Skip to content

Advertisement

  • Research
  • Open Access

Strong convergent result for quasi-nonexpansive mappings in Hilbert spaces

Fixed Point Theory and Applications20112011:88

https://doi.org/10.1186/1687-1812-2011-88

  • Received: 16 October 2011
  • Accepted: 25 November 2011
  • Published:

Abstract

In this article, we study an iterative method over the class of quasi-nonexpansive mappings which are more general than nonexpansive mappings in Hilbert spaces. Our strong convergent theorems include several corresponding authors' results.

2000 MSC: 58E35; 47H09; 65J15.

Keywords

  • quasi-nonexpansive mapping
  • Lipschitzian continuous
  • strongly monotone
  • nonlinear operator
  • fixed point
  • viscosity method

1. Introduction

Let H be a real Hilbert space with inner product 〈·,·〉, and induced norm ||·||. A mapping T: HH is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x,y H. The set of the fixed points of T is denoted by Fix(T) := {x H: Tx = x}.

The viscosity approximation method was first introduced by Moudafi [1] in 2000. Starting with an arbitrary initial x0 H, define a sequence {x n } generated by
x n + 1 = ε n 1 + ε n f ( x n ) + 1 1 + ε n T x n , n 0 ,
(1.1)

where f is a contraction with a coefficient α [0,1) on H, i.e., ||f(x) - f(y)|| ≤ α||x - y|| for all x,y H, T is nonexpansive, and {ε n } is a sequence in (0,1) satisfying the following given conditions:

(i1) limn→∞ε n = 0;

(i2) n = 0 ε n = ;

(i3) lim n ( 1 ε n - 1 ε n + 1 ) = 0 .

It is proved that the sequence {x n } generated by (1.1) converges strongly to the unique solution x* C(C := Fix(T)) of the variational inequality:
( I - f ) x * , x - x * 0 , x F i x ( T ) .
In 2003, Xu [2] proved that the sequence {x n } defined by the below process where T is also nonexpansive, started with an arbitrary initial x0 H:
x n + 1 = α n b + ( I - α n A ) T x n , n 0 ,
(1.2)
converges strongly to the unique solution of the minimization problem (1.3) when the sequence {α n } satisfies certain conditions:
min x C 1 2 A x , x - x , b ,
(1.3)

where C is the set of fixed points set of T on H and b is a given point in H.

In 2006, Marino and Xu [3] combined the iterative method (1.2) with the viscosity approximation method (1.1) and considered the following general iterative method:
x n + 1 = α n γ f ( x n ) + ( I - α n A ) T x n , n 0 .
(1.4)
It is proved that if the sequence {α n } satisfies appropriate conditions, the sequence {x n } generated by (1.4) converges strongly to the unique solution of the variational inequality:
( γ f - A ) x ̃ , x - x ̃ 0 , x C ,
(1.5)

or equivalently x ̃ = P F i x ( T ) ( I - A + γ f ) x ̃ , where C is the fixed point set of a nonexpansive mapping T.

In 2009, Maingè [4] considered the viscosity approximation method (1.1), and expanded the strong convergence to quasi-nonexpansive mappings in Hilbert space.

In 2010, Tian [5] considered the following general iterative method under the frame of nonexpansive mappings:
x n + 1 = α n γ f ( x n ) + ( I - μ α n F ) T x n , n 0 ,
(1.6)

and gave some strong convergent theorems.

Very recently, Tian [6] extended (1.6) to a more general scheme, that is: the mapping f: H → H is no longer a contraction but a L-Lipschitzian continuous operator with coefficient L > 0, and proved that if the sequence {α n } satisfies appropriate conditions, the sequence {x n } generated by xn+1= α n γf(x n ) + (I - μα n F)Tx n converges strongly to the unique solution x ̃ F i x ( T ) of the variational inequality where T is still nonexpansive:
( γ f - μ F ) x ̃ , x - x ̃ 0 , x F i x ( T ) .
(1.7)
Motivated by Maingè [4] and Tian [6], we consider the following iterative process:
x 0 = x H arbitrarily chosen , x n + 1 = α n γ f ( x n ) + ( I - α n μ F ) T ω x n , n 0 ,
(1.8)

where f is L-Lipschitzian, T ω = (1 - ω)I + ωT, and T is a quasi-nonexpansive mapping. Under some appropriate conditions on ω and {α n }, we obtain strong convergence over the class of quasi-nonexpansive mappings in Hilbert spaces. Our result is more general than Maingè's [4] conclusion.

