Open Access

Strong convergence theorems for solutions of fixed point and variational inequality problems

Journal of Inequalities and Applications20142014:215

https://doi.org/10.1186/1029-242X-2014-215

Received: 16 March 2014

Accepted: 21 May 2014

Published: 29 May 2014

Abstract

The purpose of this paper is to investigate viscosity approximation methods for finding a common element in the set of fixed points of a strict pseudocontraction and in the set of solutions of a generalized variational inequality in the framework of Banach spaces.

Keywords

sunny nonexpansive retractioninverse-strongly accretive mappingnonexpansive mappingvariational inequality

1 Introduction

Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let P C be the metric projection of H onto C. Recall that a mapping A : C H is said to be monotone iff
A x A y , x y 0 x , y C .
Recall that a mapping A : C H is said to be inverse-strongly monotone iff there exists a positive real number α > 0 such that
A x A y , x y α A x A y 2 x , y C .

For such a case, A is said to be α-inverse-strongly monotone.

Recall that the classical variational inequality problem, denoted by VI ( C , A ) , is to find u C such that
A u , v u 0 v C .
(1.1)

It is clear that variational inequality problem (1.1) is equivalent to a fixed point problem. u is a solution of the above inequality iff it is a fixed point of the mapping P C ( I r A ) , where I is the identity and r is some positive real number.

Variational inequality problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, and network. Recently, many authors studied the solutions of inequality (1.1) based on iterative methods; see [117] and the references therein.

Let S : C C be a mapping. In this paper, we denote by F ( S ) the set of fixed points of the mapping S.

Recall that S is said to be nonexpansive iff
S x S y x y x , y C .
Recall that S is said to be a strict pseudocontraction iff there exits a positive constant λ such that
S x S y 2 x y 2 + λ ( I S ) x ( I S ) y 2 x , y C .

It is clear that the class of strict pseudocontractions includes the class of nonexpansive mappings as a special case.

Recently, many authors have investigated the problems of finding a common element in the set of solution of variational inequalities for an inverse-strongly monotone mapping and in the set of fixed points of nonexpansive mappings or strict pseudocontractions; see [1825] and the references therein. However, most of the results are in the framework of Hilbert spaces. In this paper, we investigate a common element problem in the framework of Banach spaces. A strong convergence theorem for common solutions to fixed point problems of strict pseudocontractions and solution problems of variational inequality (1.1) is established in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [5] and Hao [26].

2 Preliminaries

Let C be a nonempty closed and convex subset of a Banach space E. Let E be the dual space of E, and let , denote the pairing between E and E . For q > 1 , the generalized duality mapping J q : E 2 E is defined by
J q ( x ) = { f E : x , f = x q , f = x q 1 }
for all x E . In particular, J = J 2 is called the normalized duality mapping. It is known that J q ( x ) = x q 2 J ( x ) for all x E . If E is a Hilbert space, then J = I , the identity mapping. The normalized duality mapping J has the following properties:
  1. (1)

    if E is smooth, then J is single-valued;

     
  2. (2)

    if E is strictly convex, then it is one-to-one and x y , x y > 0 holds for all ( x , x ) , ( y , y ) J with x y ;

     
  3. (3)

    if E is reflexive, then J is surjective;

     
  4. (4)

    if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

     
Let U = { x X : x = 1 } . A Banach space E is said to be uniformly convex if, for any ϵ ( 0 , 2 ] , there exists δ > 0 such that, for any x , y U ,
x y ϵ implies x + y 2 1 δ .
It is known that a uniformly convex Banach space is reflexive and strictly convex. Hilbert spaces are 2-uniformly convex, while L p is max { p , 2 } -uniformly convex for every p > 1 . A Banach space E is said to be smooth if the limit
lim t 0 x + t y x t
(2.1)
exists for all x , y U . It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for x , y U . The norm of E is said to be Fréchet differentiable if, for any x U , the limit (2.1) is attained uniformly for all y U . The modulus of smoothness of E is defined by
ρ ( τ ) = sup { 1 2 ( x + y + x y ) 1 : x , y X , x = 1 , y = τ } ,

where ρ : [ 0 , ) [ 0 , ) is a function. It is known that E is uniformly smooth if and only if lim τ 0 ρ ( τ ) τ = 0 . Let q be a fixed real number with 1 < q 2 . A Banach space E is said to be q-uniformly smooth if there exists a constant c > 0 such that ρ ( τ ) c τ q for all τ > 0 .

