Open Access

Convergence theorems for mixed type asymptotically nonexpansive mappings

Fixed Point Theory and Applications20122012:224

DOI: 10.1186/1687-1812-2012-224

Received: 27 April 2012

Accepted: 16 November 2012

Published: 11 December 2012

Abstract

In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Keywords

mixed type asymptotically nonexpansive mapping strong and weak convergence common fixed point uniformly convex Banach space

1 Introduction

Let K be a nonempty subset of a real normed linear space E. A mapping T : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq1_HTML.gif is said to be asymptotically nonexpansive if there exists a sequence { k n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq2_HTML.gif with lim n k n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq3_HTML.gif such that
T n x T n y k n x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ1_HTML.gif
(1.1)

for all x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq4_HTML.gif and n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif.

In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.

Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [216]), which extend and improve the result of Goebel and Kirk in several ways.

Recently, Chidume et al. [10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

Definition 1.1 [10]

Let K be a nonempty subset of a real normed linear space E. Let P : E K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq6_HTML.gif be a nonexpansive retraction of E onto K. A nonself-mapping T : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq7_HTML.gif is said to be asymptotically nonexpansive if there exists a sequence { k n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq8_HTML.gif with k n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq9_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq10_HTML.gif such that
T ( P T ) n 1 x T ( P T ) n 1 y k n x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ2_HTML.gif
(1.2)

for all x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq4_HTML.gif and n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif.

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.

In 2003, also, Chidume et al. [10] studied the following iteration scheme:
{ x 1 K , x n + 1 = P ( ( 1 α n ) x n + α n T 1 ( P T 1 ) n 1 x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ3_HTML.gif
(1.3)

for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif, where { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq11_HTML.gif is a sequence in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq12_HTML.gif and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

In 2006, Wang [11] generalized the iteration process (1.3) as follows:
{ x 1 K , x n + 1 = P ( ( 1 α n ) x n + α n T 1 ( P T 1 ) n 1 y n ) , y n = P ( ( 1 β n ) x n + β n T 2 ( P T 2 ) n 1 x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ4_HTML.gif
(1.4)

for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif, where T 1 , T 2 : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq13_HTML.gif are two asymptotically nonexpansive nonself-mappings and { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq11_HTML.gif, { β n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq14_HTML.gif are real sequences in [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq15_HTML.gif, and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

2 Preliminaries

Let E be a real Banach space, K be a nonempty closed convex subset of E and P : E K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq6_HTML.gif be a nonexpansive retraction of E onto K. Let S 1 , S 2 : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq16_HTML.gif be two asymptotically nonexpansive self-mappings and T 1 , T 2 : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq17_HTML.gif be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:
{ x 1 K , x n + 1 = P ( ( 1 α n ) S 1 n x n + α n T 1 ( P T 1 ) n 1 y n ) , y n = P ( ( 1 β n ) S 2 n x n + β n T 2 ( P T 2 ) n 1 x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ5_HTML.gif
(2.1)

for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif, where { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq11_HTML.gif, { β n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq14_HTML.gif are two sequences in [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq15_HTML.gif.

If S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif and S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

We denote the set of common fixed points of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif by F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq22_HTML.gif and denote the distance between a point z and a set A in E by d ( z , A ) = inf x A z x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq23_HTML.gif.

Now, we recall some well-known concepts and results.

Let E be a real Banach space, E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq24_HTML.gif be the dual space of E and J : E 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq25_HTML.gif be the normalized duality mapping defined by
J ( x ) = { f E : x , f = x f , f = x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equa_HTML.gif

for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq26_HTML.gif, where , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq27_HTML.gif denotes duality pairing between E and E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq24_HTML.gif. A single-valued normalized duality mapping is denoted by j.

A subset K of a real Banach space E is called a retract of E [10] if there exists a continuous mapping P : E K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq6_HTML.gif such that P x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq28_HTML.gif for all x K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq29_HTML.gif. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq30_HTML.gif is called a retraction if P 2 = P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq31_HTML.gif. It follows that if a mapping P is a retraction, then P y = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq32_HTML.gif for all y in the range of P.

