Open Access

Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

Fixed Point Theory and Applications20132013:76

DOI: 10.1186/1687-1812-2013-76

Received: 5 December 2012

Accepted: 10 March 2013

Published: 28 March 2013

Abstract

In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

MSC:47H10.

Keywords

new multi-step iteration S-iteration data dependence contractive-like operator

1 Introduction

Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area, viz., [112] among others.

Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost importance. Therefore it is natural to see many works related to the convergence of iteration methods such as [1322].

Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points, etc. One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [23], Rus et al. [24, 25], Berinde [26], Espínola and Petruşel [27], Markin [28], Chifu and Petruşel [29], Olantiwo [30, 31], Şoltuz [32, 33], Şoltuz and Grosan [34], Chugh and Kumar [35] and the references therein.

This paper is organized as follows. In Section 1 we present a brief survey of some known contractive mappings and iterative schemes and collect some preliminaries that will be used in the proofs of our main results. In Section 2 we show that the convergence of a new multi-step iteration, which is a special case of the Jungck multistep-SP iterative process defined in [36], and S-iteration (due to Agarwal et al.) can be used to approximate the fixed points of contractive-like operators. Motivated by the works of Şoltuz [32, 33], Şoltuz and Grosan [34], and Chugh and Kumar [35], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators.

As a background of our exposition, we now mention some contractive mappings and iteration schemes.

In [37] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq1_HTML.gif, there exist real numbers a, b, c satisfying 0 < a < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq2_HTML.gif, 0 < b , c < 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq3_HTML.gif such that, for each pair x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq4_HTML.gif, at least one of the following is true:
{ ( z 1 ) T x T y a x y , ( z 2 ) T x T y b ( x T x + y T y ) , ( z 3 ) T x T y c ( x T y + y T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ1_HTML.gif
(1.1)
A mapping T satisfying the contractive conditions (z1), (z2) and (z3) in (1.1) is called a Zamfirescu operator. An operator satisfying condition (z2) is called a Kannan operator, while the mapping satisfying condition (z3) is called a Chatterjea operator. As shown in [13], the contractive condition (1.1) leads to
{ ( b 1 ) T x T y δ x y + 2 δ x T x  if one uses  ( z 2 ) ,  and ( b 2 ) T x T y δ x y + 2 δ x T y  if one uses  ( z 3 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ2_HTML.gif
(1.2)

for all x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq5_HTML.gif, where δ : = max { a , b 1 b , c 1 c } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq6_HTML.gif, δ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq7_HTML.gif, and it was shown that this class of operators is wider than the class of Zamfirescu operators. Any mapping satisfying condition (b1) or (b2) is called a quasi-contractive operator.

Extending the above definition, Osilike and Udomene [20] considered operators T for which there exist real numbers L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq8_HTML.gif and δ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq7_HTML.gif such that for all x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq5_HTML.gif,
T x T y δ x y + L x T x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ3_HTML.gif
(1.3)
Imoru and Olantiwo [38] gave a more general definition: An operator T is called a contractive-like operator if there exists a constant δ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq7_HTML.gif and a strictly increasing and continuous function φ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq9_HTML.gif, with φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq10_HTML.gif, such that for each x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq11_HTML.gif,
T x T y δ x y + φ ( x T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ4_HTML.gif
(1.4)
A map satisfying (1.4) need not have a fixed point, even if E is complete. For example, let E = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq12_HTML.gif and define T by
T x = { 1.0 , 0 x 0.8 , 0.6 , 0.8 < x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equa_HTML.gif

WLOG, assume that x < y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq13_HTML.gif. Then, for 0 x < y 0.8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq14_HTML.gif or 0.8 < x < y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq15_HTML.gif, T x T y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq16_HTML.gif, and (1.4) is automatically satisfied.

If 0 x 0.8 < y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq17_HTML.gif, then T x T y = 0.4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq18_HTML.gif.

Define φ by φ ( t ) = L t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq19_HTML.gif for any L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq20_HTML.gif. Then φ is increasing, continuous, and φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq10_HTML.gif. Also, x T x = 1 x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq21_HTML.gif so that φ ( x T x ) = L ( 1 x ) 0.2 L 0.4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq22_HTML.gif.

