)ϕ(x, z).

Lemma 2.1. [9] Let E be a uniformly convex Banach space and let r > 0. Then, there exists a strictly increasing, continuous, and convex function g: [0, 2r] → [0, ∞) such that g(0) = 0 and

$||ax+by+cz+dw|{|}^{2}\le a||x|{|}^{2}+b||y|{|}^{2}+c||z|{|}^{2}+d||w|{|}^{2}-abg\left(||x-y||\right)$

for all x, y, z, w B r and a, b, c, d [0,1] with a + b + c + d = 1, where B r : = {z E: ||z|| ≤ r}.

Lemma 2.2. [10] Let E be a uniformly convex Banach space and let r > 0. Then, there exists a strictly increasing, continuous, and convex function g: [0, 2r] → [0, ∞) such that g(0) = 0 and

$\varphi \left(x,{J}^{-1}\left(\lambda Jy+\left(1-\lambda \right)Jz\right)\right)\le \lambda \varphi \left(x,y\right)+\left(1-\lambda \right)\varphi \left(x,z\right)-\lambda \left(1-\lambda \right)g\left(||Jy-Jz||\right)$

for all x, y, z B r and λ [0,1], where B r : = {z E: ||z|| ≤ r}.

Lemma 2.3. [11] Let E be a uniformly convex Banach space, let {α n } be a sequence of real numbers such that 0 < bα n c < 1 for all n , and let {x n } and {y n } be sequences of E such that lim supn→∞||x n || ≤ a, lim supn→∞||y n || ≤ a, and limn→∞||α n x n + (1 - α n )y n || = a for some a ≥ 0. Then, limn→∞||x n - y n || = 0.

Lemma 2.4. [12] Let E be a smooth and uniformly convex Banach space, and let {x n } and {y n } be sequences in E such that either {x n } or {y n } is bounded. If limn→∞ϕ(x n , y n ) = 0, then limn→∞||x n - y n || = 0.

Remark 2.1. [13] Let E be a uniformly convex and uniformly smooth Banach space. If {x n } and {y n } are bounded sequences in E, then

$\underset{n\to \infty }{lim}\varphi \left({x}_{n},{y}_{n}\right)=0⇔\underset{n\to \infty }{lim}||{x}_{n}-{y}_{n}||=0⇔\underset{n\to \infty }{lim}||J{x}_{n}-J{y}_{n}||=0.$

Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. For an arbitrary point x of E, the set

$\left\{z\in C:\varphi \left(z,x\right)=\underset{y\in C}{min}\varphi \left(y,x\right)\right\}$

is always nonempty and a singleton [14]. Let us define the mapping Π C from E onto C by Π C x = z, that is,

$\varphi \left({\Pi }_{C}x,x\right)=\underset{y\in C}{min}\varphi \left(y,x\right)$

for every x E. Such Π C is called the generalized projection from E onto C[14].

Lemma 2.5. [14, 15] Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and let (x, z) E × C. Then:

1. (i)

z = Π C x if and only if 〈y - z, Jx - Jz〉 ≤ 0 for all y C;

2. (ii)

ϕ(z, Π C x) + ϕ C x, x) ≤ ϕ(z, x).

Lemma 2.6. [16] Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and T: CC is a nonexpansive mapping. Let {x n } be a sequence in C with x n x C and limn→∞||x n - Tx n || = 0. Then, Tx = x.

Lemma 2.7. [1] Let C be a nonempty subset of a Banach space E with the Opial property. Assume that T: CE satisfies condition (C). Let {x n } be a sequence in C with x n x C and limn→∞||x n - Tx n || = 0. Then, Tx = x.

Lemma 2.8. [1] Let T be a mapping on a closed subset C of a Banach space E. Assume that T satisfies condition (C). Then, F(T) is a closed set. Moreover, if E is strictly convex and C is convex, then F(T) is also convex.

Lemma 2.9. [17] Let C be a nonempty closed convex subset of a strictly convex Banach space E, and T: CC be a nonexpansive mapping. Then, F(T) is a closed convex subset of C.

## 3 Weak convergence theorems

Lemma 3.1. Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n } be sequences with 0 < aa n , b n , c n , d n b < 1 and a n + b n + c n + d n = 1. Suppose that $\Omega :=F\left(S\right)\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary,}}\hfill \\ \hfill {x}_{n}={a}_{n}{x}_{n-1}+{b}_{n}S{x}_{n-1}+{c}_{n}{T}_{1}{x}_{n}+{d}_{n}{T}_{2}{x}_{n}.\hfill \end{array}\right\$

Then, we have:

1. (i)

$\underset{n\to \infty }{lim}||{x}_{n}-p||$ exists for each p Ω.

2. (ii)

$\underset{n\to \infty }{lim}||{x}_{n}-S{x}_{n}||=\underset{n\to \infty }{lim}||{x}_{n}-{T}_{1}{x}_{n}||=\underset{n\to \infty }{lim}||{x}_{n}-{T}_{2}{x}_{n}||=0$.

Proof. First, we show that {x n } is well-defined. Now, let f(x): = a1x0+b1Sx0+c1T1x+d1T2x. Then,

$||f\left(x\right)-f\left(y\right)||\le {c}_{1}||{T}_{1}x-{T}_{1}y||+{d}_{1}||{T}_{2}x-{T}_{2}y||\le \left({c}_{1}+{d}_{1}\right)||x-y||\le \left(1-2a\right)||x-y||.$

By Banach contraction principle, the existence of x1 is established. Similarly, the existence of {x n } is well-defined.

1. (i)

For each p Ω and n , we have:

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-p||+{b}_{n}||S{x}_{n-1}-p||+{c}_{n}||{T}_{1}{x}_{n}-p||+{d}_{n}||{T}_{2}{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-p||+{b}_{n}||{x}_{n-1}-p||+\left({c}_{n}+{d}_{n}\right)||{x}_{n}-p||.\phantom{\rule{2em}{0ex}}\end{array}$

This implies that (1 - c n - d n )||x n - p|| ≤ (a n + b n )||xn-1-p||. Hence, ||x n -p||≤ ||xn-1-p||, limn →∞||x n -p|| exists, and {x n } is a bounded sequence.

1. (ii)

Take any p Ω and let p be fixed. Suppose that $\underset{n\to \infty }{lim}||{x}_{n}-p||=d$. Clearly,$\underset{n\to \infty }{limsup}||{T}_{2}{x}_{n}-p||\le d$, and we have:

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{lim}||{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{lim}||{a}_{n}{x}_{n-1}+{b}_{n}S{x}_{n-1}+{c}_{n}{T}_{1}{x}_{n}+{d}_{n}{T}_{2}{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{lim}∥\left(1-{d}_{n}\right)\left[\frac{{a}_{n}}{1-{d}_{n}}\left({x}_{n-1}-p\right)+\frac{{b}_{n}}{1-{d}_{n}}\left(S{x}_{n-1}-p\right)+\frac{{c}_{n}}{1-{d}_{n}}\left({T}_{1}{x}_{n}-p\right)\right]+{d}_{n}\left({T}_{2}{x}_{n}-p\right)∥.\phantom{\rule{2em}{0ex}}\end{array}$