2. Preliminaries

Throughout this article, we write x n x to indicate that the sequence {x n } converges weakly to x. x n → x implies that the sequence {x n } converges strongly to x. The following lemmas are useful for our article.

The following statements are valid in a Hilbert space H: for each x,y H, t [0,1]
  1. (i)

    ||x + y|| ≤ ||x||2 + 2〈y, x + y〉;

     
  2. (ii)

    ||(1 - t)x + ty||2 = (1 - t)||x||2 + t||y||2 - (1 - t)t||x - y||2;

     
  3. (iii)

    x , y = - 1 2 x - y 2 + 1 2 x 2 + 1 2 y 2 .

     
Lemma 2.1. Let f: HH be a L-Lipschitzian continuous operator with coefficient L > 0. F: HH is a κ-Lipschitzian continuous and η-strongly monotone operator with κ > 0 and η > 0. Then, for 0 < γμη/L,
x - y , ( μ F - γ f ) x - ( μ F - γ f ) y ( μ η - γ L ) x - y 2 .
(2.1)

That is, μF - γf is strongly monotone with coefficient μ η - γ L .

Lemma 2.2.[4]Let T ω := (1 - ω)I + ωT, with T quasi-nonexpansive on H, Fix(T) ≠ , and ω (0,1]. Then, the following statements are reached:

(a1) Fix(T) = Fix(T ω );

(a2) T ω is quasi-nonexpansive;

(a3) ||T ω x - q||2 ≤ ||x - q||2 - ω(1 - ω)||Tx - x||2for all x H and q Fix(T);

(a4) x - T ω x , x - q ω 2 x - T x 2 for all x H and q Fix(T).

Proposition 2.3. From the equality (iii) and the fact that T is quasi-nonexpansive, we have
x - T x , x - q = - 1 2 T x - q 2 + 1 2 x - T x 2 + 1 2 x - q 2 1 2 x - T x 2 .

(a4) is easily deduced by I-T ω = ω(I-T) and the previous inequality.

Lemma 2.4.[7]Let n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exist a subsequence { Γ n j } j 0 of n } which satisfies Γ n j < Γ n j + 1 for all j ≥ 0. Also, consider the sequence of integers { τ ( n ) } n n 0 defined by
τ ( n ) = max { k n Γ k < Γ k + 1 } .
Then, { τ ( n ) } n n 0 is a nondecreasing sequence verifying limn→∞τ(n) = ∞ and for all nn0, it holds that Γτ(n)< Γτ(n)+1and we have
Γ n Γ τ ( n ) + 1 .
Recall the metric projection P K from a Hilbert space H to a closed convex subset K of H is defined: for each x H the unique element P K x K such that
x - P K x : = inf { x - y : y K } .
Lemma 2.5. Let K be a closed convex subset of H. Given x H, and z K, z = P K x, if and only if there holds the inequality:
x - z , y - z 0 , y K .
Lemma 2.6. If x* is the solution of the variational inequality (1.7) with T: HH demi-closed and {y n } H is a bounded sequence such that ||Ty n - y n || → 0, then
lim inf n ( μ F - γ f ) x * , y n - x * 0 .
(2.2)
Proof. We assume that there exists a subsequence { y n j } of {y n } such that y n j . From the given conditions T y n - y n 0 and T: HH demi-closed, we have that any weak cluster point of {y n } belongs to the fixed point set Fix(T). Hence, we conclude that F i x ( T ) , and also have that
lim inf n ( μ F - γ f ) x * , y n - x * = lim j ( μ F - γ f ) x * , y n j - x * .
Recalling (1.7), we immediately obtain
lim inf n ( μ F - γ f ) x * , y n - x * = ( μ F - γ f ) x * , - x * 0 .