Therefore
0.4 = T x T y L x T x δ x y + L x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equb_HTML.gif

for any 0 δ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq23_HTML.gif, and (1.4) is satisfied for 0 x 0.8 < y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq17_HTML.gif. But T has no fixed point.

However, using (1.4) it is obvious that if T has a fixed point, then it is unique.

From now on, we demand that denotes the set of all nonnegative integers. Let X be a Banach space, let E X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq24_HTML.gif be a nonempty closed, convex subset of X, and let T be a self-map on E. Define F T : = { p X : p = T p } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq25_HTML.gif to be the set of fixed points of T. Let { α n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq26_HTML.gif, { β n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq27_HTML.gif, { γ n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq28_HTML.gif and { β n i } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq29_HTML.gif, i = 1 , k 2 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq30_HTML.gif, k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq31_HTML.gif be real sequences in [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq32_HTML.gif satisfying certain conditions.

In [5] Rhoades and Şoltuz introduced a multi-step iterative procedure given by
{ x 0 E , y n k 1 = ( 1 β n k 1 ) x n + β n k 1 T x n , k 2 , y n i = ( 1 β n i ) x n + β n i T y n i + 1 , i = 1 , k 2 ¯ , x n + 1 = ( 1 α n ) x n + α n T y n 1 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ5_HTML.gif
(1.5)
The sequence { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif defined by
{ x 0 E , x n + 1 = ( 1 α n ) T x n + α n T y n , y n = ( 1 β n ) x n + β n T x n , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ6_HTML.gif
(1.6)

is known as the S-iteration process (see [12, 17, 39]).

Thianwan [6] defined a two-step iteration { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq34_HTML.gif by
{ x 0 E , x n + 1 = ( 1 α n ) y n + α n T y n , y n = ( 1 β n ) x n + β n T x n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ7_HTML.gif
(1.7)
Recently Phuengrattana and Suantai [7] introduced an SP iteration method defined by
{ x 0 E , x n + 1 = ( 1 α n ) y n + α n T y n , y n = ( 1 β n ) z n + β n T z n , z n = ( 1 γ n ) x n + γ n T x n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ8_HTML.gif
(1.8)
We shall employ the following iterative process. For an arbitrary fixed order k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq31_HTML.gif,
{ x 0 E , x n + 1 = ( 1 α n ) y n 1 + α n T y n 1 , y n 1 = ( 1 β n 1 ) y n 2 + β n 1 T y n 2 , y n 2 = ( 1 β n 2 ) y n 3 + β n 2 T y n 3 , , y n k 2 = ( 1 β n k 2 ) y n k 1 + β n k 2 T y n k 1 , y n k 1 = ( 1 β n k 1 ) x n + β n k 1 T x n , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ9_HTML.gif
(1.9)
or, in short,
{ x 0 E , x n + 1 = ( 1 α n ) y n 1 + α n T y n 1 , y n i = ( 1 β n i ) y n i + 1 + β n i T y n i + 1 , i = 1 , k 2 ¯ , y n k 1 = ( 1 β n k 1 ) x n + β n k 1 T x n , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ10_HTML.gif
(1.10)
where
{ α n } n = 0 [ 0 , 1 ) , n = 0 α n = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ11_HTML.gif
(1.11)
and
{ β n i } n = 0 [ 0 , 1 ) , i = 1 , k 1 ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ12_HTML.gif
(1.12)

Remark 1 If each γ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq35_HTML.gif, then SP iteration (1.8) reduces to two-step iteration (1.7). By taking k = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq36_HTML.gif and k = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq37_HTML.gif in (1.10), we obtain iterations (1.8) and (1.7), respectively.

We shall need the following definition and lemma in the sequel.

Definition 1 [40]

Let T , T ˜ : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq38_HTML.gif be two operators. We say that T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif is an approximate operator for T if, for some ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq40_HTML.gif, we have
T x T ˜ x ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equc_HTML.gif

for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq41_HTML.gif.