Besides,

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{limsup}∥\frac{{a}_{n}}{1-{d}_{n}}\left({x}_{n-1}-p\right)+\frac{{b}_{n}}{1-{d}_{n}}\left(S{x}_{n-1}-p\right)+\frac{{c}_{n}}{1-{d}_{n}}\left({T}_{1}{x}_{n}-p\right)∥\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}}{1-{d}_{n}}||{x}_{n-1}-p||+\frac{{b}_{n}}{1-{d}_{n}}||S{x}_{n-1}-p||+\frac{{c}_{n}}{1-{d}_{n}}||{T}_{1}{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}}{1-{d}_{n}}||{x}_{n-1}-p||+\frac{{b}_{n}}{1-{d}_{n}}||S{x}_{n-1}-p||+\frac{{c}_{n}}{1-{d}_{n}}||{T}_{1}{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}+{b}_{n}}{1-{d}_{n}}||{x}_{n-1}-p||+\frac{{c}_{n}}{1-{d}_{n}}||{x}_{n}-p||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}+{b}_{n}+{c}_{n}}{1-{d}_{n}}||{x}_{n-1}-p||=d.\phantom{\rule{2em}{0ex}}\end{array}$

By Lemma 2.3,

$\underset{n\to \infty }{lim}∥\frac{{a}_{n}}{1-{d}_{n}}\left({x}_{n-1}-p\right)+\frac{{b}_{n}}{1-{d}_{n}}\left(S{x}_{n-1}-p\right)+\frac{{c}_{n}}{1-{d}_{n}}\left({T}_{1}{x}_{n}-p\right)-\left({T}_{2}{x}_{n}-p\right)∥=0.$

This implies that limn→∞||x n - T2x n || = 0. Similarly, limn→∞||x n - T1x n || = 0.

Since {x n } is bounded, there exists r > 0 such that 2 sup{||x n -p||:n }≤ r.

By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2r] → [0, ∞) such that g(0) = 0 and

$\begin{array}{c}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}||{x}_{n}-p|{|}^{2}\\ \le {a}_{n}||{x}_{n-1}-p|{|}^{2}+{b}_{n}||S{x}_{n-1}-p|{|}^{2}+{c}_{n}||{T}_{1}{x}_{n}-p|{|}^{2}+{d}_{n}||{T}_{2}{x}_{n}-p|{|}^{2}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-{a}_{n}{b}_{n}g\left(||{x}_{n-1}-S{x}_{n-1}||\right)\\ \le \left({a}_{n}+{b}_{n}\right)||{x}_{n-1}-p|{|}^{2}+\left({c}_{n}+{d}_{n}\right)||{x}_{n}-p|{|}^{2}-{a}_{n}{b}_{n}g\left(||{x}_{n-1}-S{x}_{n-1}||\right).\end{array}$

This implies that

${a}_{n}{b}_{n}g\left(||{x}_{n-1}-S{x}_{n-1}||\right)\le \left({a}_{n}+{b}_{2}\right)\left(||{x}_{n-1}-p|{|}^{2}-||{x}_{n}-p|{|}^{2}\right).$

By the properties of g and limn→∞||x n - p|| = d, we get limn→∞||x n - Sx n || = 0.

Theorem 3.1. Let E be a uniformly convex Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n } be sequences with 0 < aa n , b n , c n , d n b < 1 and a n + b n + c n + d n = 1. Suppose that $\Omega :=F\left(S\right)\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}},\hfill \\ \hfill {x}_{n}={a}_{n}{x}_{n-1}+{b}_{n}S{x}_{n-1}+{c}_{n}{T}_{1}{x}_{n}+{d}_{n}{T}_{2}{x}_{n}.\hfill \end{array}\right\$

Then, x n z for some z Ω.

Proof. By Lemma 3.1, {x n } is a bounded sequence. Then, there exists a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of {x n } and z C such that ${x}_{{n}_{k}}⇀z$. By Lemmas 2.6, 2.7, and 3.1, we know that z Ω. Since E has Opial's condition, it is easy to see that x n z.

Hence, the proof is completed.

Remark 3.1. The conclusion of Theorem 3.1 is still true if S: CC is a quasi-nonexpansive mapping, and I - S is demiclosed at zero, that is, x n x and (I - S)x n 0 implies that (I - S)x = 0.

In Theorem 3.1, if S = I, then we get the following result. Hence, Theorem 3.1 generalizes Theorem 4 in [5].

Corollary 3.1. [5] Let E be a uniformly convex Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings. Let {a n }, {b n }, and {c n } be sequences with 0 < aa n , b n , c n b < 1 and a n + b n + c n = 1. Suppose that $\Omega :=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$.

Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}},\hfill \\ \hfill {x}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{x}_{n}+{c}_{n}{T}_{2}{x}_{n}.\hfill \end{array}\right\$

Then, x n z for some z Ω.

Besides, it is easy to get the following result from Theorem 3.1.

Corollary 3.2. Let E be a uniformly convex Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let S: CC be a mapping with condition (C). Let {a n } be a sequence with 0 < aa n b < 1. Suppose that $F\left(S\right)\ne 0̸$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}},\hfill \\ \hfill {x}_{n}={a}_{n}{x}_{n-1}+\left(1-{a}_{n}\right)S{x}_{n-1}.\hfill \end{array}\right\$

Then, x n z for some z F(S).

Proof. Let T1 = T2 = I, where I is the identity mapping. For each n , we know that

${x}_{n}=\frac{{a}_{n}}{2}{x}_{n-1}+\frac{1-{a}_{n}}{2}S{x}_{n-1}+\frac{1}{4}{T}_{1}{x}_{n}+\frac{1}{4}{T}_{2}{x}_{n}.$

By Theorem 3.1, it is easy to get the conclusion.

Theorem 3.2. Let E be a uniformly convex Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n } be sequences with 0 < aa n , b n , c n , d n b < 1 and a n + b n + c n = 1. Suppose that $\Omega :=F\left(S\right)\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary,}}\hfill \\ \hfill {y}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n},\hfill \\ \hfill {x}_{n}={d}_{n}{y}_{n}+\left(1-{d}_{n}\right)S{y}_{n}.\hfill \end{array}\right\$

Then, x n z for some z Ω.

Proof. Following the same argument as in Lemma 3.1, we know that {y n } is well-defined. Take any w Ω and let w be fixed. Then, for each n , we have

$\begin{array}{ll}\hfill ||{y}_{n}-w||& =||{a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w||+{b}_{n}||{T}_{1}{y}_{n}-w||+{c}_{n}||{T}_{2}{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w||+\left({b}_{n}+{c}_{n}\right)||{y}_{n}-w||.\phantom{\rule{2em}{0ex}}\end{array}$

This implies that ||y n - w|| ≤ ||xn-1- w|| for each n . Besides, we also have

$\begin{array}{ll}\hfill ||{x}_{n}-w||& =||{d}_{n}{y}_{n}+\left(1-{d}_{n}\right)S{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}||{y}_{n}-w||+\left(1-{d}_{n}\right)||S{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le ||{y}_{n}-w||.\phantom{\rule{2em}{0ex}}\end{array}$

Hence, ||x n - w|| ≤ ||y n - w|| ≤ ||xn-1- w|| for each n . So, limn→∞||x n - w|| and limn→∞||y n - w|| exist, and {x n }, {y n } are bounded sequences.