This completes the proof.   □

3. Main results

Let H be a real Hilbert space, let F be a κ-Lipschitzian and η-strongly monotone operator on H with k > 0, η > 0, and let T be a quasi-nonexpansive mapping on H, and f is a L-Lipschitzian mapping with coefficient L > 0 for all x,y H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix(T) is closed and convex.

Theorem 3.1. Let 0 < μ < 2 η κ 2 , 0 < γ < μ ( η - μ κ 2 2 ) L = τ L , and start with an arbitrary chosen x0 H, let the sequence {x n } be generated by
x n + 1 = α n γ f ( x n ) + ( I - α n μ F ) T ω x n ,
(3.1)

where the sequence {α n } (0,1) satisfies limn→∞α n = 0, and n = 0 α n = . Also ω ( 0 , 1 2 ) , T ω := (1 - ω)I + ωI with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x H, and q Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demi-closed on H; that is: if {y k } H, y k z, and (I - T)y k → 0, then z Fix(T).

Then, {x n } converges strongly to the x* Fix(T) which is the unique solution of the VIP:
( μ F - γ f ) x * , x - x * 0 , x F i x ( T ) .
(3.2)

Proof. First, we show that {x n } is bounded.

Take any p Fix(T), by Lemma 2.2 (a3), we have
x n + 1 - p = α n γ f ( x n ) + ( I - α n μ F ) T ω x n - p = α n γ ( f ( x n ) - f ( p ) ) + α n ( γ f ( p ) - μ F p ) + ( I - α n μ F ) T ω x n - ( I - α n μ F ) p α n γ L x n - p + α n γ f ( p ) - μ F p + ( 1 - α n τ ) x n - p ( 1 - α n ( τ - γ L ) ) x n - p + α n γ f ( p ) - μ F p .
(3.3)
By induction, we have
x n - p max x 0 - p , γ f ( p ) - μ F p τ - γ L , n 0 .

Hence, {x n } is bounded, so are the {f(x n )} and {F(x n )}.

From (3.1), we have
x n + 1 - x n + α n ( μ F x n - γ f ( x n ) ) = ( I - α n μ F ) T ω x n - ( I - α n μ F ) x n .
(3.4)
Since x* Fix(T), from Lemma 2.2 (a4), and together with (3.4), we obtain
x n + 1 - x n + α n ( μ F ( x n ) - γ f ( x n ) ) , x n - x * = ( I - α n μ F ) T ω x n - ( I - α n μ F ) x n , x n - x * = ( 1 - α n ) T ω x n - x n , x n - x * + α n ( I - μ F ) T ω x n - ( I - μ F ) x n , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + α n ( I - μ F ) T ω x n - ( I - μ F ) x n x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + α n ( 1 - τ ) T ω x n - x n x n - x * = - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( 1 - τ ) T x n - x n x n - x * ,
it follows from the previous inequality that
- x n - x n + 1 , x n - x * - α n ( μ F - γ f ) x n , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( 1 - τ ) T x n - x n x n - x * .
(3.5)
From (iii), we obviously have
x n - x n + 1 , x n - x * = - 1 2 x n + 1 - x * 2 + 1 2 x n - x * 2 + 1 2 x n + 1 - x n 2 .
(3.6)
Set Γ n : = 1 2 x n - x * 2 , and combine (3.5) with (3.6), it follows that
Γ n + 1 - Γ n - 1 2 x n + 1 - x n 2 - α n ( μ F - γ f ) x n , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( 1 - τ ) T x n - x n x n - x * .
(3.7)

Now, we calculate ||x n +1 - x n ||.

From the given condition: T ω := (1 - ω)I + ωT, it is easy to deduce that ||T ω x n - x n || = ω||x n - Tx n ||. Thus, it follows from (3.4) that
x n + 1 - x n 2 = α n ( γ f ( x n ) - μ F x n ) + ( I - α n μ F ) T ω x n - ( I - α n μ F ) x n 2 2 α n 2 γ f ( x n ) - μ F x n 2 + 2 ( 1 - α n τ ) 2 T ω x n - x n 2 = 2 α n 2 γ f ( x n ) - μ F x n 2 + 2 ω 2 ( 1 - α n τ ) 2 T x n - x n 2 .
(3.8)
Then, from (3.7) and (3.8), we have
Γ n + 1 - Γ n + ω 2 ( 1 - α n ) - ω 2 ( 1 - α n τ ) 2 x n - T x n 2 α n [ α n γ f ( x n ) - μ F x n 2 - ( μ F - γ f ) x n , x n - x * + ω ( 1 - τ ) T x n - x n x n - x * ] .
(3.9)

Finally, we prove x n x*. To this end, we consider two cases.

