Open Access

Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings

Fixed Point Theory and Applications20132013:273

https://doi.org/10.1186/1687-1812-2013-273

Received: 15 October 2012

Accepted: 25 September 2013

Published: 8 November 2013

Abstract

In this paper, we introduce a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudo-contractive mapping and the set of solutions of a monotone mapping. We also prove that the common element is the unique solution of certain variational inequality. The strong convergence theorems are obtained under some mild conditions. The results presented in this paper extend and unify most of the results that have been proposed for this class of nonlinear mappings.

MSC:47H09, 47H10, 47L25.

Keywords

pseudo-contractive mappings monotone mappings fixed point variational inequalities viscosity approximation

1 Introduction

Let C be a closed convex subset of a real Hilbert space H. A mapping A : C H is called monotone if and only if
x y , A x A y 0 , x , y C .
(1.1)
A mapping A : C H is called α-inverse strongly monotone if there exists a positive real number α > 0 such that
x y , A x A y α A x A y 2 , x , y C .
(1.2)

Obviously, the class of monotone mappings includes the class of the α-inverse strongly monotone mappings.

A mapping T : C H is called pseudo-contractive if x , y C , we have
T x T y , x y x y 2 .
(1.3)
A mapping T : C H is called κ-strict pseudo-contractive, if there exists a constant 0 κ 1 such that
x y , T x T y x y 2 κ ( I T ) x ( I T ) y 2 , x , y C .
(1.4)
A mapping T : C C is called non-expansive if
T x T y x y , x , y C .
(1.5)

Clearly, the class of pseudo-contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote by F ( T ) the set of fixed points of T, that is, F ( T ) = { x C : T x = x } .

A mapping f : C C is called contractive with a contraction coefficient if there exists a constant ρ ( 0 , 1 ) such that
f ( x ) f ( y ) ρ x y , x , y C .
(1.6)
For finding an element of the set of fixed points of the non-expansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967
x n + 1 = α n + 1 u + ( 1 α n + 1 ) T ( x n ) .
(1.7)
In 2000, Moudafi [2] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions
x n + 1 = α n f ( x n ) + ( 1 α n ) T ( x n ) .
(1.8)

Viscosity approximation methods are very important, because they are applied to convex optimization, linear programming, monotone inclusions and elliptic differential equations. In a Hilbert space, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results, see [318].

Suppose that A is a monotone mapping from C into H. The classical variational inequality problem is formulated as finding a point u C such that v u , A u 0 , v C . The set of solutions of variational inequality problems is denoted by VI ( C , A ) .

Takahashi [19, 20] introduced the following scheme and studied the weak and strong convergence theorem of the elements of F ( T ) VI ( C , A ) , respectively, under different conditions
x n + 1 = α n x n + ( 1 α n + 1 ) T P C ( x n λ n x n ) ,
(1.9)

where T is a non-expansive mapping, A is an α-inverse strong monotone operator.

Recently, Zegeye and Shahzad [21] introduced the algorithms and obtained the strong convergence theorems in a Hilbert space, respectively,
x n + 1 = α n f ( x n ) + ( 1 α n ) T n x n
(1.10)
and
x n + 1 = α n f ( x n ) + ( 1 α n ) T r n F r n x n ,
(1.11)

where T n are asymptotically non-expansive mappings, and T r n , F r n are non-expansive mappings.

Our concern is now the following: Is it possible to construct a new sequence that converges strongly to a common element of the intersection of the set of fixed points of a pseudo-contractive mapping and the solution set of a variational inequality problem for a monotone mapping?

2 Preliminaries

Let C be a nonempty closed and convex subset of a real Hilbert space H, a mapping P C : H C is called the metric projection, if x H , there exists a unique point in C, denote by P C x such that
x P C x x y , y C .
It is well known that P C is a non-expansive mapping, and P C x have the property as follows:
x P C x , y P C x 0 , x H , y C ,
(2.1)
x y 2 x P C x 2 + y P C x 2 , x H , y C .
(2.2)

Lemma 2.1 [6]

Let { a n } be a sequence of nonnegative real numbers satisfying the following relation:
a n + 1 ( 1 θ n ) a n + σ n , n 0 ,
where { θ n } is a sequence in (0,1) and { σ n } is a real sequence such that
  1. (i)

    n = 0 θ n = ;

     
  2. (ii)

    lim sup n σ n θ n 0 or n = 0 σ n < .

     

Then lim n a n = 0 .