Suppose that limn→∞||x n -w|| = limn→∞||y n -w|| = d. Clearly, lim supn→∞||T2y n -w|| ≤ d, and we have

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{lim}||{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{lim}||{a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{lim}∥\left(1-{c}_{n}\right)\left[\frac{{a}_{n}}{1-{c}_{n}}\left({x}_{n-1}-w\right)+\frac{{b}_{n}}{1-{c}_{n}}\left({T}_{1}{y}_{n}-w\right)\right]+{c}_{n}\left({T}_{2}{y}_{n}-w\right)∥.\phantom{\rule{2em}{0ex}}\end{array}$

Besides,

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{limsup}∥\frac{{a}_{n}}{1-{c}_{n}}\left({x}_{n-1}-w\right)+\frac{{b}_{n}}{1-{c}_{n}}\left({T}_{1}{y}_{n}-w\right)∥\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}}{1-{c}_{n}}||{x}_{n-1}-w||+\frac{{b}_{n}}{1-{c}_{n}}||{T}_{1}{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\frac{{a}_{n}}{1-{c}_{n}}||{x}_{n-1}-w||+\frac{{b}_{n}}{1-{c}_{n}}||{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}||{x}_{n-1}-w||=d.\phantom{\rule{2em}{0ex}}\end{array}$

By Lemma 2.3,

$\underset{n\to \infty }{lim}∥\frac{{a}_{n}}{1-{c}_{n}}\left({x}_{n-1}-w\right)+\frac{{b}_{n}}{1-{c}_{n}}\left({T}_{1}{y}_{n}-w\right)-\left({T}_{2}{y}_{n}-w\right)∥=0.$

This implies that limn→∞||y n - T2y n || = 0. Similarly, limn→∞||y n - T1y n || = 0.

Since {x n } and {y n } are bounded sequences, there exists r > 0 such that

$2sup\left\{||{x}_{n}||,||{y}_{n}||,||{x}_{n}-w||,||{y}_{n}-w||:n\in ℕ\right\}\le r.$

By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2r] → [0, ∞) such that g(0) = 0 and

$||{d}_{n}{y}_{n}+\left(1-{d}_{n}\right)S{y}_{n}-w|{|}^{2}\le {d}_{n}||{y}_{n}-w|{|}^{2}+\left(1-{d}_{n}\right)||S{y}_{n}-w|{|}^{2}-{d}_{n}\left(1-{d}_{n}\right)g\left(||{y}_{n}-S{y}_{n}||\right).$

This implies that

${d}_{n}\left(1-{d}_{n}\right)g\left(||{y}_{n}-S{y}_{n}||\right)\le ||{y}_{n}-w|{|}^{2}-||{x}_{n}-w|{|}^{2}.$

Since limn→∞||x n - w|| = limn→∞||y n - w|| = d, and the properties of g, we get limn→∞||y n - Sy n || = 0. Besides,

$||{x}_{n}-{y}_{n}||=||{d}_{n}{y}_{n}+\left(1-{d}_{n}\right)S{y}_{n}-{y}_{n}||=\left(1-{d}_{n}\right)||{y}_{n}-S{y}_{n}||.$

Hence, limn→∞||x n -y n || = 0. Finally, following the same argument as in the proof of Theorem 3.1, we know that x n z for some z Ω.

Next, we give the following examples for Theorems 3.1 and 3.2.

Example 3.1. Let E = , C: = [0,3], T1x = T2x = x, and let S: CC be the same as in [1]:

$Sx:=\left\{\begin{array}{ccc}\hfill 0\hfill & \hfill \mathsf{\text{if}}\hfill & \hfill x\ne 3,\hfill \\ \hfill 1\hfill & \hfill \mathsf{\text{if}}\hfill & \hfill x=3.\hfill \end{array}\right\$

For each n, let ${a}_{n}={b}_{n}={c}_{n}={d}_{n}=\frac{1}{4}$. Let x0 = 1. Then, for the sequence {x n }, in Theorem 3.1, we know that ${x}_{n}=\frac{1}{{2}^{n}}$ for all n , and x n → 0, and 0 is a common fixed point of S, T1, and T2.

Example 3.2. Let E, C, T1, T2, S be the same as in Example 3.1. For each n, let ${a}_{n}={b}_{n}={c}_{n}=\frac{1}{3}$, and ${d}_{n}=\frac{1}{2}$. Let x0 = 1. Then, for the sequence {x n } in Theorem 3.1, we know that ${x}_{n}=\frac{1}{{2}^{n}}$ for all n , and x n → 0, and 0 is a common fixed point of S, T1, and T2.

Example 3.3. Let E, C, {a n }, {b n }, {c n }, {d n }, and let S: CC be the same as in Example 3.1. Let T1x = T2x = 0 for each x C. Then, for the sequence {x n } in Theorem 3.1, we know that ${x}_{n}=\frac{1}{{4}^{n}}$ for all n .

Example 3.4. Let E, C, {a n }, {b n }, {c n }, {d n }, and let S: CC be the same as in Example 3.2. Let T1x = T2x = 0 for each x C. Then, for the sequence {x n } in Theorem 3.2, we know that ${x}_{n}=\frac{1}{{6}^{n}}$ for all n .

Remark 3.2.

1. (i)

For the rate of convergence, by Examples 3.3 and 3.4, we know that the iteration process in Theorem 3.2 may be faster than the iteration process in Theorem 3.1. But, the times of iteration process for Theorem 3.2 is much than ones in Theorem 3.1.

2. (ii)

The conclusion of Theorem 3.2 is still true if S: CC is a quasi-nonexpansive mapping, and I - S is demiclosed at zero, that is, x n x and (I - S)x n → 0 implies that (I - S)x = 0.

3. (iii)

Corollaries 3.1 and 3.2 are special cases of Theorem 3.2.

Definition 3.1. [18] Let C be a nonempty subset of a Banach space E. A mapping T:CE satisfy condition (E) if there exists μ ≥ 1 such that for all x, y C,

$||x-Ty||\le \mu ||x-Tx||+||x-y||.$

By Lemma 7 in [1], we know that if T satisfies condition (C), then T satisfies condition (E). But, the converse may be false [18, Example 1]. Furthermore, we also observe the following result.

Lemma 3.2. [18] Let C be a nonempty subset of a Banach space E. Let T: CE be a mapping. Assume that:

1. (i)

$\underset{n\to \infty }{lim}||{x}_{n}-T{x}_{n}||=0$ and x n x;

2. (ii)

T satisfies condition (E);

3. (iii)

E has Opial condition.

Then, Tx = x.

By Lemma 3.2, if S satisfies condition (E), then the conclusions of Theorems 3.1 and 3.2 are still true. Hence, we can use the following condition to replace condition (C) in Theorems 3.1 and 3.2 by Proposition 19 in [19].

Definition 3.2. [19] Let T be a mapping on a subset C of a Banach space E.

Then, T is said to satisfy (SKC)-condition if

$\frac{1}{2}||x-Tx||\le ||x-y||⇒||Tx-Ty||\le N\left(x,y\right),$

where $N\left(x,y\right):=max\left\{||x-y||,\frac{1}{2}\left(||x-Tx||+||Ty-y||\right),\frac{1}{2}\left(||Tx-y||+||x-Ty||\right)\right\}$ for all x, y C.

## 4 Strong convergence theorems (I)

Let C be a nonempty closed convex subset of a Banach space E. A point p in C is said to be an asymptotic fixed point of a mapping T: CC if C contains a sequence {x n } which converges weakly to p such that limn→∞, ||x n - Tx n || = 0. The set of asymptotic fixed points of T will be denoted by $\stackrel{^}{F}\left(T\right)$. A mapping T: CC is called relatively nonexpansive [20] if $F\left(T\right)\ne 0̸,\stackrel{^}{F}\left(T\right)=F\left(T\right)$, and ϕ(p,Tx) ≤ ϕ(p,x) for all x C and p F(T). Note that every identity mapping is a relatively nonexpansive mapping.

