Open Access

A general iterative method for quasi-nonexpansive mappings in Hilbert space

Journal of Inequalities and Applications20122012:38

https://doi.org/10.1186/1029-242X-2012-38

Received: 1 September 2011

Accepted: 20 February 2012

Published: 20 February 2012

Abstract

Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.

Keywords

quasi-nonexpansive mapping iterative analysis variational inequality 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}.

Iterative theory and methods for nonlinear mappings and variational inequalities have recently been applied to solve convex minimization problems, zero point problems and many others; see, e.g., [19] and references therein.

The viscosity approximation method was first introduced by Moudafi [10]. 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, and {ε n } is a sequence in (0,1) satisfying the following given conditions:
  1. (1)

    limn→∞ε n = 0;

     
  2. (2)

    n = 0 ε n = ;

     
  3. (3)

    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 [1], Xu proved that the sequence {x n } defined by the below process 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) provided the 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 [2], Marino and Xu 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 [11], Maingé considered the viscosity approximation method (1.1), and expanded the strong convergence to quasi-nonexpansive mappings in Hilbert space. Motivated by Marino and Xu [2] and Maingé [11], we consider the following iterative process:
x 0 = x H a r b i t r a r i l y c h o s e n , x n + 1 = α n γ f ( x n ) + ( I - α n A ) T ω x n , n 0 ,
(1.6)

where 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 Maingé's [11] conclusion, and also extends the iterative method (1.4) to quasi-nonexpansive mappings.

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

    ||x + y||2 ≤ ||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. [2] Let H be a Hilbert space H. Given x H, C is a closed convex subset of H, f : HH is a contraction with coefficient 0 < α < 1, and A is a strongly positive linear bounded operator with coefficient γ ̄ . Then for 0 < γ < γ ̄ / α ,
x - y , ( A - γ f ) x - ( A - γ f ) y ( γ ̄ - γ α ) x - y 2 , x , y H .

That is, A - γ f is strongly monotone with coefficient γ ̄ - γ α .

Lemma 2.2. [2] Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ ̄ > 0 and 0 < ρ ≤ ||A||-1. Then I - ρ A 1 - ρ γ ̄ .

Lemma 2.3. [11] Let T ω : = (1 - ω)I + ωT, with T being a quasi-nonexpansive mapping on H, F i x ( 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||2 for 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).

Remark 2.4. (a4) was revised by Wongchan and Saejung [12] (Proposition 2).

Lemma 2.5. [13] 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 form a Hilbert space H to a closed convex subset K of H is defined: for each x H, there exists a unique element P K x K such that
x - P K x : = inf { x - y : y K } .
Lemma 2.6. 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 2.7. If x* is the solution of the variational inequality (1.5) with demi-closedness of T and {y n } H is a bounded sequence such that ||Ty n - y n || → 0, then
lim inf n ( A γ f ) x * , y n x * 0.
(2.1)
Proof. We assume that there exists a subsequence { y n j } of {y n } such that y n j . From the given conditions ||Ty 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 ( A γ f ) x * , y n x * = lim j ( A γ f ) x * , y n j x * .
Recalling the (1.5), we immediately obtain
lim inf n ( A γ f ) x * , y n x * = ( A γ f ) x * , y ˜ x * 0.

This completes the proof.

3. Main results

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let T be a quasi-nonexpansive mapping on H, and f is a contraction with coefficient α; that is ||f (x) - f(y)|| ≤ α||x - y|| 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 (see [14] for more general results).

Throughout this article, we assume that A is strongly positive; that is, there exist a constant γ ̄ > 0 such that A x , x γ ̄ x 2 , for all x H. Let 0 < γ < γ ̄ / α .

Theorem 3.1. Starting 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 A ) 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 + 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 demiclosed 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:
( γ f - A ) x * , x - x * 0 , x F i x ( T ) .
(3.2)
Remark 3.2. Equivalently, from the VIP (3.2), we have
x * = P F i x ( T ) ( I - A + γ f ) x * .
(3.3)

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

Take any p Fix(T), from Lemma 2.3 (a3), we have
x n + 1 - p = α n γ f ( x n ) + ( I - α n A ) T ω x n - p = α n γ ( f ( x n ) - f ( p ) ) + α n ( γ f ( p ) - A p ) + ( I - α n A ) ( T ω x n - p ) α n γ α f ( x n ) - f ( p ) + α n γ f ( p ) - A p + ( 1 - α n γ ̄ ) x n - p ( 1 - α n ( γ ̄ - γ α ) ) x n - p + α n γ f ( p ) - A p .
(3.4)
By induction
x n - p max x 0 - p , γ f ( p ) - A p γ ̄ - γ α , n 0 .

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

Let x* = PFix(T)o(I - A + γf)x* From (3.1), we have
x n + 1 - x n + α n ( A x n - γ f ( x n ) ) = ( I - α n A ) ( T ω x n - x n ) .
(3.5)
Since x* Fix(T), from (a4), and together with (3.5), we obtain
x n + 1 - x n + α n ( A x n - γ f ( x n ) ) , x n - x * = ( I - α n A ) ( T ω x n - x n ) , x n - x * = ( 1 - α n ) T ω x n - x n , x n - x * + α n ( I - A ) ( T ω x n - x n ) , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( I - A ) ( T - I ) x n , x n - x * ,
it follows from the previous inequality that
- x n - x n + 1 , x n - x * - α n ( A - γ f ) x n , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( I - A ) ( T - I ) x n , x n - x * .
(3.6)
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 .
(3.7)
Set Γ n : = 1 2 x n - x * 2 , and combine with (3.6), it follows that
Γ n + 1 - Γ n - 1 2 x n + 1 - x n 2 - α n ( A - γ f ) x n , x n - x * - ω 2 ( 1 - α n ) x n - T x n 2 + ω α n ( I - A ) ( T - I ) x n , x n - x * .
(3.8)