A Banach space E is said to satisfy Opial’s condition [17] if, for any sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif of E, x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq34_HTML.gif weakly as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq10_HTML.gif implies that
lim sup n x n x < lim sup n x n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equb_HTML.gif

for all y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq35_HTML.gif with y x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq36_HTML.gif.

A Banach space E is said to have a Fréchet differentiable norm [18] if, for all x U = { x E : x = 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq37_HTML.gif,
lim t 0 x + t y x t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equc_HTML.gif

exists and is attained uniformly in y U https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq38_HTML.gif.

A Banach space E is said to have the Kadec-Klee property [19] if for every sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif in E, x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq34_HTML.gif weakly and x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq39_HTML.gif, it follows that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq34_HTML.gif strongly.

Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping T : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq7_HTML.gif is said to be semi-compact [11] if, for any sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif in K such that x n T x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq40_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq10_HTML.gif, there exists a subsequence { x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq41_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif such that { x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq41_HTML.gif converges strongly to some x K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq42_HTML.gif.

Lemma 2.1 [15]

Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq43_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq44_HTML.gif and { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq45_HTML.gif be three nonnegative sequences satisfying the following condition:
a n + 1 ( 1 + b n ) a n + c n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equd_HTML.gif

for each n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq46_HTML.gif, where n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq47_HTML.gif is some nonnegative integer, n = n 0 b n < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq48_HTML.gif and n = n 0 c n < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq49_HTML.gif. Then lim n a n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq50_HTML.gif exists.

Lemma 2.2 [8]

Let E be a real uniformly convex Banach space and 0 < p t n q < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq51_HTML.gif for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif. Also, suppose that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq52_HTML.gif are two sequences of E such that
lim sup n x n r , lim sup n y n r , lim n t n x n + ( 1 t n ) y n = r https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Eque_HTML.gif

hold for some r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq53_HTML.gif. Then lim n x n y n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq54_HTML.gif.

Lemma 2.3 [10]

Let E be a real uniformly convex Banach space, K be a nonempty closed convex subset of E and T : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq7_HTML.gif be an asymptotically nonexpansive mapping with a sequence { k n } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq8_HTML.gif and k n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq9_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq10_HTML.gif. Then I T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq55_HTML.gif is demiclosed at zero, i.e., if x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq34_HTML.gif weakly and x n T x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq56_HTML.gif strongly, then x F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq57_HTML.gif, where F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq58_HTML.gif is the set of fixed points of T.

Lemma 2.4 [16]

Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function γ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq59_HTML.gif with γ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq60_HTML.gif such that, for each mapping S : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq61_HTML.gif with a Lipschitz constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq62_HTML.gif,
α S x + ( 1 α ) S y S [ α x + ( 1 α ) y ] L γ 1 ( x y 1 L S x S y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equf_HTML.gif

for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq63_HTML.gif and 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq64_HTML.gif.

Lemma 2.5 [16]

Let X be a uniformly convex Banach space such that its dual space X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq65_HTML.gif has the Kadec-Klee property. Suppose { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is a bounded sequence and f 1 , f 2 W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq66_HTML.gif such that
lim n α x n + ( 1 α ) f 1 f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equg_HTML.gif

exists for all α [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq67_HTML.gif, where W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq68_HTML.gif denotes the set of all weak subsequential limits of  { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif. Then f 1 = f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq69_HTML.gif.