Lemma 4.1. [21] Let E be a strictly convex and smooth Banach space, let C be a closed convex subset of E, and let T: CC be a relatively nonexpansive mapping. Then, F(T) is a closed and convex subset of C.

The following property is motivated by the property (Q4) in [22].

Definition 4.1. Let E be a Banach space. Then, we say that E satisfies condition (Q) if for each x, y, z1, z2 E and t [0,1],

$\left\{\begin{array}{l}{x}_{0}\in C\text{chosen}\text{arbitrary}\text{and}{C}_{0}={D}_{0}=C,\hfill \\ {y}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n},\hfill \\ {z}_{n}={J}^{-1}\left({d}_{n}J{y}_{n}+\left(1-{d}_{n}\right)JS{y}_{n}\right),\hfill \\ {C}_{n}=\left\{z\in {C}_{n-1}:\varphi \left(z,{z}_{n}\right)\le \varphi \left(z,{y}_{n}\right)\right\},\hfill \\ {D}_{n}=\left\{z\in {D}_{n-1}:||{y}_{n}-z||\le ||{x}_{n-1}-z||\right\},\hfill \\ {x}_{n}={\Pi }_{{C}_{n}\cap {D}_{n}{x}_{0}}.\hfill \end{array}$

Remark 4.1. If H is a Hilbert space, then H satisfies condition (Q).

Theorem 4.1. Let E be a uniformly convex and uniformly smooth Banach space with condition (Q), and let C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a relatively nonexpansive mapping. Let {a n }, {b n }, {c n }, and {d n } be sequences in (0,1) with and a n + b n + c n = 1. Suppose that $\Omega :=F\left(S\right)\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{C}_{0}={D}_{0}=C,\hfill \\ \hfill {y}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n},\hfill \\ \hfill {z}_{n}={J}^{-1}\left({d}_{n}J{y}_{n}+\left(1-{d}_{n}\right)JS{y}_{n}\right),\hfill \\ \hfill {C}_{n}=\left\{z\in {C}_{n-1}:\varphi \left(z,{z}_{n}\right)\le \varphi \left(z,{y}_{n}\right)\right\},\hfill \\ \hfill {D}_{n}=\left\{z\in {D}_{n-1}:||{y}_{n}-z||\le ||{x}_{n-1}-z||\right\},\hfill \\ \hfill {x}_{n}={\Pi }_{{C}_{n}\cap {D}_{n}{x}_{0}}.\hfill \end{array}\right\$

Assume that lim infn→∞b n > 0, lim infn→∞c n > 0, and lim infn→∞d n (1 - d n ) > 0.

Then, limn→∞x n = limn→∞y n = limn→∞z n = ΠΩx0.

Proof. Following the same argument as in Lemma 3.1, we know that {y n } is well-defined.

Clearly, C0 and D0 are nonempty closed convex subsets of C, and C n is a closed subset of C for every n . Since ϕ(z, z n ) ≤ ϕ(z, y n ) is equivalent to

$2⟨z,J{y}_{n}-J{z}_{n}⟩\le ||{y}_{n}|{|}^{2}-||{z}_{n}|{|}^{2},$

it is easy to see that C n is a convex set for each n . Besides, by condition (Q), it is easy to see that D n is a nonempty closed convex subset of C.

Next, we want to show that Ω C n D n for each n {0}. Clearly, Ω C0. Suppose that Ω Cn-1. Let w Ω. Then, w F(S) and

$\begin{array}{ll}\hfill \varphi \left(w,{z}_{n}\right)& =\varphi \left(w,{J}^{-1}\left({d}_{n}J{y}_{n}+\left(1-{d}_{n}\right)JS{y}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}\varphi \left(w,{y}_{n}\right)+\left(1-{d}_{n}\right)\varphi \left(w,S{y}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}\varphi \left(w,{y}_{n}\right)+\left(1-{d}_{n}\right)\varphi \left(w,{y}_{n}\right)=\varphi \left(w,{y}_{n}\right).\phantom{\rule{2em}{0ex}}\end{array}$

So, Ω C n . By induction, Ω C n for each n {0}.

Clearly, Ω D0. Suppose that Ω Dn-1. Let w Ω. Then, w F(T1) F(T2) and

$\begin{array}{ll}\hfill ||{y}_{n}-w||& \le {a}_{n}||{x}_{n-1}-w||+{b}_{n}||{T}_{1}{y}_{n}-w||+{c}_{n}||{T}_{2}{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w||+{b}_{n}||{y}_{n}-w||+{c}_{n}||{y}_{n}-w||.\phantom{\rule{2em}{0ex}}\end{array}$

This implies that ||y n - w|| ≤ ||xn-1- w|| and w D n . By induction, Ω D n for each n {0}. So, Ω C n D n for each n {0}.

Since ${x}_{n}={\Pi }_{{C}_{n}\cap {D}_{n}{x}_{0}}$,

$\varphi \left({x}_{n},{x}_{0}\right)\le \varphi \left(w,{x}_{0}\right)-\varphi \left(w,{x}_{n}\right)\le \varphi \left(w,{x}_{0}\right)$

for each w Ω. Therefore, {ϕ(x n , x 0 )} is a bounded sequence. Furthermore, {x n } is a bounded sequence.

By Lemma 2.5, ${x}_{n}={\Pi }_{{C}_{n}\cap {D}_{n}{x}_{0}}$, and ${x}_{n+1}={\Pi }_{{C}_{n+1}\cap {D}_{n+1}}{x}_{0}$,

$\varphi \left({x}_{n+1},{x}_{n}\right)=\varphi \left({x}_{n+1},{\Pi }_{{C}_{n}\cap {D}_{n}}{x}_{0}\right)\le \varphi \left({x}_{n+1},{x}_{0}\right)-\varphi \left({x}_{n},{x}_{0}\right).$

Hence, ϕ(x n , x0) ≤ ϕ(xn+ 1, x0), limn→∞ϕ(x n , x0) exists, and limn→∞ϕ(xn+1, x n ) = 0. By Lemma 2.4, limn→∞, ||xn+1- x n || = 0. Since x n D n , we know that ||y n -x n || ≤ ||xn-1-x n || and limn→∞||x n -y n || = 0. Furthermore, limn→∞ϕ (x n , y n ) = 0. Since x n C n , it is easy to see that limn→∞ϕ(x n , z n ) = 0. Hence, limn→∞||x n -z n || = 0.

Take any w Ω and let w be fixed. Let r: = 2sup{||x n ||, ||x n - w||, ||y n ||, ||y n -w||: n }. By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2r] → [0, ∞) such that g(0) = 0 and

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{y}_{n}-w|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w|{|}^{2}+{b}_{n}||{T}_{1}{y}_{n}-w|{|}^{2}+{c}_{n}||{T}_{2}{y}_{n}-w|{|}^{2}-{a}_{n}{b}_{n}g\left(||{x}_{n-1}-{T}_{1}{y}_{n}||\right)\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w|{|}^{2}+{b}_{n}||{y}_{n}-w|{|}^{2}+{c}_{n}||{y}_{n}-w|{|}^{2}-{a}_{n}{b}_{n}g\left(||{x}_{n-1}-{T}_{1}{y}_{n}||\right).\phantom{\rule{2em}{0ex}}\end{array}$

This implies that

${b}_{n}g\left(||{x}_{n-1}-{T}_{1}{y}_{n}||\right)\le ||{x}_{n-1}-{y}_{n}||\left(||{x}_{n-1}-w||+||{y}_{n}-w||\right).$

So, limn→∞b n g(||xn-1- T1y n ||) = 0. By (ii), limn→∞||xn-1- T1y n || = 0. Furthermore, limn→∞||y n - T1y n || = 0. Similarly, limn→∞||y n - T2y n || = 0.