Lemma 2.2 [21]

Let C be a closed convex subset of a Hilbert space H. Let A : C H be a continuous monotone mapping, let T : C C be a continuous pseudo-contractive mapping, define mappings T r and F r as follows: x H , r ( 0 , )
T r ( x ) = { z C : y z , T z 1 r y z , ( 1 + r ) z x 0 , y C } , F r ( x ) = { z C : y z , A z + 1 r y z , z x 0 , y C } .
Then the following hold:
  1. (i)

    T r and F r are single-valued;

     
  2. (ii)

    T r and F r are firmly non-expansive mappings, i.e., T r x T r y 2 T r x T r y , x y , F r x F r y 2 F r x F r y , x y ;

     
  3. (iii)

    F ( T r ) = F ( T ) , F ( F r ) = VI ( C , A ) ;

     
  4. (iv)

    F ( T ) and VI ( C , A ) are closed convex.

     

Lemma 2.3 [22]

Let { x n } and { z n } be bounded sequence in a Banach space, and let { β n } be a sequence in [ 0 , 1 ] , which satisfies the following condition:
0 < lim inf n β n < lim sup n β n < 1 .
Suppose that
x n + 1 = β n x n + ( 1 β n ) z n , n 0
and
lim n ( z n + 1 z n x n + 1 x n ) 0 .

Then lim n z n x n = 0 .

Let C be a closed convex subset of a Hilbert space H. Let A : C H be a continuous monotone mapping, let T : C C be a continuous pseudo-contractive mapping. Then we define the mappings as follows: for x H , τ n ( 0 , )
T τ n ( x ) = { z C : y z , T z 1 τ n y z , ( 1 + τ n ) z x 0 , y C } ,
(2.3)
F τ n ( x ) = { z C : y z , A z + 1 τ n y z , z x 0 , y C } .
(2.4)

3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping, let A : C H be a continuous monotone mapping such that F = F ( T ) VI ( C , A ) , let f : C C be a contraction with a contraction coefficient ρ ( 0 , 1 ) . The mappings T τ n and F τ n are defined as (2.3) and (2.4), respectively. Let { x n } be a sequence generated by x 0 C
{ y n = λ n x n + ( 1 λ n ) F τ n x n , x n + 1 = α n f ( x n ) + β n x n + γ n T τ n y n ,
(3.1)
where λ n [ 0 , 1 ] , let { α n } , { β n } , { γ n } be sequences of nonnegative real numbers in [ 0 , 1 ] and
  1. (i)

    α n + β n + γ n = 1 , n 0 ;

     
  2. (ii)

    lim n α n = 0 , n = 1 α n = ;

     
  3. (iii)

    0 < lim inf n λ n < lim sup n λ n < 1 ;

     
  4. (iv)

    lim inf n τ n > 0 , n = 1 | τ n + 1 τ n | < .

     
Then the sequence { x n } converges strongly to x ¯ = P F f ( x ¯ ) , and also x ¯ is the unique solution of the variational inequality
f ( x ¯ ) x ¯ , y x ¯ 0 , y F .
(3.2)
Proof First, we prove that { x n } is bounded. Take p F , then we have from Lemmas 2.2 that
y n p λ n x n p + ( 1 λ n ) F τ n x n F τ n p x n p .
(3.3)
For n 0 , because T τ n and F τ n are non-expansive, and f is contractive, we have
x n + 1 p = α n ( f ( x n ) p ) + β n ( x n p ) + γ n ( T τ n y n p ) α n f ( x n ) f ( p ) + α n f ( p ) p + β n x n p + γ n T τ n y n p ρ α n x n p + α n f ( p ) p + ( 1 α n ) x n p [ 1 ( 1 ρ ) α n ] x n p + α n f ( p ) p max { x 0 p , f ( p ) p 1 ρ } .

Therefore, { x n } is bounded. Consequently, we get that { F τ n x n } , { T τ n y n } and { y n } , { f ( x n ) } are bounded.