Lemma 1 [34]

Let { a n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq42_HTML.gif be a nonnegative sequence for which one assumes that there exists an n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq43_HTML.gif such that for all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq44_HTML.gif,
a n + 1 ( 1 μ n ) a n + μ n η n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equd_HTML.gif
is satisfied, where μ n ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq45_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq46_HTML.gif, n = 0 μ n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq47_HTML.gif and η n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq48_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq49_HTML.gif. Then the following holds:
0 lim n sup a n lim n sup η n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Eque_HTML.gif

2 Main results

For simplicity we use the following notation throughout this section.

For any iterative process, { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif and { u n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq50_HTML.gif denote iterative sequences associated to T and T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif, respectively.

Theorem 1 Let T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq1_HTML.gif be a map satisfying (1.4) with F T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq51_HTML.gif, and let { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif be a sequence defined by (1.10), then the sequence { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq52_HTML.gif converges to the unique fixed point of T.

Proof The proof can be easily obtained by using the argument in the proof of ([36], Theorem 3.1). □

This result allows us to give the next theorem.

Theorem 2 Let T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq1_HTML.gif be a map satisfying (1.4) with F T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq53_HTML.gif, and let T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif be an approximate operator of T as in Definition  1. Let { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif, { u n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq54_HTML.gif be two iterative sequences defined by (1.10) with real sequences { α n } n = 0 , { β n i } n = 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq55_HTML.gif satisfying (i) 0 β n i < α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq56_HTML.gif, i = 1 , k 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq57_HTML.gif, (ii) α n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq58_HTML.gif. If p = T p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq59_HTML.gif and q = T ˜ q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq60_HTML.gif, then we have
p q k ε 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equf_HTML.gif
Proof For given x 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq61_HTML.gif and u 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq62_HTML.gif, we consider the following multi-step iteration for T and T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif:
{ x 0 E , x n + 1 = ( 1 α n ) y n 1 + α n T y n 1 , y n i = ( 1 β n i ) y n i + 1 + β n i T y n i + 1 , i = 1 , k 2 ¯ , y n k 1 = ( 1 β n k 1 ) x n + β n k 1 T x n , k 2 , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ13_HTML.gif
(2.1)
and
{ u 0 E , u n + 1 = ( 1 α n ) v n 1 + α n T ˜ v n 1 , v n i = ( 1 β n i ) v n i + 1 + β n i T ˜ v n i + 1 , i = 1 , k 2 ¯ , v n k 1 = ( 1 β n k 1 ) u n + β n k 1 T ˜ u n , k 2 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ14_HTML.gif
(2.2)
Thus, from (1.4), (2.1) and (2.2), we have the following inequalities.
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ15_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ16_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ17_HTML.gif
(2.5)
Combining (2.3), (2.4) and (2.5), we obtain