Now, we calculate ||xn+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 - Txn||. Thus, it follows from (3.5) that
x n + 1 - x n 2 = α n ( γ f ( x n ) - A x n ) + ( I - α n A ) ( T ω x n - x n ) 2 2 α n 2 γ f ( x n ) - A x n 2 + 2 ( 1 - α n γ ̄ ) 2 T ω x n - x n 2 2 α n 2 γ f ( x n ) - A x n 2 + 2 ( 1 - α n γ ̄ ) T ω x n - x n 2 2 α n 2 γ f ( x n ) - A x n 2 + 2 ω 2 ( 1 - α n γ ̄ ) T x n - x n 2 .
(3.9)
Then from (3.8) and (3.9), we have
Γ n + 1 - Γ n + ω 2 ( 1 - α n ) - ω 2 ( 1 - α n γ ̄ ) x n - T x n 2 α n α n γ f ( x n ) - A x n 2 - ( A - γ f ) x n , x n - x * + ω ( I - A ) ( T - I ) x n , x n - x * .
(3.10)

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

Case 1: Suppose that there exists n0 such that { Γ n } n n 0 is nonincreasing, it is equal to Γ n+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 .
(3.11)
It follows from (3.10), (3.11) and the fact that limn→∞α n = 0, we have limn→∞||x n -Tx n || = 0. Again, from (3.10), we have
- α n [ α n γ f ( x n ) - A x n 2 - ( A - γ f ) x n , x n - x * + ω ( I - A ) ( T - I ) x n , x n - x * ] Γ n - Γ n + 1 .
(3.12)
Then, by n = 0 α n = , we conclude that
lim inf n [ α n ( γ f A ) x n 2 ( A γ f ) x n , x n x * + ω ( I A ) ( T I ) x n , x n x * ] 0.
(3.13)
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 (3.13) that
lim inf n ( A γ f ) x n , x n x * 0.
(3.14)
From Lemma 2.1, it is obvious that
( A - γ f ) x n , x n - x * ( A - γ f ) x * , x n - x * + 2 ( γ ̄ - γ α ) Γ n .
(3.15)
Thus, from (3.14), (3.15) and the fact that limn→∞Γ n exists, we immediately obtain
lim inf n [ ( A γ f ) x * , x n x * + 2 ( γ ¯ γ α ) Γ n ] = 2 ( γ ¯ γ α ) lim n Γ n + lim inf n ( A γ f ) x * , x n x * 0 ,
(3.16)
or equivalently
2 ( γ ¯ γ α ) lim n Γ n lim inf n ( A γ f ) x * , x n x * .
(3.17)
Finally, by Lemma 2.7, we have
2 ( γ ̄ - γ α ) lim n Γ n 0 ,
(3.18)

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 2.5 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 2.5.

Invoking the (3.10) again, it follows that
Γ τ ( n ) + 1 Γ τ ( n ) + [ ω 2 ( 1 α τ ( n ) ) ω 2 ( 1 α τ ( n ) γ ¯ ) ] x τ ( n ) T x τ ( n ) 2 α τ ( n ) [ α τ ( n ) γ f ( x τ ( n ) ) A x τ ( n ) 2 ( A γ f ) x τ ( n ) , x τ ( n ) x * + ω ( I A ) ( T I ) 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 ) γ ¯ ) ] x τ ( n ) T x τ ( n ) 2 α τ ( n ) [ α τ ( n ) γ f ( x τ ( n ) ) A x τ ( n ) 2 ( A γ f ) x τ ( n ) , x τ ( n ) x * + ω ( I A ) ( T I ) x τ ( n ) , x τ ( n ) x * ] .
(3.19)
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 .
(3.20)
Hence, combining (3.19) with (3.20), we immediately deduce that
( A - γ f ) x τ ( n ) , x τ ( n ) - x * α τ ( n ) γ f ( x τ ( n ) ) - A x τ ( n ) 2 + ω ( I - A ) ( T - I ) x τ ( n ) , x τ ( n ) - x * .
(3.21)
Again, (3.15) and (3.21) yield
( A - γ f ) x * , x τ ( n ) - x * + 2 ( γ ̄ - γ α ) Γ τ ( n ) α τ ( n ) γ f ( x τ ( n ) ) - A x τ ( n ) 2 + ω ( I - A ) ( T - I ) x τ ( n ) , x τ ( n ) - x * .
(3.22)
Recall that limn→∞α τ(n)= 0 and (3.20), we immediately have
2 ( γ ̄ - γ α ) lim sup n Γ τ ( n ) - lim inf n ( A - γ f ) x * , x τ ( n ) - x *
(3.23)
By Lemma 2.7, we have
lim inf n ( A - γ f ) x * , x τ ( n ) - x * 0 .
(3.24)
Consider (3.23) again, we conclude that
lim sup n Γ τ ( n ) = 0 ,
(3.25)

which means that limn→∞Γτ(n)= 0. By Lemma 2.5, it follows that Γ n ≤ Γτ(n), thus, we get limn→∞Γ n = 0, which is equivalent to x n x*.

Corollary 3.3. [11] Let the sequence {x n } be generated by
x n + 1 = α n f ( x n ) + ( 1 - α n ) T ω x n ,
(3.26)

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 demiclosed on H; that is: if{yk} 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 (3.27):
( I - f ) x * , x - x * 0 , x F i x ( T ) .
(3.27)

Declarations

Acknowledgements

M. Tian was supported by the Fundamental Research Funds for the Central Universities (No. ZXH2011C002).

Authors’ Affiliations

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

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© Tian and Jin; licensee Springer. 2012

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