Case 1: Suppose that there exists n 0 such that { Γ n } n n 0 is nonincreasing, it is equal to Γn+1≤ Γ n for all nn0. It follows that limn→∞Γ n exists, so we conclude that
lim n ( Γ n + 1 - Γ n ) = 0 .
(3.10)
It follows from (3.9),(3.10) and combine with the fact that limn→∞α n = 0, we have limn→∞||x n - Tx n || = 0. Considering (3.9) again, from (3.10), we have
- α n [ α n γ f ( x n ) - μ F x n 2 - ( μ F - γ f ) x n , x n - x * + ω ( 1 - τ ) T x n - x n x n - x * ] Γ n - Γ n + 1 .
(3.11)
Then, by n = 0 α n = , we conclude that
lim inf n - [ α n γ f ( x n ) - μ F x n 2 - ( μ F - γ f ) x n , x n - x * + ω ( 1 - τ ) T x n - x n x n - x * ] 0 .
(3.12)
Since {f(x n )} and {x n } are both bounded, as well as α n → 0, and limn→∞||x n - Tx n || = 0, it follows from (3.12) that
lim inf n ( μ F - γ f ) x n , x n - x * 0 .
(3.13)
From Lemma 2.1, it is obvious that
( μ F - γ f ) x n , x n - x * ( μ F - γ f ) x * , x n - x * + 2 ( μ η - γ L ) Γ n .
(3.14)
Thus, from (3.14), and the fact that limn→∞Γ n exists, we immediately obtain*******
lim inf n ( μ F - γ f ) x * , x n - x * + 2 ( μ η - γ L ) Γ n = 2 ( μ η - γ L ) lim n Γ n + lim inf n ( μ F - γ f ) x * , x n - x * 0 ,
(3.15)
or equivalently
2 ( μ η - γ L ) lim n Γ n - lim inf n ( μ F - γ f ) x * , x n - x * .
(3.16)
Finally, by Lemma 2.6, we have
2 ( μ η - γ L ) lim n Γ n 0 ,
(3.17)

so we conclude that limn→∞Γ n = 0, which equivalently means that {x n } converges strongly to x*.

Case 2: Assume that there exists a subsequence { Γ n j } j 0 of {Γ n }n ≥ 0such that Γ n j < Γ n j + 1 for all j . In this case, it follows from Lemma 2.4 that there exists a subsequence {Γτ(n)} of {Γ n } such that Γτ(n)+1> Γτ(n), and {τ(n)} is defined as in Lemma 2.4.