3 Strong convergence theorems

In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

Lemma 3.1 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let S 1 , S 2 : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq70_HTML.gif be two asymptotically nonexpansive self-mappings with { k n ( 1 ) } , { k n ( 2 ) } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq71_HTML.gif and T 1 , T 2 : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq72_HTML.gif be two asymptotically nonexpansive nonself-mappings with { l n ( 1 ) } , { l n ( 2 ) } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq73_HTML.gif such that n = 1 ( k n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq74_HTML.gif and n = 1 ( l n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq75_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif, respectively, and F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq77_HTML.gif. Let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif be the sequence defined by (2.1), where { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq11_HTML.gif and { β n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq14_HTML.gif are two real sequences in [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq15_HTML.gif. Then
  1. (1)

    lim n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq78_HTML.gif exists for any q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif;

     
  2. (2)

    lim n d ( x n , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq80_HTML.gif exists.

     
Proof (1) Set h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq81_HTML.gif. For any q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif, it follows from (2.1) that
y n q ( 1 β n ) ( S 2 n x n q ) + β n ( T 2 ( P T 2 ) n 1 x n q ) ( 1 β n ) h n x n q + β n h n x n q = h n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ6_HTML.gif
(3.1)
and so
x n + 1 q ( 1 α n ) ( S 1 n x n q ) + α n ( T 1 ( P T 1 ) n 1 y n q ) ( 1 α n ) h n x n q + α n h n y n q ( 1 α n ) h n 2 x n q + α n h n 2 x n q = [ 1 + ( h n 2 1 ) ] x n q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ7_HTML.gif
(3.2)
Since n = 1 ( k n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq74_HTML.gif and n = 1 ( l n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq75_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif, we have n = 1 ( h n 2 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq82_HTML.gif. It follows from Lemma 2.1 that lim n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq78_HTML.gif exists.
  1. (2)
    Taking the infimum over all q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif in (3.2), we have
    d ( x n + 1 , F ) [ 1 + ( h n 2 1 ) ] d ( x n , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equh_HTML.gif
     

for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif. It follows from n = 1 ( h n 2 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq82_HTML.gif and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

Lemma 3.2 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let S 1 , S 2 : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq70_HTML.gif be two asymptotically nonexpansive self-mappings with { k n ( 1 ) } , { k n ( 2 ) } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq71_HTML.gif and T 1 , T 2 : K E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq72_HTML.gif be two asymptotically nonexpansive nonself-mappings with { l n ( 1 ) } , { l n ( 2 ) } [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq73_HTML.gif such that n = 1 ( k n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq74_HTML.gif and n = 1 ( l n ( i ) 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq75_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif, respectively, and F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq77_HTML.gif. Let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif be the sequence defined by (2.1) and the following conditions hold:
  1. (a)

    { α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq11_HTML.gif and { β n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq14_HTML.gif are two real sequences in [ ϵ , 1 ϵ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq83_HTML.gif for some ϵ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq84_HTML.gif;

     
  2. (b)

    x T i y S i x T i y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq85_HTML.gif for all x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq4_HTML.gif and i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif.

     

Then lim n x n S i x n = lim n x n T i x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq86_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif.