We remark that all Hilbert spaces, L p (or l p ) spaces ( p 2 ) and the Sobolev spaces W m p ( p 2 ) are 2-uniformly smooth, while L p (or l p ) and W m p spaces ( 1 < p 2 ) are p-uniformly smooth. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L p , where p > 1 . More precisely, L p is min { p , 2 } -uniformly smooth for every p > 1 .

Recall that a mapping S is said to be λ-strictly pseudocontractive iff there exist a constant λ ( 0 , 1 ) and j ( x y ) J ( x y ) such that
S x S y , j ( x y ) x y 2 λ ( I S ) x ( I S ) y 2 x , y C .
(2.2)
It is clear that (2.2) is equivalent to the following:
( I S ) x ( I S ) y , j ( x y ) λ ( I S ) x ( I S ) y 2 x , y C .
(2.3)
Next, we assume that E is a smooth Banach space. Let C be a nonempty closed convex subset of E. Recall that an operator A of C into E is said to be accretive iff
A x A y , J ( x y ) 0 x , y C .

An accretive operator A is said to be m-accretive if the range of I + r A is E for all r > 0 . In a real Hilbert space, an operator A is m-accretive if and only if A is maximal monotone.

Recall that an operator A of C into E is said to be α-inverse strongly accretive iff there exits a real constant α > 0 such that
A x A y , J ( x y ) α A x A y 2 x , y C .

Evidently, the definition of an inverse-strongly accretive operator is based on that of an inverse-strongly monotone operator.

Let D be a subset of C and Q be a mapping of C into D. Then Q is said to be sunny if
Q ( Q x + t ( x Q x ) ) = Q x ,

whenever Q x + t ( x Q x ) C for x C and t 0 . A mapping Q of C into itself is called a retraction if Q 2 = Q . If a mapping Q of C into itself is a retraction, then Q z = z for all z R ( Q ) , where R ( Q ) is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.

The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 2.1 [27]

Let E be a smooth Banach space, and let C be a nonempty subset of E. Let Q : E C be a retraction, and let J be the normalized duality mapping on E. Then the following are equivalent:
  1. (1)

    Q C is sunny and nonexpansive;

     
  2. (2)

    Q C x Q C y 2 x y , J ( Q C x Q C y ) x , y E ;

     
  3. (3)

    x Q C x , J ( y Q C x ) 0 x E , y C .

     

Proposition 2.2 [28]

Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with F ( T ) . Then the set F ( T ) is a sunny nonexpansive retract of C.

Recently, Aoyama et al. [29] considered the following generalized variational inequality problem.

Let C be a nonempty closed convex subset of E, and let A be an accretive operator of C into E. Find a point u C such that
A u , J ( v u ) 0 v C .
(2.4)

Next, we use BVI ( C , A ) to denote the set of solutions of variational inequality problem (2.4).

Aoyama et al. [29] proved that variational inequality (2.4) is equivalent to a fixed point problem. The element u C is a solution of variational inequality (2.4) iff u C is a fixed point of the mapping Q C ( I r A ) , where r > 0 is a constant and Q C is a sunny nonexpansive retraction from E onto C.

The following lemmas also play an important role in this paper.