Invoking (3.9) again, it follows that
Γ τ ( n ) + 1 - Γ τ ( n ) + ω 2 ( 1 - α τ ( n ) ) - ω 2 ( 1 - α τ ( n ) τ ) 2 x τ ( n ) - T x τ ( n ) 2 α τ ( n ) [ α τ ( n ) γ f ( x τ ( n ) ) - μ F x τ ( n ) 2 - ( μ F - γ f ) x τ ( n ) , x τ ( n ) - x * + ω ( 1 - τ ) T x τ ( n ) - x τ ( n ) x τ ( n ) - x * ] .
Recalling the fact that Γτ(n)+1> Γτ(n), we have
ω 2 ( 1 - α τ ( n ) ) - ω 2 ( 1 - α τ ( n ) τ ) 2 x τ ( n ) - T x τ ( n ) 2 α τ ( n ) [ α τ ( n ) γ f ( x τ ( n ) ) - μ F x τ ( n ) 2 - ( μ F - γ f ) x τ ( n ) , x τ ( n ) - x * + ω ( 1 - τ ) T x τ ( n ) - x τ ( n ) x τ ( n ) - x * ] .
(3.18)
From the preceding results, we get the boundedness of {x n } and α n → 0 which obviously lead to
lim n x τ ( n ) - T x τ ( n ) = 0 .
(3.19)
Hence, combining (3.18) with (3.19), we immediately deduce that
( μ F - γ f ) x τ ( n ) , x τ ( n ) - x * α τ ( n ) γ f ( x τ ( n ) ) - μ F x τ ( n ) 2 + ω ( 1 - τ ) T x τ ( n ) - x τ ( n ) x τ ( n ) - x * .
(3.20)
Again, (3.14) and (3.20) yield
( μ F - γ f ) x * , x τ ( n ) - x * + 2 ( μ η - γ L ) Γ τ ( n ) α τ ( n ) γ f ( x τ ( n ) ) - μ F x τ ( n ) 2 + ω ( 1 - τ ) T x τ ( n ) - x τ ( n ) x τ ( n ) - x * .
(3.21)
Recall that limn→∞ατ(n)= 0, from (3.19) and (3.21), we immediately have
2 ( μ η - γ L ) lim sup n Γ τ ( n ) - lim inf n ( μ F - γ f ) x * , x τ ( n ) - x * .
(3.22)
By Lemma 2.6, we have
lim inf n ( μ F - γ f ) x * , x τ ( n ) - x * 0 .
(3.23)
Consider (3.22) again, we conclude that
lim sup n Γ τ ( n ) = 0 ,
(3.24)

which means that limn→∞Γτ(n)= 0. By Lemma 2.4, it follows that Γ n ≤ Γτ(n), thus, we get limn→∞Γ n = 0, which is equivalent to x n x*.   □

Remark 3.2. Corollary 3.3 is only valid for ω ( 0 , 1 2 ) . This is revised by Wongchan and Saejung [8].

corollary 3.3.[4]Let the sequence {x n } be generated by
x n + 1 = α n f ( x n ) + ( 1 - α n ) T ω x n ,
(3.25)

where the sequence {α n } (0,1) satisfies limn→∞α n = 0, and n = 0 α n = . Also ω ( 0 , 1 2 ) , and T ω := (1 - ω)I + ωT with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x H, and q Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demi-closed on H; that is: if {y k } H, y k z, and (I - T)y k → 0, z Fix(T).

Then, {x n } converges strongly to the x* Fix(T) which is the unique solution of the VIP(3.26):
( I - f ) x * , x - x * 0 , x F i x ( T ) .
(3.26)

Declarations

Acknowledgements

The first author was supported by the Fundamental Research Funds for the Central Universities (No. ZXH2011C002).

Authors’ Affiliations

(1)
College of Science, Civil Aviation University of China, Tianjin, 300300, China

References

  1. Moudafi A: Viscosity approximation methods for fixed-points problems. J Math Anal Appl 2000, 241: 46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
  2. Xu HK: An iterative approach to quadratic optimizaton. J Optim Theory Appl 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
  3. Marino G, Xu HK: An general iterative method for nonexpansive mapping in Hilbert space. J Math Anal Appl 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar
  4. Maingé PE: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput Math Appl 2009,59(1):74–79.View ArticleMathSciNetMATHGoogle Scholar
  5. Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal 2010, 73: 689–694. 10.1016/j.na.2010.03.058MathSciNetView ArticleMATHGoogle Scholar
  6. Tian M: A general iterative method based on the hybrid steepest descent scheme for nonexpansive mappings in Hilbert spaces. 2010 International Conference on Computational Intelligence and Software Engineering, CiSE 2010 2010. art. no. 5677064Google Scholar
  7. Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 2008,16(7–8):899–912. 10.1007/s11228-008-0102-zMathSciNetView ArticleMATHGoogle Scholar
  8. Wongchan K, Saejung S: On the strong convergence of viscosity approximation process of quasi-nonexpansive mappings in Hilbert spaces. J Abstr Appl Anal 2011, 2011: 9. Article ID 385843MathSciNetMATHGoogle Scholar

Copyright

© Tian and Jin; licensee Springer. 2011

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.

Advertisement