Next, we show that x n + 1 x n 0 .
y n + 1 y n λ n + 1 x n + 1 x n + ( 1 λ n + 1 ) F τ n + 1 x n + 1 F τ n x n + | λ n + 1 λ n | x n F τ n x n .
(3.4)
Let v n = F τ n x n , v n + 1 = F τ n + 1 x n + 1 , by the definition of the mapping F τ n , we have that
y v n , A v n + 1 τ n y v n , v n x n 0 , y C ,
(3.5)
y v n + 1 , A v n + 1 + 1 τ n + 1 y v n + 1 , v n + 1 x n + 1 0 , y C .
(3.6)
Putting y : = v n + 1 in (3.5), and letting y : = v n in (3.6), we have that
v n + 1 v n , A v n + 1 τ n v n + 1 v n , v n x n 0 ,
(3.7)
v n v n + 1 , A v n + 1 + 1 τ n + 1 v n v n + 1 , v n + 1 x n + 1 0 .
(3.8)
Adding (3.7) and (3.8), we have that
v n + 1 v n , A v n A v n + 1 + v n + 1 v n , v n x n τ n v n + 1 x n + 1 τ n + 1 0 .
Since A is a monotone mapping, which implies that
v n + 1 v n , v n x n τ n v n + 1 x n + 1 τ n + 1 0 .
Therefore, we have that
v n + 1 v n , v n x n τ n ( v n + 1 x n + 1 ) τ n + 1 + v n + 1 v n + 1 0 ,
i.e.,
v n + 1 v n 2 v n + 1 v n , x n + 1 x n + ( 1 τ n τ n + 1 ) ( v n + 1 x n + 1 ) v n + 1 v n { x n + 1 x n + | 1 τ n τ n + 1 | v n + 1 x n + 1 } .
(3.9)
Without loss of generality, let b be a real number such that τ n > b > 0 , n N , then we have that
v n + 1 v n x n + 1 x n + | 1 τ n τ n + 1 | v n + 1 x n + 1 x n + 1 x n + 1 b | τ n + 1 τ n | K ,
(3.10)
where K = sup v n + 1 x n + 1 . Then we have from (3.10) and (3.4) that
y n + 1 y n x n + 1 x n + ( 1 λ n + 1 ) | τ n + 1 τ n | b K + | λ n + 1 λ n | x n F τ n x n .
(3.11)
On the other hand, let u n = T τ n y n , u n + 1 = T τ n + 1 y n + 1 , we have that
y u n , T u n 1 τ n y u n , ( 1 + τ n ) u n y n 0 , y C ,
(3.12)
y u n + 1 , T u n + 1 1 τ n + 1 y u n + 1 , ( 1 + τ n + 1 ) u n + 1 y n + 1 0 , y C .
(3.13)
Let y : = u n + 1 in (3.12), and let y : = u n in (3.13), we have that
u n + 1 u n , T u n 1 τ n u n + 1 u n , ( 1 + τ n ) u n y n 0 ,
(3.14)
u n u n + 1 , T u n + 1 1 τ n + 1 u n u n + 1 , ( 1 + τ n + 1 ) u n + 1 y n + 1 0 .
(3.15)
Adding (3.14) and (3.15), and because T is pseudo-contractive, we have that
u n + 1 u n , u n y n τ n u n + 1 y n + 1 τ n + 1 0 .
Therefore, we have
u n + 1 u n , u n y n τ n ( u n + 1 y n + 1 ) τ n + 1 + u n + 1 u n + 1 0 .
Hence we have that
u n + 1 u n y n + 1 y n + 1 b | τ n + 1 τ n | M ,
(3.16)

where M = sup { u n y n : n N } .

Let x n + 1 = β n x n + ( 1 β n ) z n , hence we have that
z n + 1 z n = α n + 1 1 β n + 1 ( f ( x n + 1 ) f ( x n ) ) + ( α n + 1 1 β n + 1 α n 1 β n ) f ( x n ) + γ n + 1 1 β n + 1 ( u n + 1 u n ) + ( γ n + 1 1 β n + 1 γ n 1 β n ) u n .
(3.17)
Hence we have from (3.17), (3.16), (3.11) and condition (iii) that
z n + 1 z n x n + 1 x n ( ρ 1 ) α n + 1 1 β n + 1 x n + 1 x n + | α n + 1 1 β n + 1 α n 1 β n | { f ( x n ) + u n } + γ n + 1 1 β n + 1 | τ n + 1 τ n | b ( ( 1 λ n + 1 ) K + M ) + γ n + 1 1 β n + 1 | λ n + 1 λ n | x n F τ n x n .
(3.18)
Notice conditions (ii) and (iv), we have that
lim sup n ( z n + 1 z n x n + 1 x n ) = 0 .
(3.19)
Hence we have from Lemma 2.3 that
lim sup n z n x n = 0 .
(3.20)
Therefore, we have that
x n + 1 x n = | 1 β n | z n x n 0 .
(3.21)
Hence we have from (3.10) and (3.16) that
y n + 1 y n 0 , u n + 1 u n 0 , v n + 1 v n 0 .
(3.22)
Since x n = α n 1 f ( x n 1 ) + β n 1 x n 1 + γ n 1 v n 1 , so we have that
x n v n x n v n 1 + v n 1 v n [ 1 α n 1 ( 1 ρ ) ] x n 1 v n 1 + v n 1 v n .
According to Lemma 2.1, we have that
x n v n 0 .
In the same way, we have that
x n y n 0 , x n u n 0 .
Consequently, we have that
y n u n 0 .