Proof Set h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq81_HTML.gif. For any given q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif, lim n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq78_HTML.gif exists by Lemma 3.1. Now, we assume that lim n x n q = c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq87_HTML.gif. It follows from (3.2) and n = 1 ( h n 2 1 ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq82_HTML.gif that
lim n ( 1 α n ) ( S 1 n x n q ) + α n ( T 1 ( P T 1 ) n 1 y n q ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equi_HTML.gif
and
lim sup n S 1 n x n q lim sup n k n ( 1 ) x n q = c . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equj_HTML.gif
Taking lim sup on both sides in (3.1), we obtain lim sup n y n q c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq88_HTML.gif and so
lim sup n T 1 ( P T 1 ) n 1 y n q lim sup n l n ( 1 ) y n q c . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equk_HTML.gif
Using Lemma 2.2, we have
lim n S 1 n x n T 1 ( P T 1 ) n 1 y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ8_HTML.gif
(3.3)
By the condition (b), it follows that
x n T 1 ( P T 1 ) n 1 y n S 1 n x n T 1 ( P T 1 ) n 1 y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equl_HTML.gif
and so, from (3.3), we have
lim n x n T 1 ( P T 1 ) n 1 y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ9_HTML.gif
(3.4)
Since
x n q x n T 1 ( P T 1 ) n 1 y n + T 1 ( P T 1 ) n 1 y n q x n T 1 ( P T 1 ) n 1 y n + l n ( 1 ) y n q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equm_HTML.gif
Taking lim inf on both sides in the inequality above, we have
lim inf n y n q c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equn_HTML.gif
by (3.4) and so
lim n y n q = c . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equo_HTML.gif
Using (3.1), we have
lim n ( 1 β n ) ( S 2 n x n q ) + β n ( T 2 ( P T 2 ) n 1 x n q ) = c . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equp_HTML.gif
In addition, we have
lim sup n S 2 n x n q lim sup n k n ( 2 ) x n q = c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equq_HTML.gif
and
lim sup n T 2 ( P T 2 ) n 1 x n q lim sup n l n ( 2 ) x n q = c . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equr_HTML.gif
It follows from Lemma 2.2 that
lim n S 2 n x n T 2 ( P T 2 ) n 1 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ10_HTML.gif
(3.5)
Now, we prove that
lim n x n T 1 x n = lim n x n T 2 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equs_HTML.gif
Indeed, since x n T 2 ( P T 2 ) n 1 x n S 2 n x n T 2 ( P T 2 ) n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq89_HTML.gif by the condition (b). It follows from (3.5) that
lim n x n T 2 ( P T 2 ) n 1 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ11_HTML.gif
(3.6)
Since S 2 n x n = P ( S 2 n x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq90_HTML.gif and P : E K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq6_HTML.gif is a nonexpansive retraction of E onto K, we have
y n S 2 n x n β n S 2 n x n T 2 ( P T 2 ) n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equt_HTML.gif
and so
lim n y n S 2 n x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ12_HTML.gif
(3.7)
Furthermore, we have
y n x n y n S 2 n x n + S 2 n x n T 2 ( P T 2 ) n 1 x n + T 2 ( P T 2 ) n 1 x n x n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equu_HTML.gif
Thus it follows from (3.5), (3.6) and (3.7) that
lim n x n y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ13_HTML.gif
(3.8)
Since x n T 1 ( P T 1 ) n 1 x n S 1 n x n T 1 ( P T 1 ) n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq91_HTML.gif by the condition (b) and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equv_HTML.gif
Using (3.3) and (3.8), we have
lim n S 1 n x n T 1 ( P T 1 ) n 1 x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ14_HTML.gif
(3.9)
and
lim n x n T 1 ( P T 1 ) n 1 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ15_HTML.gif
(3.10)
It follows from
x n + 1 S 1 n x n = P [ ( 1 α n ) S 1 n x n + α n T 1 ( P T 1 ) n 1 y n ] P ( S 1 n x n ) α n S 1 n x n T 1 ( P T 1 ) n 1 y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equw_HTML.gif
and (3.3) that
lim n x n + 1 S 1 n x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ16_HTML.gif
(3.11)
In addition, we have
x n + 1 T 1 ( P T 1 ) n 1 y n x n + 1 S 1 n x n + S 1 n x n T 1 ( P T 1 ) n 1 y n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equx_HTML.gif
Using (3.3) and (3.11), we obtain that
lim n x n + 1 T 1 ( P T 1 ) n 1 y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ17_HTML.gif
(3.12)
Thus, using (3.9), (3.10) and the inequality
S 1 n x n x n S 1 n x n T 1 ( P T 1 ) n 1 x n + T 1 ( P T 1 ) n 1 x n x n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equy_HTML.gif
we have lim n S 1 n x n x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq92_HTML.gif. It follows from (3.6) and the inequality
S 1 n x n T 2 ( P T 2 ) n 1 x n S 1 n x n x n + x n T 2 ( P T 2 ) n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equz_HTML.gif
that
lim n S 1 n x n T 2 ( P T 2 ) n 1 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ18_HTML.gif
(3.13)
Since
x n + 1 T 2 ( P T 2 ) n 1 y n x n + 1 S 1 n x n + S 1 n x n T 2 ( P T 2 ) n 1 x n + l n ( 2 ) x n y n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaa_HTML.gif
from (3.8), (3.11) and (3.13), it follows that
lim n x n + 1 T 2 ( P T 2 ) n 1 y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ19_HTML.gif
(3.14)
Again, since ( P T i ) ( P T i ) n 2 y n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq93_HTML.gif, x n K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq94_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif and T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif, T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif are two asymptotically nonexpansive nonself-mappings, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ20_HTML.gif
(3.15)
for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif. It follows from (3.12), (3.14) and (3.15) that
lim n T i ( P T i ) n 1 y n 1 T i x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ21_HTML.gif
(3.16)
for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif. Moreover, we have
x n + 1 y n x n + 1 T 1 ( P T 1 ) n 1 y n + T 1 ( P T 1 ) n 1 y n x n + x n y n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equab_HTML.gif
Using (3.4), (3.8) and (3.12), we have
lim n x n + 1 y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ22_HTML.gif
(3.17)
In addition, we have
x n T i x n x n T i ( P T i ) n 1 x n + T i ( P T i ) n 1 x n T i ( P T i ) n 1 y n 1 + T i ( P T i ) n 1 y n 1 T i x n x n T i ( P T i ) n 1 x n + max { sup n 1 l n ( 1 ) , sup n 1 l n ( 2 ) } x n y n 1 + T i ( P T i ) n 1 y n 1 T i x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equac_HTML.gif
for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq95_HTML.gif. Thus it follows from (3.6), (3.10), (3.16) and (3.17) that
lim n x n T 1 x n = lim n x n T 2 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equad_HTML.gif
Finally, we prove that
lim n x n S 1 x n = lim n x n S 2 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equae_HTML.gif
In fact, by the condition (b), we have
x n S i x n x n T i ( P T i ) n 1 x n + S i x n T i ( P T i ) n 1 x n x n T i ( P T i ) n 1 x n + S i n x n T i ( P T i ) n 1 x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaf_HTML.gif
for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq95_HTML.gif. Thus it follows from (3.5), (3.6), (3.9) and (3.10) that
lim n x n S 1 x n = lim n x n S 2 x n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equag_HTML.gif