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n 2 ( 1 δ ) ] y n 3 v n 3 + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] β n 2 φ ( y n 3 T y n 3 ) + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] β n 2 ε + [ 1 α n ( 1 δ ) ] β n 1 φ ( y n 2 T y n 2 ) + [ 1 α n ( 1 δ ) ] β n 1 ε + α n φ ( y n 1 T y n 1 ) + α n ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ18_HTML.gif
(2.6)
Thus, by induction, we get
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 2 ( 1 δ ) ] y n k 1 v n k 1 + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 3 ( 1 δ ) ] β n k 2 φ ( y n k 1 T y n k 1 ) + + [ 1 α n ( 1 δ ) ] β n 1 φ ( y n 2 T y n 2 ) + α n φ ( y n 1 T y n 1 ) + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 3 ( 1 δ ) ] β n k 2 ε + + [ 1 α n ( 1 δ ) ] β n 1 ε + α n ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ19_HTML.gif
(2.7)
Again, using (1.4), (2.1) and (2.2), we get
y n k 1 v n k 1 = ( 1 β n k 1 ) ( x n u n ) + β n k 1 ( T x n T ˜ u n ) ( 1 β n k 1 ) x n u n + β n k 1 T x n T ˜ u n ( 1 β n k 1 ) x n u n + β n k 1 T x n T u n + β n k 1 T u n T ˜ u n [ 1 β n k 1 ( 1 δ ) ] x n u n + β n k 1 φ ( x n T x n ) + β n k 1 ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ20_HTML.gif
(2.8)
Substituting (2.8) in (2.7), we have
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 1 ( 1 δ ) ] x n u n + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 2 ( 1 δ ) ] β n k 1 φ ( x n T x n ) + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 3 ( 1 δ ) ] β n k 2 φ ( y n k 1 T y n k 1 ) + + [ 1 α n ( 1 δ ) ] β n 1 φ ( y n 2 T y n 2 ) + α n φ ( y n 1 T y n 1 ) + [ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 2 ( 1 δ ) ] β n k 1 ε + + [ 1 α n ( 1 δ ) ] β n 1 ε + α n ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ21_HTML.gif
(2.9)
Since δ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq7_HTML.gif and { α n } n = 0 , { β n i } n = 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq63_HTML.gif, for i = 1 , k 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq57_HTML.gif, we have
[ 1 α n ( 1 δ ) ] [ 1 β n 1 ( 1 δ ) ] [ 1 β n i ( 1 δ ) ] [ 1 α n ( 1 δ ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ22_HTML.gif
(2.10)
From inequality (2.10) and assumption (i) in (2.9), it follows
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] x n u n + α n φ ( x n T x n ) + α n φ ( y n k 1 T y n k 1 ) + + α n φ ( y n 2 T y n 2 ) + α n φ ( y n 1 T y n 1 ) + α n ε + α n ε + + α n ε + α n ε = [ 1 α n ( 1 δ ) ] x n u n + α n ( 1 δ ) { φ ( x n T x n ) + φ ( y n k 1 T y n k 1 ) 1 δ + + φ ( y n 1 T y n 1 ) + k ε 1 δ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ23_HTML.gif
(2.11)
Define
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equg_HTML.gif
From Theorem 1 it follows that lim n x n p = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq64_HTML.gif. Since T satisfies condition (1.4) and T p = p F T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq65_HTML.gif,
0 x n T x n x n p + T p T x n x n p + δ p x n + φ ( p T p ) = ( 1 + δ ) x n p 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ24_HTML.gif
(2.12)
Since β n i [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq66_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq67_HTML.gif, i = 1 , k 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq57_HTML.gif, using (1.4) and (1.10), we have
0 y n 1 T y n 1 = y n 1 p + p T y n 1 y n 1 p + T p T y n 1 y n 1 p + δ p y n 1 + φ ( p T p ) = ( 1 + δ ) y n 1 p = ( 1 + δ ) ( 1 β n 1 ) y n 2 + β n 1 T y n 2 p ( 1 β n 1 + β n 1 ) ( 1 + δ ) { ( 1 β n 1 ) y n 2 p + β n 1 T y n 2 T p } ( 1 + δ ) { ( 1 β n 1 ) y n 2 p + β n 1 δ y n 2 p } = ( 1 + δ ) [ 1 β n 1 ( 1 δ ) ] y n 2 p ( 1 + δ ) [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 2 ( 1 δ ) ] y n k 1 p ( 1 + δ ) [ 1 β n 1 ( 1 δ ) ] [ 1 β n k 1 ( 1 δ ) ] x n p ( 1 + δ ) x n p 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ25_HTML.gif
(2.13)