Now, we show that lim sup n f ( x ¯ ) x ¯ , x n + 1 x ¯ 0 .

Since sequence { x n } is bounded, then there exists a sub-sequence { x n k } of { x n } and w C such that x n k w . Next, we show that w F = F ( T ) VI ( C , A ) .

Since v n = F τ n x n , we have that
y v n k , A v n k + 1 τ n k y v n k , v n k x n k 0 , y C .
Let v t = t v + ( 1 t ) w , t ( 0 , 1 ) , v C , then we have that
v t v n k , A v t v t v n k , A v t A v n k v t v n k , v n k x n k τ n k .
Since x n k v n k 0 , and also A is monotone, we have that
0 lim k v t v n k , A v t = v t w , A v t .
Consequently,
v t w , A v t 0 .

If t 0 , by the continuity of A, we have v w , A w 0 , v C . Thus, w VI ( C , A ) .

In addition, since u n = F τ n y n , we have that
y u n k , T u n k 1 τ n k y u n k , ( 1 + τ n k ) u n k y n k 0 , y C .
Let v t = t v + ( 1 t ) w , t ( 0 , 1 ) , v C , then we have that
u n k v t , T v t v t u n k , T u n k T v t v t u n k , 1 + τ n k τ n k u n k y n k v t u n k , v t 1 τ n k v t u n k , u n k y n k .
Since y n k u n k 0 , if t 0 , by the continuity of T, we have that
v w , T w v w , w , v C .

Let v = T w , we have w = T w , thus, w F ( T ) . Consequently, we conclude that w F = F ( T ) VI ( C , A ) .

Because x ¯ = P F f ( x ¯ ) , we have from (2.1) that
lim sup n f ( x ¯ ) x ¯ , x n x ¯ = f ( x ¯ ) P F f ( x ¯ ) , w P F f ( x ¯ ) 0 .
(3.23)
Next, we show that x n x ¯ . From formula (3.3), we have that
x n + 1 x ¯ 2 = α n ( f ( x n ) x ¯ ) + β n ( x n x ¯ ) + γ n ( u n x ¯ ) 2 α n 2 f ( x n ) x ¯ 2 + β n ( x n x ¯ ) + γ n ( u n x ¯ ) 2 + 2 α n f ( x n ) x ¯ , β n ( x n x ¯ ) + γ n ( u n x ¯ ) α n 2 f ( x n ) x ¯ 2 + ( 1 α n ) 2 x n x ¯ 2 + 2 α n β n f ( x n ) x ¯ , x n x ¯ + 2 α n γ n f ( x n ) x ¯ u n x ¯ α n 2 f ( x n ) x ¯ 2 + ( 1 α n ) 2 x n x ¯ 2 + 2 α n β n f ( x ¯ ) x ¯ , x n x ¯ + 2 α n β n f ( x n ) f ( x ¯ ) , x n x ¯ + 2 α n γ n f ( x n ) x ¯ u n x ¯ = [ ( 1 α n ) 2 + 2 ρ α n β n ] x n x ¯ 2 + α n 2 f ( x n ) x ¯ 2 + 2 α n β n f ( x ¯ ) x ¯ , x n x ¯ + 2 α n γ n f ( x n ) x ¯ u n x ¯ = ( 1 2 α n ( 1 ρ β n ) ) x n x ¯ 2 + α n 2 [ f ( x n ) x ¯ 2 + x n x ¯ 2 ] + 2 α n γ n f ( x n ) x ¯ u n x ¯ + 2 α n β n f ( x ¯ ) x ¯ , x n x ¯ .

Let θ n = 2 α n ( 1 ρ β n ) , σ n = α n 2 [ f ( x n ) x ¯ 2 + x n x ¯ 2 ] + 2 α n γ n f ( x ¯ ) x ¯ u n x ¯ + 2 α n β n f ( x ¯ ) x ¯ , x n x ¯ . According to Lemma 2.1 and formula (3.23), we have that lim n x n x ¯ = 0 , i.e., the sequence { x n } converges strongly to x ¯ F .

According to formula (3.23), we conclude that x ¯ is the solution of the variational inequality (3.2). Now, we show that x ¯ is the unique solution of the variational inequality (3.2).