Lemma 2.3 [30]

Assume that { α n } is a sequence of nonnegative real numbers such that
α n + 1 ( 1 γ n ) α n + δ n + e n ,
where { γ n } is a sequence in ( 0 , 1 ) , { e n } and { δ n } are sequences such that
  1. (1)

    n = 1 γ n = ;

     
  2. (2)

    n = 1 e n < ;

     
  3. (3)

    lim sup n δ n / γ n 0 or n = 1 | δ n | < .

     

Then lim n α n = 0 .

Lemma 2.4 [31]

Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
x + y 2 x 2 + 2 y , J x + 2 K y 2 x , y E .

Lemma 2.5 [29]

Let C be a nonempty closed convex subset of a smooth Banach space E. Let Q C be a sunny nonexpansive retraction from E onto C, and let A be an accretive operator of C into E. Then, for all λ > 0 ,
BVI ( C , A ) = F ( Q C ( I λ A ) ) .

Lemma 2.6 [32]

Let C be a closed convex subset of a real strictly convex Banach space E and S i : C C ( i = 1 , 2 ) be two nonexpansive mappings such that F = F ( S 1 ) F ( S 2 ) . Define S x = δ S 1 x + ( 1 δ ) S 2 x , where δ ( 0 , 1 ) . Then S : C C is a nonexpansive mapping with F ( S ) = F .

Lemma 2.7 [33]

Let C be a nonempty subset of a real 2-uniformly smooth Banach space E, and let T : C C be a κ-strict pseudocontraction. For α ( 0 , 1 ) , we define T α x = ( 1 α ) x + α T x for every x C . Then, as α ( 0 , κ K 2 ] , T α is nonexpansive such that F ( T α ) = F ( T ) .

Lemma 2.8 [34]

Let E be a real uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let T : C C be a nonexpansive mapping with a fixed point, and let f : C C be a contraction. For each t ( 0 , 1 ) , let z t be the unique solution of the equation x = t f ( x ) + ( 1 t ) T x . Then { z t } converges to a fixed point of T as t 0 and Q ( f ) = s - lim t 0 z t defines the unique sunny nonexpansive retraction from C onto F ( T ) .

3 Main results

Theorem 3.1 Let E be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty, closed and convex subset of E. Let Q C be a sunny nonexpansive retraction from E onto C, and let A : C E be an α-inverse strongly accretive mapping. Let S : C C be a λ-strict pseudocontraction with a fixed point. Assume that F : = F ( S ) BVI ( C , A ) . Let { α n } , { β n } and { γ n } be real number sequences in ( 0 , 1 ) . Suppose that x 1 = x C and that { x n } is given by
x n + 1 = α n f ( x n ) + β n [ μ S t x n + ( 1 μ ) Q C ( x n λ A x n ) ] + γ n Q C e n , n 1 ,
where S t = ( 1 t ) x + t S x , t ( 0 , λ K 2 ] , f : C C is a κ-contractive mapping, { e n } is a bounded computational error in E, λ ( 0 , α / K 2 ] and μ ( 0 , 1 ) . Assume that the following restrictions are satisfied:
  1. (a)

    lim n α n = 0 , n = 1 α n = and n = 1 | α n + 1 α n | < ;

     
  2. (b)

    n = 1 γ n < .

     

Then { x n } converges strongly to x = Q F f ( x ) , where Q F is a sunny nonexpansive retraction from C onto F.

