Open Access

The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Fixed Point Theory and Applications20122012:206

DOI: 10.1186/1687-1812-2012-206

Received: 11 May 2012

Accepted: 29 October 2012

Published: 22 November 2012

Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq1_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq2_HTML.gif. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq4_HTML.gif, { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif, { d n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq6_HTML.gif be four real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy the conditions (i) a n , b n , d n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq8_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif and c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif; (ii) n = 0 a n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq11_HTML.gif. For some x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif, { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq14_HTML.gif be any bounded sequences in D, and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC:47H10.

Keywords

generalized Lipschitz mapping Φ-hemi-contractive mapping Ishikawa iterative sequence with errors uniformly smooth real Banach space

1 Introduction and preliminary

Let E be a real Banach space and E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq16_HTML.gif be its dual space. The normalized duality mapping J : E 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq17_HTML.gif is defined by
J ( x ) = { f E : x , f = x 2 = f 2 } , x E , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equa_HTML.gif
where , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq18_HTML.gif denotes the generalized duality pairing. It is well known that
  1. (i)

    If E is a smooth Banach space, then the mapping J is single-valued;

     
  2. (ii)

    J ( α x ) = α J ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq19_HTML.gif for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif and α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq21_HTML.gif;

     
  3. (iii)

    If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

     

Definition 1.1 ([1])

Let D be a nonempty closed convex subset of E, T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq1_HTML.gif be a mapping.
  1. (1)
    T is called strongly pseudocontractive if there is a constant k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq22_HTML.gif such that for all x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq23_HTML.gif,
    T x T y , j ( x y ) k x y 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equb_HTML.gif
     
  2. (2)
    T is called ϕ-strongly pseudocontractive if for all x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq23_HTML.gif, there exist j ( x y ) J ( x y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq24_HTML.gif and a strictly increasing continuous function ϕ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq25_HTML.gif with ϕ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq26_HTML.gif such that
    T x T y , j ( x y ) x y 2 ϕ ( x y ) x y ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equc_HTML.gif
     
  3. (3)
    T is called Φ-pseudocontractive if for all x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq23_HTML.gif, there exist j ( x y ) J ( x y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq24_HTML.gif and a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq27_HTML.gif with Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq28_HTML.gif such that
    T x T y , j ( x y ) x y 2 Φ ( x y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equd_HTML.gif
     

It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

Definition 1.2 ([1])

Let T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq1_HTML.gif be a mapping and F ( T ) = { x D : T x = x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq29_HTML.gif.
  1. (1)
    T is called ϕ-strongly-hemi-pseudocontractive if for all x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq30_HTML.gif, q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq31_HTML.gif, there exist j ( x q ) J ( x q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq32_HTML.gif and a strictly increasing continuous function ϕ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq25_HTML.gif with ϕ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq26_HTML.gif such that
    T x T q , j ( x q ) x q 2 ϕ ( x q ) x q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Eque_HTML.gif
     
  2. (2)
    T is called Φ-hemi-pseudocontractive if for all x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq30_HTML.gif, q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq31_HTML.gif, there exist j ( x q ) J ( x q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq32_HTML.gif and the strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq27_HTML.gif with Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq28_HTML.gif such that
    T x T q , j ( x q ) x q 2 Φ ( x q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equf_HTML.gif
     

Closely related to the class of pseudocontractive-type mappings are those of accretive type.

Definition 1.3 ([1])

Let N ( T ) = { x E : T x = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq33_HTML.gif. The mapping T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq34_HTML.gif is called strongly quasi-accretive if for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif, x N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq35_HTML.gif, there exist j ( x x ) J ( x x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq36_HTML.gif and a constant k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq37_HTML.gif such that T x T x , j ( x x ) k x x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq38_HTML.gif; T is called ϕ-strongly quasi-accretive if for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif, x N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq35_HTML.gif, there exist j ( x x ) J ( x x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq36_HTML.gif and a strictly increasing continuous function ϕ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq25_HTML.gif with ϕ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq26_HTML.gif such that T x T x , j ( x x ) ϕ ( x x ) x x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq39_HTML.gif; T is called Φ-quasi-accretive if for all x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif, x N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq35_HTML.gif, there exist j ( x x ) J ( x x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq36_HTML.gif and a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq27_HTML.gif with Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq28_HTML.gif such that T x T x , j ( x x ) Φ ( x x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq40_HTML.gif.