By Lemma 2.2, there exists a strictly increasing, continuous, and convex function g': [0, 2r] → [0, ∞) such that g'(0) = 0 and

$\begin{array}{ll}\hfill \varphi \left(w,{z}_{n}\right)& \le {d}_{n}\varphi \left(w,{y}_{n}\right)+\left(1-{d}_{n}\right)\varphi \left(w,S{y}_{n}\right)-{d}_{n}\left(1-{d}_{n}\right)g\prime \left(||J{y}_{n}-JS{y}_{n}||\right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left(w,{y}_{n}\right)-{d}_{n}\left(1-{d}_{n}\right)g\prime \left(||J{y}_{n}-JS{y}_{n}||\right).\phantom{\rule{2em}{0ex}}\end{array}$

Hence,

$\begin{array}{l}\phantom{\rule{1em}{0ex}}{d}_{n}\left(1-{d}_{n}\right)g\prime \left(||J{y}_{n}-JS{y}_{n}||\right)\phantom{\rule{2em}{0ex}}\\ \le \varphi \left(w,{y}_{n}\right)-\varphi \left(w,{z}_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\left(||w|{|}^{2}+||{y}_{n}|{|}^{2}-2⟨w,J{y}_{n}⟩\right)-\left(||w|{|}^{2}+||{z}_{n}|{|}^{2}-2⟨w,J{z}_{n}⟩\right)\phantom{\rule{2em}{0ex}}\\ =||{y}_{n}|{|}^{2}-||{z}_{n}|{|}^{2}+2⟨w,J{z}_{n}-J{y}_{n}⟩\phantom{\rule{2em}{0ex}}\\ =||{y}_{n}-{z}_{n}||\left(||{y}_{n}||+||{z}_{n}||\right)+2||w||\cdot ||J{z}_{n}-J{y}_{n}||.\phantom{\rule{2em}{0ex}}\end{array}$

By Remark 2.1, limn→∞d n (1 - d n )g(||Jy n - JSy n ||) = 0. By assumptions and the properties of g, limn→∞||Jy n - JSy n || = 0. Furthermore, limn→∞||y n - Sy n || = 0.

Since {y n } is a bounded sequence, there exists a subsequence $\left\{{y}_{{n}_{k}}\right\}$ of {y n } and $\stackrel{̄}{x}\in C$ such that ${y}_{{n}_{k}}⇀\stackrel{̄}{x}$. By Lemma 2.6, $\stackrel{̄}{x}\in F\left({T}_{1}\right)\cap F\left({T}_{2}\right)$. Besides, since S is a relatively nonexpansive mapping, $\stackrel{̄}{x}\in \stackrel{^}{F}\left(S\right)=F\left(S\right)$. So, $\stackrel{̄}{x}\in \Omega$.

Finally, we want to show that y n → ΠΩx0. Let q = ΠΩx0. Then, q Ω C n D n for each n . So,

$\varphi \left({x}_{n},{x}_{0}\right)=\underset{y\in {C}_{n}\cap {D}_{n}}{min}\varphi \left(y,{x}_{0}\right)\le \varphi \left(q,{x}_{0}\right).$

On the other hand, from weakly lower semicontinuity of the norm and limn→∞||x n -y n || = 0, we have

$\begin{array}{ll}\hfill \varphi \left(\stackrel{̄}{x},{x}_{0}\right)& =||\stackrel{̄}{x}|{|}^{2}-2⟨\stackrel{̄}{x},J{x}_{0}⟩+||{x}_{0}|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{liminf}\left(||{y}_{{n}_{k}}|{|}^{2}-2⟨{y}_{{n}_{k}},J{x}_{0}⟩+||{x}_{0}|{|}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{liminf}\left(||{x}_{{n}_{k}}|{|}^{2}-2⟨{x}_{{n}_{k}},J{x}_{0}⟩+||{x}_{0}|{|}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{liminf}\varphi \left({x}_{{n}_{k}},{x}_{0}\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\varphi \left({x}_{{n}_{k}},{x}_{0}\right)\le \varphi \left(q,{x}_{0}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Since $q={\Pi }_{\Omega }{x}_{0},\stackrel{̄}{x}=q$. Hence, $\underset{n\to \infty }{lim}\varphi \left({x}_{{n}_{k}},{x}_{0}\right)=\varphi \left(\stackrel{̄}{x},{x}_{0}\right)$. So, we have $\underset{n\to \infty }{lim}||{x}_{{n}_{k}}||=||\stackrel{̄}{x}||$. Using the Kadec-Klee property of E, we obtain that $\underset{k\to \infty }{lim}{x}_{{n}_{k}}=q={\Pi }_{\Omega }{x}_{0}$.

Furthermore, for each weakly convergence subsequence $\left\{{x}_{{n}_{m}}\right\}$ of {x n }, we know that $\underset{m\to \infty }{lim}{x}_{{n}_{m}}=q={\Pi }_{\Omega }{y}_{1}$ by following the same argument as the above conclusion. Therefore,

$\underset{n\to \infty }{lim}{x}_{n}=\underset{n\to \infty }{lim}{y}_{n}=\underset{n\to \infty }{lim}{z}_{n}={\Pi }_{\Omega }{x}_{0}.$

Hence, the proof is completed.

Remark 4.2. Since nonspreading mappings with fixed points in a strictly convex Banach space with a uniformly Gateaux differentiable norm are relatively nonex-pansive mappings [[23], Theorem 3.3], we know that the conclusion of Theorem 4.1 is still true if S is replaced by a nonspreading mapping.

Next, we give an easy example for Theorem 4.1.

Example 4.1. Let E = , C: = [0,3], T1x = T2x = x, and let S: CC be the as in [1]:

$Sx:=\left\{\begin{array}{ccc}\hfill 0\hfill & \hfill \mathsf{\text{if}}\hfill & \hfill x\ne 3,\hfill \\ \hfill 1\hfill & \hfill \mathsf{\text{if}}\hfill & \hfill x=3.\hfill \end{array}\right\$

For each n, let ${a}_{n}={b}_{n}={c}_{n}=\frac{1}{3}$ and ${d}_{n}=\frac{1}{2}$. Let x0 = 1. Hence, we have

1. (a)

y n = x n-1for each n ;

2. (b)

${z}_{n}=\frac{1}{2}{y}_{n}$ for each n ;

3. (c)

${C}_{n}:=\left\{z\in {C}_{n-1}:|z-{z}_{n}|\le |z-{y}_{n}|\right\}=0\left[0,\frac{{y}_{n}+zn}{2}\right]$;

4. (d)

D n : = {z D n-1: |z - y n | ≤ |z - x n-1|} = [0,3];

5. (e)

${x}_{n}=\frac{1}{2}\left({y}_{n}+{z}_{n}\right)=\frac{1}{2}\left({x}_{n-1}+\frac{1}{2}{x}_{n-1}\right)=\frac{3}{4}{x}_{n-1}$.