It is easy to see from (2.13) that this result is also valid for T y n 2 y n 2 , , T y n k 1 y n k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq68_HTML.gif.

Since φ is continuous, we have
lim n φ ( x n T x n ) = lim n φ ( y n 1 T y n 1 ) = = lim n φ ( y n k 1 T y n k 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ26_HTML.gif
(2.14)
Hence an application of Lemma 1 to (2.11) leads to
p q k ε 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ27_HTML.gif
(2.15)

 □

As shown by Hussain et al. ([22], Theorem 8), in an arbitrary Banach space X, the S-iteration { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif given by (1.6) converges to the fixed point of T, where T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq1_HTML.gif is a mapping satisfying condition (1.3).

Theorem 3 Let T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq1_HTML.gif be a map satisfying (1.4) with F T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq53_HTML.gif, and let { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif be defined by (1.6) with real sequences { β n } n = 0 , { α n } n = 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq69_HTML.gif satisfying n = 0 α n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq70_HTML.gif. Then the sequence { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq33_HTML.gif converges to the unique fixed point of T.

Proof The argument is similar to the proof of Theorem 8 of [22], and is thus omitted. □

We now prove the result on data dependence for the S-iterative procedure by utilizing Theorem 3.

Theorem 4 Let T, T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif be two operators as in Theorem  2. Let { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq34_HTML.gif, { u n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq50_HTML.gif be S-iterations defined by (1.6) with real sequences { β n } n = 0 , { α n } n = 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq71_HTML.gif satisfying (i) 1 2 α n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq72_HTML.gif, n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq67_HTML.gif, and (ii) n = 0 α n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq70_HTML.gif. If p = T p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq59_HTML.gif and q = T ˜ q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq60_HTML.gif, then we have
p q 3 ε 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equh_HTML.gif
Proof For a given x 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq61_HTML.gif and u 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq62_HTML.gif, we consider the following iteration for T and T ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq39_HTML.gif:
{ x 0 E , x n + 1 = ( 1 α n ) T x n + α n T y n , y n = ( 1 β n ) x n + β n T x n , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ28_HTML.gif
(2.16)
and
{ u 0 E , u n + 1 = ( 1 α n ) T ˜ u n + α n T ˜ v n , v n = ( 1 β n ) u n + β n T ˜ u n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ29_HTML.gif
(2.17)
Using (1.4), (2.16) and (2.17), we obtain the following estimates:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ30_HTML.gif
(2.18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ31_HTML.gif
(2.19)
Combining (2.18) and (2.19), we get
x n + 1 u n + 1 { ( 1 α n ) δ + α n δ [ 1 β n ( 1 δ ) ] } x n u n + { 1 α n + α n δ β n } φ ( x n T x n ) + α n φ ( y n T y n ) + α n δ β n ε + ( 1 α n ) ε + α n ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ32_HTML.gif
(2.20)
For { α n } n = 0 , { β n } n = 0 [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq73_HTML.gif and δ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq7_HTML.gif,
( 1 α n ) δ < 1 α n , 1 β n ( 1 δ ) < 1 , α n δ β n < α n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ33_HTML.gif
(2.21)
It follows from assumption (i) that
1 α n < α n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ34_HTML.gif
(2.22)
Therefore, combining (2.22) and (2.21) to (2.20) gives
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] x n u n + 2 α n φ ( x n T x n ) + α n φ ( y n T y n ) + α n ε + α n ε + α n ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ35_HTML.gif
(2.23)
or, equivalently,
x n + 1 u n + 1 [ 1 α n ( 1 δ ) ] x n u n + α n ( 1 δ ) { 2 φ ( x n T x n ) + φ ( y n T y n ) + 3 ε } 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ36_HTML.gif
(2.24)
Now define
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equi_HTML.gif
From Theorem 3, we have lim n x n p = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq64_HTML.gif. Since T satisfies condition (1.4), and T p = p F T https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq65_HTML.gif, using an argument similar to that in the proof of Theorem 2,
lim n x n T x n = lim n y n T y n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ37_HTML.gif
(2.25)
Using the fact that φ is continuous, we have
lim n φ ( x n T x n ) = lim n φ ( y n T y n ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ38_HTML.gif
(2.26)
An application of Lemma 1 to (2.24) leads to
p q 3 ε 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_Equ39_HTML.gif
(2.27)

 □

3 Conclusion

Since the iterative schemes (1.7) and (1.8) are special cases of the iterative process (1.10), Theorem 1 generalizes Theorem 2.1 of [19] and Theorem 2.1 of [18]. By taking k = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq36_HTML.gif and k = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq37_HTML.gif in Theorem 2, data dependence results for the iterative schemes (1.8) and (1.7) can be easily obtained. For k = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-76/MediaObjects/13663_2012_Article_411_IEq36_HTML.gif, Theorem 2 reduces to Theorem 3.2 of [35]. Since condition (1.4) is more general than condition (1.3), Theorem 3 generalizes Theorem 8 of [22].

Declarations

Acknowledgements

The first two authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Unit under project number BAPK 2012-07-03-DOP02 for financial support during the preparation of this manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University
(2)
Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University
(3)
Department of Mathematics, Indiana University

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