Definition 1.4 ([2])

For arbitrary given x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, the Ishikawa iterative process with errors { x n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq41_HTML.gif is defined by
{ y n = ( 1 b n d n ) x n + b n T x n + d n v n , n 0 , x n + 1 = ( 1 a n c n ) x n + a n T y n + c n u n , n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ1_HTML.gif
(1.1)
where { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif, { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq14_HTML.gif are any bounded sequences in D; { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq4_HTML.gif, { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif, { d n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq6_HTML.gif are four real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy a n + c n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq42_HTML.gif, b n + d n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq43_HTML.gif, for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq44_HTML.gif. If b n = d n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq45_HTML.gif, then the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif defined by
x n + 1 = ( 1 a n c n ) x n + a n T x n + c n u n , n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ2_HTML.gif
(1.2)

is called the Mann iterative process with errors.

Definition 1.5 ([3, 4])

A mapping T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq1_HTML.gif is called generalized Lipschitz if there exists a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq46_HTML.gif such that T x T y L ( 1 + x y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq47_HTML.gif, x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq48_HTML.gif.

The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

Lemma 1.6 ([5])

Let E be a uniformly smooth real Banach space, and let J : E 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq49_HTML.gif be a normalized duality mapping. Then
x + y 2 x 2 + 2 y , J ( x + y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ3_HTML.gif
(1.3)

for all x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq50_HTML.gif.

Lemma 1.7 ([6])

Let { ρ n } n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq51_HTML.gif be a nonnegative sequence which satisfies the following inequality:
ρ n + 1 ( 1 λ n ) ρ n + σ n , n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ4_HTML.gif
(1.4)

where λ n [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq52_HTML.gif with n = 0 λ n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq53_HTML.gif, σ n = o ( λ n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq54_HTML.gif. Then ρ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq55_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif.

2 Main results

Theorem 2.1 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq56_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq2_HTML.gif. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq4_HTML.gif, { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif, { d n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq6_HTML.gif be four real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy the conditions (i) a n , b n , d n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq57_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif and c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif; (ii) n = 0 a n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq11_HTML.gif. For some x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif, { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq14_HTML.gif be any bounded sequences in D, and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed point q of T.

Proof Since T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq1_HTML.gif is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq27_HTML.gif with Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq28_HTML.gif such that
T x T q , J ( x q ) x q 2 Φ ( x q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equg_HTML.gif
i.e.,
x T x , J ( x q ) Φ ( x q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equh_HTML.gif
and
T x T y L ( 1 + x y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equi_HTML.gif

for any x , y D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq58_HTML.gif and q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq31_HTML.gif.

Step 1. There exists x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif and x 0 T x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq59_HTML.gif such that r 0 = x 0 T x 0 x 0 q R ( Φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq60_HTML.gif (range of Φ). Indeed, if Φ ( r ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq61_HTML.gif as r + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq62_HTML.gif, then r 0 R ( Φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq63_HTML.gif; if sup { Φ ( r ) : r [ 0 , + ) } = r 1 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq64_HTML.gif with r 0 < r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq65_HTML.gif, then for q D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq66_HTML.gif, there exists a sequence { w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq67_HTML.gif in D such that w n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq68_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif with w n q https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq69_HTML.gif. Furthermore, we obtain that { w n T w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq70_HTML.gif is bounded. Hence, there exists a natural number n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq71_HTML.gif such that w n T w n w n q < r 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq72_HTML.gif for n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq73_HTML.gif, then we redefine x 0 = w n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq74_HTML.gif and x 0 T x 0 x 0 q R ( Φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq75_HTML.gif.