By (e) and x0 = 1, we know that ${x}_{n}={\left(\frac{3}{4}\right)}^{n}$ for each n {0}, limn→∞x n = 0, and 0 is a common fixed point of S, T1, and T2.

The following results are special cases of Theorem 4.1.

Corollary 4.1. Let E be a uniformly convex and uniformly smooth Banach space with condition (Q), and let C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings. Let {a n }, {b n }, {c n } be sequences in (0,1) with and a n + b n + c n = 1. Suppose that $\Omega :=F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{D}_{0}=C,\hfill \\ \hfill {y}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n},\hfill \\ \hfill {D}_{n}=\left\{z\in {D}_{n-1}:||{y}_{n}-z||\le ||{x}_{n-1}-z||\right\},\hfill \\ \hfill {x}_{n}={\Pi }_{{D}_{n}}{x}_{0.}\hfill \end{array}\right\$

Assume that lim infn→∞b n > 0, lim infn→∞c n > 0. Then, limn→∞x n = limn→∞, y n = ∏Ωx0.

Corollary 4.2. Let E be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E, and let S: CC be a relatively nonexpansive mapping. Let {d n } be a sequence in (0,1). Suppose that $F\left(S\right)\ne 0̸$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{C}_{0}=C,\hfill \\ \hfill {z}_{n}={J}^{-1}\left({d}_{n}J{x}_{n-1}+\left(1-{d}_{n}\right)JS{x}_{n-1}\right),\hfill \\ \hfill {C}_{n}=\left\{z\in {C}_{n-1}:\varphi \left(z,{z}_{n}\right)\le \varphi \left(z,{x}_{n-1}\right)\right\},\hfill \\ \hfill {x}_{n}={\Pi }_{{C}_{n}}{x}_{0.}\hfill \end{array}\right\$

Assume that lim infn→∞d n (1 - d n ) > 0. Then, limn→∞x n = limn→∞z n = ΠF(s)x0.

Remark 4.3. Corollary 4.2 is a generalization of Theorem 4.1 in [24]. But, it is a special case of Theorem 3.1 in [25].

## 5 Strong convergence theorems (II)

In this section, we need the following important lemmas.

Lemma 5.1. [26] Let E be a reflexive Banach space and f: E {+∞} be a convex and lower semicontinuous function. Let C be a nonempty bounded and closed convex subset of E. Then, the function f attains its minimum on C. That is, there exists x* C such that f(x*) ≤ f(x) for all x C.

Lemma 5.2. In a Banach space E, there holds the inequality

$||x+y|{|}^{2}\le ||x|{|}^{2}+2⟨y,j\left(x+y\right)⟩,\phantom{\rule{1em}{0ex}}x,y\in E,$

where j(x+y) J(x+y).

Lemma 5.3. [27] Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gâteaux differentiable norm. Let {x n } be a bounded sequence of E and let μ n be a Banach limit and z C. Then,

${\mu }_{n}||{x}_{n}-z|{|}^{2}=\underset{y\in C}{min}{\mu }_{n}||{x}_{n}-y|{|}^{2}⇔{\mu }_{n}⟨y-z,J\left({x}_{n}-z\right)⟩\le 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

Lemma 5.4. [28] Let α be a real number and (x0, x1,...) 2 such that μ n x n α for all Banach μ n . If lim supn→∞(xn+1- x n ) ≤ 0, then lim supn→∞x n α.

Lemma 5.5. [29] Assume that {a n }nis a sequence of nonnegative real numbers such that an+1< (1 - γ n )a n + δ n , n , where {γ n } (0,1) and δ n is a sequence in such that (i) ${\sum }_{n=1}^{\infty }{\gamma }_{n}=\infty$; (ii) $lim\underset{n\to \infty }{sup}\frac{\delta n}{\gamma n}\le 0$ or ${\sum }_{n=1}^{\infty }|{\delta }_{n}|<\infty$. Then, limn→∞a n = 0.

Theorem 5.1. Let E be a uniformly convex and uniformly smooth Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let S: CC be a mapping with condition (C). Let {a n }, {b n }, and {c n } be sequences in (a, b) for some 0 < a, b < 1 with a n + b n + c n = 1. Let {d n } be a sequence in [0,1]. Suppose that $\Omega :=F\left(S\right)\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne \varnothing$. Define a sequence {x n } by

$\left\{\begin{array}{c}\hfill {x}_{0}\in C\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{chosen}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{arbitrary}},\hfill \\ \hfill {y}_{n}={a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n},\hfill \\ \hfill {x}_{n}={d}_{n}{x}_{0}+\left(1-{d}_{n}\right)S{y}_{n}.\hfill \end{array}\right\$

Assume that:

1. (i)

$\underset{n\to \infty }{lim}{d}_{n}=0,\phantom{\rule{2.77695pt}{0ex}}\sum _{n=1}^{\infty }{d}_{n}=\infty$, and $\underset{n\to \infty }{lim}\frac{|{d}_{n+1}-{d}_{n}|}{{d}_{n}}=0$;

2. (ii)

$\underset{n\to \infty }{lim}\left({a}_{n+1}-{a}_{n}\right)=\underset{n\to \infty }{lim}\left({b}_{n+1}-{b}_{n}\right)=\underset{n\to \infty }{lim}\left({c}_{n+1}-{c}_{n}\right)=0$.

Then, $\underset{n\to \infty }{lim}{x}_{n}=\underset{n\to \infty }{lim}{y}_{n}=\stackrel{̄}{x}$ for some $\stackrel{̄}{x}\in \Omega$.

Proof. Following the same argument as in Lemma 3.1, we know that {y n } is well-defined. Take any w Ω: = F(S) F(T1) F(T2) and let w be fixed. Then, for each n , we have

This implies that ||y n - w|| ≤ ||xn-1- w|| for each n . Next, we have

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{x}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}||{x}_{0}-w||+\left(1-{d}_{n}\right)||S{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}||{x}_{0}-w||+\left(1-{d}_{n}\right)||{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le {d}_{n}||{x}_{0}-w||+\left(1-{d}_{n}\right)||{x}_{n-1}-w||\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}⋮\phantom{\rule{2em}{0ex}}\\ \le max\left\{||{x}_{0}-w||,||{x}_{1}-w||\right\}.\phantom{\rule{2em}{0ex}}\end{array}$

Then, {x n } is a bounded sequence. Furthermore, {y n }, {Sy n }, {T1y n }, {T2y n } are bounded sequences. Define M as

$M:=sup\left\{||{x}_{n}||,||{y}_{n}||,||{T}_{1}{y}_{n}||,||{T}_{2}{y}_{n}||,||S{y}_{n}||,||{x}_{n}-w||,||{y}_{n}-w||:n\in ℕ\right\}.$

Besides, we know that

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{limsup}||{x}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}\left({d}_{n}||{x}_{0}-w||+\left(1-{d}_{n}\right)||{y}_{n}-w||\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{limsup}{d}_{n}||{x}_{0}-w||+\underset{n\to \infty }{limsup}||{y}_{n}-w||\phantom{\rule{2em}{0ex}}\\ \le \underset{n←\infty }{limsup}||{x}_{n-1}-w||.\phantom{\rule{2em}{0ex}}\end{array}$

This implies that

$\underset{n\to \infty }{limsup}||{x}_{n}-w||=\underset{n\to \infty }{limsup}||{y}_{n}-w||.$