This completes the proof. □

Now, we find two mappings, S 1 = S 2 = S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq96_HTML.gif and T 1 = T 2 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq97_HTML.gif, satisfying the condition (b) in Lemma 3.2 as follows.

Example 3.1 [20]

Let be the real line with the usual norm | | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq98_HTML.gif and let K = [ 1 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq99_HTML.gif. Define two mappings S , T : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq100_HTML.gif by
T x = { 2 sin x 2 , if  x [ 0 , 1 ] , 2 sin x 2 , if  x [ 1 , 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equah_HTML.gif
and
S x = { x , if  x [ 0 , 1 ] , x , if  x [ 1 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equai_HTML.gif
Now, we show that T is nonexpansive. In fact, if x , y [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq101_HTML.gif or x , y [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq102_HTML.gif, then we have
| T x T y | = 2 | sin x 2 sin y 2 | | x y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaj_HTML.gif
If x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq103_HTML.gif and y [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq104_HTML.gif or x [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq105_HTML.gif and y [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq106_HTML.gif, then we have
| T x T y | = 2 | sin x 2 + sin y 2 | = 4 | sin x + y 4 cos x y 4 | | x + y | | x y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equak_HTML.gif

This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with k n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq107_HTML.gif for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif. Similarly, we can show that S is an asymptotically nonexpansive mapping with l n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq108_HTML.gif for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif.

Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Case 1. Let x , y [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq101_HTML.gif. Then we have
| x T y | = | x + 2 sin y 2 | = | S x T y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equal_HTML.gif
Case 2. Let x , y [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq102_HTML.gif. Then we have
| x T y | = | x 2 sin y 2 | | x 2 sin y 2 | = | S x T y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equam_HTML.gif
Case 3. Let x [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq105_HTML.gif and y [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq109_HTML.gif. Then we have
| x T y | = | x + 2 sin y 2 | | x + 2 sin y 2 | = | S x T y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equan_HTML.gif
Case 4. Let x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq103_HTML.gif and y [ 1 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq104_HTML.gif. Then we have
| x T y | = | x 2 sin y 2 | = | S x T y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equao_HTML.gif

Therefore, the condition (b) in Lemma 3.2 is satisfied.

Theorem 3.1 Under the assumptions of Lemma  3.2, if one of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif is completely continuous, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Without loss of generality, we can assume that S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif is completely continuous. Since { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is bounded by Lemma 3.1, there exists a subsequence { S 1 x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq110_HTML.gif of { S 1 x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq111_HTML.gif such that { S 1 x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq110_HTML.gif converges strongly to some q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq112_HTML.gif. Moreover, we know that
lim j x n j S 1 x n j = lim j x n j S 2 x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equap_HTML.gif
and
lim j x n j T 1 x n j = lim j x n j T 2 x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaq_HTML.gif
by Lemma 3.2, which imply that
x n j q x n j S 1 x n j + S 1 x n j q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equar_HTML.gif
as j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq113_HTML.gif and so x n j q K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq114_HTML.gif. Thus, by the continuity of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif, we have
q S i q = lim j x n j S i x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equas_HTML.gif
and
q T i q = lim j x n j T i x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equat_HTML.gif

for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif. Thus it follows that q F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq115_HTML.gif. Furthermore, since lim n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq116_HTML.gif exists by Lemma 3.1, we have lim n x n q = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq117_HTML.gif. This completes the proof. □

Theorem 3.2 Under the assumptions of Lemma  3.2, if one of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif is semi-compact, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Since lim n x n S i x n = lim n x n T i x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq86_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq95_HTML.gif by Lemma 3.2 and one of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif is semi-compact, there exists a subsequence { x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq41_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif such that { x n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq41_HTML.gif converges strongly to some q K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq118_HTML.gif. Moreover, by the continuity of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif, we have q S i q = lim j x n j S i x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq119_HTML.gif and q T i q = lim j x n j T i x n j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq120_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif. Thus it follows that q F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq115_HTML.gif. Since lim n x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq116_HTML.gif exists by Lemma 3.1, we have lim n x n q = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq117_HTML.gif. This completes the proof. □

Theorem 3.3 Under the assumptions of Lemma  3.2, if there exists a nondecreasing function f : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq121_HTML.gif with f ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq122_HTML.gif and f ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq123_HTML.gif for all r ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq124_HTML.gif such that
f ( d ( x , F ) ) x S 1 x + x S 2 x + x T 1 x + x T 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equau_HTML.gif

for all x K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq29_HTML.gif, where F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq22_HTML.gif, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges strongly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Since lim n x n S i x n = lim n x n T i x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq86_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq95_HTML.gif by Lemma 3.2, we have lim n f ( d ( x n , F ) ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq125_HTML.gif. Since f : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq121_HTML.gif is a nondecreasing function satisfying f ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq122_HTML.gif, f ( r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq123_HTML.gif for all r ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq124_HTML.gif and lim n d ( x n , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq80_HTML.gif exists by Lemma 3.1, we have lim n d ( x n , F ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq126_HTML.gif.