Step 2. For any n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq44_HTML.gif, { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif is bounded. Set R = Φ 1 ( r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq76_HTML.gif, then from Definition 1.2(2), we obtain that x 0 q R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq77_HTML.gif. Denote B 1 = { x D : x q R } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq78_HTML.gif, B 2 = { x D : x q 2 R } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq79_HTML.gif. Since T is generalized Lipschitz, so T is bounded. We may define M = sup x B 2 { T x q + 1 } + sup n { u n q } + sup n { v n q } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq80_HTML.gif. Next, we want to prove that x n B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq81_HTML.gif. If n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq82_HTML.gif, then x 0 B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq83_HTML.gif. Now, assume that it holds for some n, i.e., x n B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq81_HTML.gif. We prove that x n + 1 B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq84_HTML.gif. Suppose it is not the case, then x n + 1 q > R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq85_HTML.gif. Since J is uniformly continuous on a bounded subset of E, then for ϵ 0 = Φ ( R 4 ) 24 L ( 1 + 2 R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq86_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq87_HTML.gif such that J x J y < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq88_HTML.gif when x y < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq89_HTML.gif, x , y B 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq90_HTML.gif. Now, denote
τ 0 = min { 1 , R 2 [ L ( 1 + 2 R ) + 2 R + M ] , R 4 [ L ( 1 + R ) + 2 R + M ] , δ 2 [ L ( 1 + 2 R ) + 2 R + M ] , Φ ( R 4 ) 24 R 2 , Φ ( R 4 ) 24 L ( 1 + 2 R ) , Φ ( R 4 ) 48 M R } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equj_HTML.gif
Owing to a n , b n , c n , d n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq91_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif, without loss of generality, assume that 0 a n , b n , c n , d n τ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq92_HTML.gif for any n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq44_HTML.gif. Since c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif, denote c n < a n τ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq93_HTML.gif. So, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ5_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ6_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ7_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ8_HTML.gif
(2.4)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ9_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ10_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ11_HTML.gif
(2.7)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ12_HTML.gif
(2.8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ13_HTML.gif
(2.9)
Therefore,
J ( x n q ) J ( y n q ) < ϵ 0 ; J ( x n + 1 q ) J ( x n q ) < ϵ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equk_HTML.gif
Using Lemma 1.6 and the above formulas, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ14_HTML.gif
(2.10)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ15_HTML.gif
(2.11)
Substitute (2.11) into (2.10)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ16_HTML.gif
(2.12)

this is a contradiction. Thus, x n + 1 B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq84_HTML.gif, i.e., { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq94_HTML.gif is a bounded sequence. So, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq95_HTML.gif, { T y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq96_HTML.gif, { T x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq97_HTML.gif are all bounded sequences.

Step 3. We want to prove x n q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq98_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif. Set M 1 = max { sup n x n q , sup n y n q , sup n T x n q , sup n T y n q , sup n u n q , sup n v n q } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq99_HTML.gif.

By (2.10), (2.11), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ17_HTML.gif
(2.13)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ18_HTML.gif
(2.14)

where A n = J ( x n + 1 q ) J ( x n q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq100_HTML.gif, B n = J ( x n q ) J ( y n q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq101_HTML.gif and A n , B n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq102_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif.

Taking (2.14) into (2.13),
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ19_HTML.gif
(2.15)

where C n = a n 2 M 1 2 + M 1 A n + 3 M 1 B n + 2 b n M 1 2 + 2 d n M 1 2 + c n M 1 2 a n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq103_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif.