By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2M] → such that

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{y}_{n}-w|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w|{|}^{2}+{b}_{n}||{T}_{1}{y}_{n}-w|{|}^{2}+{c}_{n}||{T}_{2}{y}_{n}-w|{|}^{2}-{a}_{n}{b}_{n}||{x}_{n-1}-{T}_{1}{y}_{n}|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {a}_{n}||{x}_{n-1}-w|{|}^{2}+{b}_{n}||{y}_{n}-w|{|}^{2}+{c}_{n}||{y}_{n}-w|{|}^{2}-{a}_{n}{b}_{n}||{x}_{n-1}-{T}_{1}{y}_{n}|{|}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

Then,

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{y}_{n}-w|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}||{y}_{n}-w|{|}^{2}+a||{x}_{n-1}-{T}_{1}{y}_{n}|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}||{y}_{n}-w|{|}^{2}+{b}_{n}||{x}_{n-1}-{T}_{1}{y}_{n}|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}||{x}_{n-1}-w|{|}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

This implies that

$\underset{n\to \infty }{lim}||{x}_{n-1}-{T}_{1}{y}_{n}||=0.$

Similar, we have

$\underset{n\to \infty }{lim}||{x}_{n-1}-{T}_{2}{y}_{n}||=0.$

By (i),

$\underset{n\to \infty }{lim}||{x}_{n}-S{y}_{n}||=\underset{n\to \infty }{lim}{d}_{n}||{x}_{0}-S{y}_{n}||=0$

and

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{x}_{n+1}-{x}_{n}||\phantom{\rule{2em}{0ex}}\\ =||{d}_{n+1}{x}_{0}+\left(1-{d}_{n+1}\right)S{y}_{n+1}-{d}_{n}{x}_{0}-\left(1-{d}_{n}\right)S{y}_{n}||\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}||{d}_{n+1}{x}_{0}+\left(1-{d}_{n+1}\right)S{y}_{n+1}-{d}_{n}{x}_{0}-\left(1-{d}_{n}\right)S{y}_{n+1}||\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+||{d}_{n}{x}_{0}+\left(1-{d}_{n}\right)S{y}_{n+1}-{d}_{n}{x}_{0}-\left(1-{d}_{n}\right)S{y}_{n}||\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{2.77695pt}{0ex}}|{d}_{n+1}-{d}_{n}|\cdot ||{x}_{0}||+|{d}_{n+1}-{d}_{n}|\cdot ||S{y}_{n+1}||+\left(1-{d}_{n}\right)\cdot ||S{y}_{n+1}-S{y}_{n}||\phantom{\rule{2em}{0ex}}\\ \le |{d}_{n+1}-{d}_{n}|\cdot ||{x}_{0}||+|{d}_{n+1}-{d}_{n}|\cdot ||S{y}_{n+1}||+\left(1-{d}_{n}\right)\cdot ||{x}_{n+1}-{d}_{n+1}{x}_{0}-{x}_{n}+{d}_{n}{x}_{0}||\phantom{\rule{2em}{0ex}}\\ \le 2M\cdot |{d}_{n+1}-{d}_{n}|+\left(1-{d}_{n}\right)\cdot \left(||{x}_{n+1}-{x}_{n}||+|{d}_{n+1}-{d}_{n}|\cdot ||{x}_{0}||\right).\phantom{\rule{2em}{0ex}}\end{array}$

So,

$||{x}_{n+1}-{x}_{n}||\le \frac{3M\cdot |{d}_{n+1}-{d}_{n}|}{{d}_{n}}.$

By (i),

$\underset{n\to \infty }{lim}||{x}_{n+1}-{x}_{n}||=0.$

Furthermore,

$\underset{n\to \infty }{lim}||S{y}_{n+1}-S{y}_{n}||=0.$

Next, we have

$\begin{array}{l}||{y}_{n+1}-{y}_{n}||\\ =||\left({a}_{n+1}{x}_{n}+{b}_{n+1}{T}_{1}{y}_{n+1}+{c}_{n+1}{T}_{2}{y}_{n+1}\right)-\left({a}_{n}{x}_{n-1}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n}\right)||\\ \le ||\left({a}_{n+1}{x}_{n}+{b}_{n+1}{T}_{1}{y}_{n+1}+{c}_{n+1}{T}_{2}{y}_{n+1}\right)-\left({a}_{n}{x}_{n}+{b}_{n}{T}_{1}{y}_{n+1}+{c}_{n}{T}_{2}{y}_{n+1}\right)||\\ +||\left({a}_{n}{x}_{n}+{b}_{n}{T}_{1}{y}_{n+1}+{c}_{n}{T}_{2}{y}_{n+1}\right)-\left({a}_{n}{x}_{n-1}\text{+}{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{T}_{2}{y}_{n}\right)||\\ \le |{a}_{n+1}-{a}_{n}|·||{x}_{n}||+|{b}_{n+1}-{b}_{n}|·||{T}_{1}{y}_{n+1}||+|{c}_{n+1}-{c}_{n}|·||{T}_{2}{y}_{n+1}\right)||\\ +{a}_{n}||{x}_{n}-{x}_{n-1}||+{b}_{n}||{T}_{1}{y}_{n+1}-{T}_{1}{y}_{n}||+{c}_{n}||{T}_{2}{y}_{n+1}-{T}_{2}{y}_{n}||\\ \le M·\left(|{a}_{n+1}-{a}_{n}|+|{b}_{n+1}-{b}_{n}|+|{c}_{n+1}-{c}_{n}|\right)\\ +{a}_{n}||{x}_{n}-{x}_{n-1}||+{b}_{n}||{y}_{n+1}-{y}_{n}||+{c}_{n}||{y}_{n+1}-{y}_{n}||.\end{array}$

This implies that

$||{y}_{n+1}-{y}_{n}||\phantom{\rule{2.77695pt}{0ex}}\le \frac{M\cdot \left(|{a}_{n+1}-{a}_{n}|+|{b}_{n+1}-{b}_{n}|+|{c}_{n+1}-{c}_{n}|\right)}{{a}_{n}}+||{x}_{n}-{x}_{n-1}||.$

So,

$\underset{n\to \infty }{lim}||{y}_{n+1}-{y}_{n}||=0.$

Besides,

$\underset{n\to \infty }{lim}||{y}_{n}-{x}_{n-1}||=\underset{n\to \infty }{lim}||{b}_{n}\left({T}_{1}{y}_{n}-{x}_{n-1}\right)+{c}_{n}\left({T}_{2}{y}_{n}-{x}_{n-1}\right)||=0$

and

$\underset{n\to \infty }{lim}||{y}_{n}-S{y}_{n}||=\underset{n\to \infty }{lim}||{y}_{n}-{T}_{1}{y}_{n}||=\underset{n\to \infty }{lim}||{y}_{n}-{T}_{2}{y}_{n}||=0.$

Let φ: C be defined by φ(u): = μ n ||x n - u|| for each u C. Clearly, φ is convex and continuous. Taking p Ω and defining a subset D of C by

$D:=\left\{x\in C:||x-p||\le r\right\},$

where r: = max{||x0 - p||, ||x1 - p||}. Then, D is a nonempty closed bounded convex subset of C and {x n } D. By Lemma 5.1,

${C}_{min}:=\left\{z\in D:\phi \left(z\right):=\underset{y\in D}{min}\phi \left(y\right)\right\}\ne 0̸.$

Obviously, Cmin is a bounded closed convex subset. Following the property of Banach limit μ n , for all z Cmin, we have