Suppose that y ¯ F is another solution of the variational inequality (3.2). Because x ¯ is the solution of the variational inequality (3.2), i.e., f ( x ¯ ) x ¯ , y x ¯ 0 , y F . Because y ¯ F , then we have
f ( x ¯ ) x ¯ , y ¯ x ¯ 0 .
(3.24)
On the other hand, to the solution y ¯ F , since x ¯ F , so
f ( y ¯ ) y ¯ , x ¯ y ¯ 0 .
(3.25)
Adding (3.24) and (3.25), we have that
x ¯ y ¯ ( f ( x ¯ ) f ( y ¯ ) ) , x ¯ y ¯ 0 ,
i.e.,
x ¯ y ¯ ( f ( x ¯ ) f ( y ¯ ) ) , x ¯ y ¯ f ( x ¯ ) f ( y ¯ ) , x ¯ y ¯ .
Hence
x ¯ y ¯ 2 f ( x ¯ ) f ( y ¯ ) , x ¯ y ¯ ρ x ¯ y ¯ 2 .

Because ρ ( 0 , 1 ) , hence we conclude that x ¯ = y ¯ , the uniqueness of the solution is obtained. □

Theorem 3.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping, let A : C H be a continuous monotone mapping such that F = F ( T ) VI ( C , A ) , let f : C C be a contraction with a contraction coefficient ρ ( 0 , 1 ) . The mappings T τ n and F τ n are defined as (2.3) and (2.4), respectively. Let { x n } be a sequence generated by x 0 C
x n + 1 = α n f ( x n ) + β n x n + γ n T τ n F τ n x n ,
(3.26)
where { α n } , { β n } , { γ n } are sequences of nonnegative real numbers in [ 0 , 1 ] and
  1. (i)

    α n + β n + γ n = 1 , n 0 ;

     
  2. (ii)

    lim n α n = 0 , n = 1 α n = ;

     
  3. (iii)

    lim inf n τ n > 0 , n = 1 | τ n + 1 τ n | < .

     
Then the sequence { x n } converges strongly to x ¯ = P F f ( x ¯ ) , and also x ¯ is the unique solution of the variational inequality
f ( x ¯ ) x ¯ , y x ¯ 0 , y F .
(3.27)

Proof Putting λ n = 0 in Theorem 3.1, we can obtain the result. □

If in Theorem 3.1 and Theorem 3.2, let f : u C be a constant mapping, we have the following theorems.

Theorem 3.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping, let A : C H be a continuous accretive mapping such that F = F ( T ) VI ( C , A ) . The mappings T τ n and F τ n are defined as (2.3) and (2.4), respectively. Let { x n } be a sequence generated by
{ x 0 = x C , y n = λ n x n + ( 1 λ n ) F τ n x n , x n + 1 = α n u + β n x n + γ n T τ n y n ,
(3.28)
where λ n [ 0 , 1 ] , and let { α n } , { β n } , { γ n } be sequences of nonnegative real numbers in [ 0 , 1 ] and
  1. (i)

    α n + β n + γ n = 1 , n 0 ;

     
  2. (ii)

    lim n α n = 0 , n = 1 α n = ;

     
  3. (iii)

    0 < lim inf n λ n < lim sup n λ n < 1 ;

     
  4. (iv)

    lim inf n τ n > 0 , n = 1 | τ n + 1 τ n | < .

     
Then the sequence { x n } converges strongly to x ¯ = P F u , and also x ¯ is the unique solution of the variational inequality
u x ¯ , y x ¯ 0 , y F .
(3.29)
Theorem 3.4 Let C be a nonempty closed convex subset of a Hilbert space H. Let T : C C be a continuous pseudo-contractive mapping, let A : C H be a continuous monotone mapping such that F = F ( T ) VI ( C , A ) . The mappings T τ n and F τ n are defined as (2.3) and (2.4), respectively. Let { x n } be a sequence generated by x 0 C
x n + 1 = α n u + β n x n + γ n T τ n F τ n x n ,
(3.30)
where { α n } , { β n } , { γ n } are sequences of nonnegative real numbers in [ 0 , 1 ] and
  1. (i)

    α n + β n + γ n = 1 , n 0 ;

     
  2. (ii)

    lim n α n = 0 , n = 1 α n = ;

     
  3. (iii)

    lim inf n τ n > 0 , n = 1 | τ n + 1 τ n | < .

     
Then the sequence { x n } converges to x ¯ = P F u , and also x ¯ is the unique solution of the variational inequality
u x ¯ , y x ¯ 0 , y F .
(3.31)

Declarations

Acknowledgements

This work was supported by the Natural Science Foundation Project of Chongqing (CSTC, 2012jjA00039) and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712, KJ130731).

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University

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© Tang; licensee Springer. 2013

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