Now, we show that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is a Cauchy sequence in K. In fact, from (3.2), we have
x n + 1 q [ 1 + ( h n 2 1 ) ] x n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equav_HTML.gif
for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq5_HTML.gif, where h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq127_HTML.gif and q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif. For any m, n, m > n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq128_HTML.gif, we have
x m q [ 1 + ( h m 1 2 1 ) ] x m 1 q e h m 1 2 1 x m 1 q e h m 1 2 1 e h m 2 2 1 x m 2 q e i = n m 1 ( h i 2 1 ) x n q M x n q , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaw_HTML.gif
where M = e i = 1 ( h i 2 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq129_HTML.gif. Thus, for any q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif, we have
x n x m x n q + x m q ( 1 + M ) x n q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equax_HTML.gif
Taking the infimum over all q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif, we obtain
x n x m ( 1 + M ) d ( x n , F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equay_HTML.gif

Thus it follows from lim n d ( x n , F ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq126_HTML.gif that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is a Cauchy sequence. Since K is a closed subset of E, the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges strongly to some q K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq118_HTML.gif. It is easy to prove that F ( S 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq130_HTML.gif, F ( S 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq131_HTML.gif, F ( T 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq132_HTML.gif and F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq133_HTML.gif are all closed and so F is a closed subset of K. Since lim n d ( x n , F ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq126_HTML.gif, q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq134_HTML.gif, the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges strongly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif. This completes the proof. □

4 Weak convergence theorems

In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

Lemma 4.1 Under the assumptions of Lemma  3.1, for all q 1 , q 2 F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq135_HTML.gif, the limit
lim n t x n + ( 1 t ) q 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equaz_HTML.gif

exists for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq136_HTML.gif, where { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is the sequence defined by (2.1).

Proof Set a n ( t ) = t x n + ( 1 t ) q 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq137_HTML.gif. Then lim n a n ( 0 ) = q 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq138_HTML.gif and, from Lemma 3.1, lim n a n ( 1 ) = lim n x n q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq139_HTML.gif exists. Thus it remains to prove Lemma 4.1 for any t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq140_HTML.gif.

Define the mapping G n : K K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq141_HTML.gif by
G n x = P [ ( 1 α n ) S 1 n x + α n T 1 ( P T 1 ) n 1 P ( ( 1 β n ) S 2 n x + β n T 2 ( P T 2 ) n 1 x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equba_HTML.gif
for all x K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq29_HTML.gif. It is easy to prove that
G n x G n y h n 4 x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ23_HTML.gif
(4.1)
for all x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq4_HTML.gif, where h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq127_HTML.gif. Letting h n = 1 + v n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq142_HTML.gif, it follows from 1 j = n h j 4 e 4 j = n v j https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq143_HTML.gif and n = 1 v n < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq144_HTML.gif that lim n j = n h j 4 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq145_HTML.gif. Setting
S n , m = G n + m 1 G n + m 2 G n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ24_HTML.gif
(4.2)
for each m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq146_HTML.gif, from (4.1) and (4.2), it follows that
S n , m x S n , m y ( j = n n + m 1 h j 4 ) x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbb_HTML.gif
for all x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq4_HTML.gif and S n , m x n = x n + m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq147_HTML.gif, S n , m q = q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq148_HTML.gif for any q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq79_HTML.gif. Let
b n , m = t S n , m x n + ( 1 t ) S n , m q 1 S n , m ( t x n + ( 1 t ) q 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ25_HTML.gif
(4.3)
Then, using (4.3) and Lemma 2.4, we have
b n , m ( j = n n + m 1 h j 4 ) γ 1 ( x n q 1 ( j = n n + m 1 h j 4 ) 1 S n , m x n S n , m q 1 ) ( j = n h j 4 ) γ 1 ( x n q 1 ( j = n h j 4 ) 1 x n + m q 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbc_HTML.gif
It follows from Lemma 3.1 and lim n j = n h j 4 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq149_HTML.gif that lim n b n , m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq150_HTML.gif uniformly for all m. Observe that
a n + m ( t ) S n , m ( t x n + ( 1 t ) q 1 ) q 2 + b n , m = S n , m ( t x n + ( 1 t ) q 1 ) S n , m q 2 + b n , m ( j = n n + m 1 h j 4 ) t x n + ( 1 t ) q 1 q 2 + b n , m ( j = n h j 4 ) a n ( t ) + b n , m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbd_HTML.gif