Set inf n 0 Φ ( y n q ) 1 + x n + 1 q 2 = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq104_HTML.gif, then λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq105_HTML.gif. If it is not the case, we assume that λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq106_HTML.gif. Let 0 < γ < min { 1 , λ } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq107_HTML.gif, then Φ ( y n q ) 1 + x n + 1 q 2 γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq108_HTML.gif, i.e., Φ ( y n q ) γ + γ x n + 1 q 2 γ x n + 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq109_HTML.gif. Thus, from (2.15) it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ20_HTML.gif
(2.16)
This implies that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ21_HTML.gif
(2.17)
Let ρ n = x n q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq110_HTML.gif, λ n = 2 a n γ 1 + 2 a n γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq111_HTML.gif, σ n = 2 a n C n 1 + 2 a n γ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq112_HTML.gif. Then we get that
ρ n + 1 ( 1 λ n ) ρ n + σ n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equl_HTML.gif
Applying Lemma 1.7, we get that ρ n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq55_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif. This is a contradiction and so λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq105_HTML.gif. Therefore, there exists an infinite subsequence such that Φ ( y n i q ) 1 + x n i + 1 q 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq113_HTML.gif as i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq114_HTML.gif. Since 0 Φ ( y n i q ) 1 + M 1 2 Φ ( y n i q ) 1 + x n i + 1 q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq115_HTML.gif, then Φ ( y n i q ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq116_HTML.gif as i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq114_HTML.gif. In view of the strictly increasing continuity of Φ, we have y n i q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq117_HTML.gif as i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq114_HTML.gif. Hence, x n i q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq118_HTML.gif as i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq114_HTML.gif. Next, we want to prove x n q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq98_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif. Let ε ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq119_HTML.gif, there exists n i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq120_HTML.gif such that x n i q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq121_HTML.gif, a n , c n < ϵ 8 M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq122_HTML.gif, b n , d n < ϵ 16 M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq123_HTML.gif, C n < Φ ( ϵ ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq124_HTML.gif, for any n i , n n i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq125_HTML.gif. First, we want to prove x n i + 1 q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq126_HTML.gif. Suppose it is not the case, then x n i + 1 q ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq127_HTML.gif. Using (1.1), we may get the following estimates:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ22_HTML.gif
(2.18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ23_HTML.gif
(2.19)
Since Φ is strictly increasing, then (2.19) leads to Φ ( y n i q ) Φ ( ϵ 4 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq128_HTML.gif. From (2.15), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ24_HTML.gif
(2.20)

which is a contradiction. Hence, x n i + 1 q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq126_HTML.gif. Suppose that x n i + m q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq129_HTML.gif holds. Repeating the above course, we can easily prove that x n i + m + 1 q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq130_HTML.gif holds. Therefore, for any m, we obtain that x n i + m q < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq129_HTML.gif, which means x n q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq98_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif. This completes the proof. □

Theorem 2.2 Let E be an arbitrary uniformly smooth real Banach space, and T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq34_HTML.gif be a generalized Lipschitz Φ-quasi-accretive mapping with N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq131_HTML.gif. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq4_HTML.gif, { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif, { d n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq6_HTML.gif be four real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy the conditions (i) a n , b n , d n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq57_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif and c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif; (ii) n = 0 a n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq11_HTML.gif. For some x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif, { v n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq14_HTML.gif be any bounded sequences in E, and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be an Ishikawa iterative sequence with errors defined by
{ y n = ( 1 b n d n ) x n + b n S x n + d n v n , n 0 , x n + 1 = ( 1 a n c n ) x n + a n S y n + c n u n , n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ25_HTML.gif
(2.21)

where S : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq132_HTML.gif is defined by S x = x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq133_HTML.gif for any x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif. Then { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif converges strongly to the unique solution of the equation T x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq134_HTML.gif (or the unique fixed point of S).

Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that
T x T y L ( 1 + x y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equm_HTML.gif
i.e.,
S x S y L 1 ( 1 + x y ) , L 1 = 1 + L ; T x T q , J ( x q ) Φ ( x q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equn_HTML.gif
i.e.,
S x S q , J ( x q ) x q 2 Φ ( x q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equo_HTML.gif

for all x , y E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq50_HTML.gif, q N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq135_HTML.gif. The rest of the proof is the same as that of Theorem 2.1. □

Corollary 2.3 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T : D D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq56_HTML.gif be a generalized Lipschitz Φ-hemi-contractive mapping with q F ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq2_HTML.gif. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif be two real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy the conditions (i) a n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq136_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif and c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif; (ii) n = 0 a n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq11_HTML.gif. For some x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif be any bounded sequence in D, and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed point q of T.

Corollary 2.4 Let E be an arbitrary uniformly smooth real Banach space, and T : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq34_HTML.gif be a generalized Lipschitz Φ-quasi-accretive mapping with N ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq131_HTML.gif. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { d n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq6_HTML.gif be two real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif and satisfy the conditions (i) a n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq136_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq9_HTML.gif and c n = o ( a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq10_HTML.gif; (ii)  n = 0 a n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq11_HTML.gif. For some x 0 D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq12_HTML.gif, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif be any bounded sequence in E, and { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be the Mann iterative sequence with errors defined by
x n + 1 = ( 1 a n c n ) x n + a n S x n + c n u n , n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ26_HTML.gif
(2.22)

where S : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq132_HTML.gif is defined by S x = x T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq133_HTML.gif for any x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq20_HTML.gif. Then { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif converges strongly to the unique solution of the equation T x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq134_HTML.gif (or the unique fixed point of S).

Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

Theorem CC ([[1], Theorem 3.1])

Let E be a uniformly smooth real Banach space and A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq137_HTML.gif be a mapping with N ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq138_HTML.gif. Suppose A is a generalized Lipschitz Φ-quasi-accretive mapping. Let { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq3_HTML.gif, { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq4_HTML.gif and { c n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq5_HTML.gif be real sequences in [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq7_HTML.gif satisfying the following conditions: (i) a n + b n + c n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq139_HTML.gif; (ii) n = 0 ( b n + c n ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq140_HTML.gif; (iii) n = 0 c n < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq141_HTML.gif; (iv) lim n b n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq142_HTML.gif. Let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif be generated iteratively from arbitrary x 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq143_HTML.gif by
x n + 1 = a n x n + b n S x n + c n u n , n 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_Equ27_HTML.gif
(2.23)

where S : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq132_HTML.gif is defined by S x : = f + x A x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq144_HTML.gif, x E https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq145_HTML.gif and { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq13_HTML.gif is an arbitrary bounded sequence in E. Then, there exists γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq146_HTML.gif such that if b n + c n γ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq147_HTML.gif, n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq148_HTML.gif, the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq15_HTML.gif converges strongly to the unique solution of the equation A u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq149_HTML.gif.

However, there exists a gap in the proof process of above Theorem CC. Here, c n = min { ϵ 4 β , 1 2 σ Φ ( ϵ 2 ) α n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq150_HTML.gif ( α n = b n + c n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq151_HTML.gif) does not hold in line 14 of Claim 2 on page 248, i.e., c n 1 2 σ Φ ( ϵ 2 ) α n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq152_HTML.gif is a wrong case. For instance, set the iteration parameters: a n = 1 b n c n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq153_HTML.gif, where { b n } : b 1 = 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq154_HTML.gif, b n = 1 n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq155_HTML.gif, n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq156_HTML.gif; { c n } : 1 4 , 1 2 2 , 1 3 2 , 1 4 , 1 5 2 , , 1 8 2 , 1 9 , 1 10 2 , , 1 15 2 , 1 16 , 1 17 2 , , 1 24 2 , 1 25 , 1 26 2 , , 1 35 2 , 1 36 , 1 37 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq157_HTML.gif . Then n = 0 c n < + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq158_HTML.gif, but c n o ( b n + c n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-206/MediaObjects/13663_2012_Article_322_IEq159_HTML.gif. Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!

Declarations

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Shijiazhuang Tiedao University
(2)
Department of Mathematics, Indiana University

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Copyright

© Xue et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.