$\begin{array}{ll}\hfill \phi \left(Sz\right)& ={\mu }_{n}||{x}_{n}-Sz|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\mu }_{n}{\left(||{x}_{n}-{y}_{n}||+||{y}_{n}-Sz||\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\mu }_{n}{\left(||{x}_{n}-{y}_{n}||+3||{y}_{n}-S{y}_{n}||+||{y}_{n}-z||\right)}^{2}\phantom{\rule{2em}{0ex}}\\ ={\mu }_{n}||{y}_{n}-z|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\mu }_{n}{\left(||{y}_{n}-{x}_{n}||+||{x}_{n}-z||\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\mu }_{n}||{x}_{n}-z|{|}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

Then, Sz Cmin. By Theorem 4 in [1], there exists $\stackrel{̄}{x}\in {C}_{min}$ such that $S\stackrel{̄}{x}=\stackrel{̄}{x}$. By Lemma 5.3,

${\mu }_{n}⟨y-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩\le 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

Take any y C and let y be fixed. Since limn→∞||xn+1- x n || = 0, then it follows from the norm-weak* uniformly continuity of the duality mapping J that

$\underset{n\to \infty }{lim}\left(⟨y-\stackrel{̄}{x},J\left({x}_{n+1}-\stackrel{̄}{x}\right)⟩-⟨y-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩\right)=0.$

By Lemma 5.4,

$\underset{n\to \infty }{lim}⟨y-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩\le 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

By Lemma 5.2,

$\begin{array}{l}\phantom{\rule{1em}{0ex}}||{x}_{n}-\stackrel{̄}{x}|{|}^{2}\phantom{\rule{2em}{0ex}}\\ =||{d}_{n}\left({x}_{0}-\stackrel{̄}{x}\right)+\left(1-{d}_{n}\right)\left(S{y}_{n}-\stackrel{̄}{x}\right)|{|}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\left(1-{d}_{n}\right)}^{2}||S{y}_{n}-\stackrel{̄}{x}|{|}^{2}+2{d}_{n}⟨{x}_{0}-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩\phantom{\rule{2em}{0ex}}\\ \le {\left(1-{d}_{n}\right)}^{2}||{y}_{n}-\stackrel{̄}{x}|{|}^{2}+2{d}_{n}⟨{x}_{0}-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩\phantom{\rule{2em}{0ex}}\\ \le \left(1-{d}_{n}\right)||{x}_{n-1}-\stackrel{̄}{x}|{|}^{2}+2{d}_{n}⟨{x}_{0}-\stackrel{̄}{x},J\left({x}_{n}-\stackrel{̄}{x}\right)⟩.\phantom{\rule{2em}{0ex}}\end{array}$

By Lemma 5.5, $\underset{n\to \infty }{lim}||{x}_{n}-\stackrel{̄}{x}||=0$. Furthermore, since T1 and T2 are nonexpan-sive mappings, we know that $\stackrel{̄}{x}$ is also a fixed point of T1 and T2. Therefore, the proof is completed.

The following is a special case of Theorem 5.1 when T1 and T2 are identity mappings.

Theorem 5.2. Let E be a uniformly convex and uniformly smooth Banach space with Opial's condition, C be a nonempty closed convex subset of E, and let S: CC be a mapping with condition (C). Let {d n } be a sequence in (0,1). Suppose that $F\left(S\right)\ne 0̸$. Define a sequence {x n } by

Assume that $\underset{n\to \infty }{lim}{d}_{n}=0,{\sum }_{n=1}^{\infty }{d}_{n}=\infty$, and $\underset{n\to \infty }{lim}\frac{|{d}_{n+1}-{d}_{n}|}{{d}_{n}}=0.$. Then, $\underset{n\to \infty }{lim}{x}_{n}=\underset{n\to \infty }{lim}{y}_{n}=\stackrel{̄}{x}$ for some $\stackrel{̄}{x}\in F\left(S\right)$.

## 6 Application

Let E be a reflexive, strictly convex, and smooth Banach space and let A E × E* be a set-valued mapping with range R(A): = {x* E*: x* Ax} and domain $D\left(A\right)=\left\{x\in E:Ax\ne 0̸\right\}$. Then, the mapping A is said to be monotone if 〈x -y,x* - y*〉 ≥ 0 whenever (x, x*), (y, y*) A. It is also said to be maximal monotone if A is monotone and there is no monotone operator from E into E* whose graph properly contains the graph of A. It is known that if A E × E* is maximal monotone, then A-10 is closed and convex.

Lemma 6.1. [30] Let E be a reflexive, strictly convex, and smooth Banach space and let A E × E* be a monotone operator. Then, A is maximal monotone if and only if R(J + rA) = E* for all r > 0.

By Lemma 6.1, for every r > 0 and x E, there exists a unique x r D(A) such that Jx Jx r + rAx r . Hence, define a single valued mapping J r : ED(A) by J r x = x r , that is, J r = (J + rA)-1J and such J r is called the relative resolvent of A. We know that A-10 = F(J r ) for all r > 0 [8].

Lemma 6.2. [21] Let E be a uniformly convex and uniformly smooth Banach space and let A E × E* be a maximal monotone operator. Let J r be the relative resolvent of A, where r > 0. If A-10 is nonempty, then J r is a relatively nonexpansive mapping on E.

By Theorem 4.1 and Lemma 6.2, it is easy to get the following result.

Theorem 6.1. Let E be a uniformly convex and uniformly smooth Banach space with property (Q), and let C be a nonempty closed convex subset of E, and let T1, T2 : CC be two nonexpansive mappings, and let A E × E* be a maximal monotone operator. Let {a n }, {b n }, {c n }, and {d n } be sequences in (0,1) with and a n + b n + c n = 1. Suppose that $\Omega :={A}^{-1}0\cap F\left({T}_{1}\right)\cap F\left({T}_{2}\right)\ne 0̸$. Define a sequence {x n } by

Assume that lim infn→∞b n > 0, lim infn→∞c n > 0, and lim infn→∞d n (1 - d n ) > 0.

Then, limn→∞x n = limn→∞y n = limn→∞z n = ΠΩx0.

## References

1. Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J Math Anal Appl 2008, 340: 1088–1095. 10.1016/j.jmaa.2007.09.023

2. Mann WR: Mean value methods in iteration. Proc Am Math Soc 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

3. Soltuz SM: The backward Mann iteration. Octogon Math Mag 2001,9(2):797–800.

4. Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Numer Funct Anal Optim 2001, 22: 767–773. 10.1081/NFA-100105317

5. Khan SH, Yildirim I, Ozdemir M: Convergence of an implicit algorithm for two families of nonexpansive mappings. Comput Math Appl 2010, 59: 3084–3091. 10.1016/j.camwa.2010.02.029

6. Chidume CE, Shahzad N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. Nonlinear Anal 2005, 65: 1149–1156.

7. Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asympotically quasi-nonexpansive mappings and applications. J Math Anal Appl 2007, 328: 821–829. 10.1016/j.jmaa.2006.05.068

8. Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and its Applications. Yokohama Publishers Inc, Yokohama; 2000.

9. Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl 2010: Article ID 218573, 11

10. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

11. Schu J: Weak and strong convergence to fixed points of asymptotically non-expansive mappings. Bull Austral Math Soc 1991, 43: 153–159. 10.1017/S0004972700028884

12. James RC: Inner products in normed linear spaces. Bull Am Math Soc 1947, 53: 559–566. 10.1090/S0002-9904-1947-08831-5