Thus we have lim sup n a n ( t ) lim inf n a n ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq151_HTML.gif, that is, lim n t x n + ( 1 t ) q 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq152_HTML.gif exists for all t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq140_HTML.gif. This completes the proof. □

Lemma 4.2 Under the assumptions of Lemma  3.1, if E has a Fréchet differentiable norm, then, for all q 1 , q 2 F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq153_HTML.gif, the limit
lim n x n , j ( q 1 q 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Eqube_HTML.gif

exists, where { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is the sequence defined by (2.1). Furthermore, if W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq68_HTML.gif denotes the set of all weak subsequential limits of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif, then x y , j ( q 1 q 2 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq154_HTML.gif for all q 1 , q 2 F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq155_HTML.gif and x , y W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq156_HTML.gif.

Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

Theorem 4.1 Under the assumptions of Lemma  3.2, if E has a Fréchet differentiable norm, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Since E is a uniformly convex Banach space and the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif is bounded by Lemma 3.1, there exists a subsequence { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq157_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif which converges weakly to some q K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq158_HTML.gif. By Lemma 3.2, we have
lim k x n k S i x n k = lim k x n k T i x n k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbf_HTML.gif

for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq76_HTML.gif. It follows from Lemma 2.3 that q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq159_HTML.gif.

Now, we prove that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif such that { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif converges weakly to some q 1 K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq161_HTML.gif. Then, by the same method given above, we can also prove that q 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq162_HTML.gif. So, q , q 1 F W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq163_HTML.gif. It follows from Lemma 4.2 that
q q 1 2 = q q 1 , j ( q q 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbg_HTML.gif

Therefore, q 1 = q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq164_HTML.gif, which shows that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

Theorem 4.2 Under the assumptions of Lemma  3.2, if the dual space E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq24_HTML.gif of E has the Kadec-Klee property, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq157_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif which converges weakly to some q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq159_HTML.gif.

Now, we prove that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif such that { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif converges weakly to some q K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq118_HTML.gif. Then, as for q, we have q F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq134_HTML.gif. It follows from Lemma 4.1 that the limit
lim n t x n + ( 1 t ) q q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbh_HTML.gif

exists for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq136_HTML.gif. Again, since q , q W w ( { x n } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq165_HTML.gif, q = q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq166_HTML.gif by Lemma 2.5. This shows that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

Theorem 4.3 Under the assumptions of Lemma  3.2, if E satisfies Opial’s condition, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif defined by (2.1) converges weakly to a common fixed point of S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq18_HTML.gif, S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq19_HTML.gif, T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq20_HTML.gif and T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq21_HTML.gif.

Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq157_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif which converges weakly to some q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq159_HTML.gif.

Now, we prove that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. Suppose that there exists a subsequence { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif of { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif such that { x m j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq160_HTML.gif converges weakly to some q ¯ K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq167_HTML.gif and q ¯ q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq168_HTML.gif. Then, as for q, we have q ¯ F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq169_HTML.gif. Using Lemma 3.1, we have the following two limits exist:
lim n x n q = c , lim n x n q ¯ = c 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbi_HTML.gif
Thus, by Opial’s condition, we have
c = lim sup k x n k q < lim sup k x n k q ¯ = lim sup j x m j q ¯ < lim sup j x m j q = c , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equbj_HTML.gif

which is a contradiction and so q = q ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq170_HTML.gif. This shows that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_IEq33_HTML.gif converges weakly to q. This completes the proof. □

Declarations

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Authors’ Affiliations

(1)
School of Mathematics and Physics, Suzhou University of Science and Technology
(2)
Department of Mathematics Education and the RINS College of Education, Gyeongsang National University
(3)
Department of Aerospace Engineering and Mechanics, University of Minnesota

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© Guo et al.; licensee Springer. 2012

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