Proof Fixing x F , we find that x = Q C ( x λ A x ) and S x = x . It follows from Lemma 2.7 that S t x = x . Put y n = Q C ( x n λ A x n ) . In view of Lemma 2.4, we find that
y n x 2 ( x n x ) λ ( A x n A x ) 2 x n x 2 2 λ A x n A x , J ( x n x ) + 2 K 2 λ 2 A x n A x 2 x n x 2 2 λ α A x n A x 2 + 2 K 2 λ 2 A x n A x 2 = x n x 2 + 2 λ ( λ K 2 α ) A x n A x 2 .
Since λ ( 0 , α / K 2 ] , we have that
y n x x n x .
This implies that Q C ( I λ A ) is a nonexpansive mapping. Hence, we have
x n + 1 x = α n f ( x n ) + β n [ μ S x n + ( 1 μ ) Q C ( x n λ A x n ) ] + γ n Q C e n x α n f ( x n ) x + β n μ S x n + ( 1 μ ) Q C ( x n λ A x n ) x + γ n Q C e n x α n f ( x n ) x + β n μ x n x + β n ( 1 μ ) x n x + γ n Q C e n x α n κ x n x + α n f ( x ) x + β n x n x + γ n e n x ( 1 α n ( 1 κ ) ) x n x + α n f ( x ) x + γ n e n x max { x n x , f ( x ) x 1 κ } + γ n e n x ,
which implies that the sequence { x n } is bounded, so is { y n } . Define
t n = μ S t x n + ( 1 μ ) Q C ( x n λ A x n ) .
It follows that
t n t n 1 = μ S t x n + ( 1 μ ) Q C ( I λ A ) x n [ μ S t x n 1 + ( 1 μ ) Q C ( I λ A ) x n 1 ] μ S t x n S t x n 1 + ( 1 μ ) Q C ( I λ A ) x n Q C ( I λ A ) x n 1 μ x n x n 1 + ( 1 μ ) x n x n 1 = x n x n 1 .
(3.1)
On the other hand, we have
x n + 1 x n α n f ( x n ) f ( x n 1 ) + | α n α n 1 | f ( x n 1 ) + β n t n t n 1 + | β n β n 1 | t n 1 + γ n Q C e n Q C e n 1 + | γ n γ n 1 | Q C e n .
(3.2)
Substituting (3.1) into (3.2), we see that
x n + 1 x n ( 1 α n ( 1 κ ) ) x n x n 1 + | α n α n 1 | f ( x n 1 ) + | β n β n 1 | t n 1 + γ n Q C e n Q C e n 1 + | γ n γ n 1 | Q C e n ( 1 α n ( 1 κ ) ) x n x n 1 + | α n α n 1 | ( f ( x n 1 ) + t n 1 ) + | γ n γ n 1 | ( t n 1 + Q C e n ) + γ n Q C e n Q C e n 1 .
In view of Lemma 2.3, we find from the restrictions (a) and (b) that
lim n x n + 1 x n = 0 .
(3.3)
Note that
x n t n x n x n + 1 + x n + 1 t n x n x n + 1 + α n f ( x n ) t n + γ n Q C e n t n .
Using (3.3), we find from the restrictions (a) and (b) that
lim n x n t n = 0 .
(3.4)
Define a mapping V by
V x = μ S t x + ( 1 μ ) Q C ( I λ A ) x x C .
Using Lemma 2.6, we see that the mapping V is a nonexpansive mapping with
F ( V ) = F ( S t ) F ( Q C ( I λ A ) ) = F ( S t ) BVI ( C , A ) = F ( S ) BVI ( C , A ) = F .
From (3.4), we see that
lim n x n V x n = 0 .
(3.5)
Next, we show that
lim sup n f ( x ) x , J ( x n x ) 0 ,
(3.6)
where x = Q F f ( x ) , and Q F is a sunny nonexpansive retraction from C onto F, the strong limit of the sequence z t defined by
z t = t f ( z t ) + ( 1 t ) V z t .
It follows that
z t x n = ( 1 t ) ( V z t x n ) + t ( f ( z t ) x n ) .
For any t ( 0 , 1 ) , we see that
z t x n 2 ( 1 t ) 2 V z t x n 2 + 2 t f ( z t ) x n , j ( z t x n ) ( 1 t ) 2 ( V z t V x n 2 + V x n x n 2 + 2 V z t V x n V x n x n ) + 2 t f ( z t ) z t , j ( z t x n ) + 2 t z t x n , j ( z t x n ) ( 1 t ) 2 z t x n 2 + λ n ( t ) + 2 t f ( z t ) z t , j ( z t x n ) + 2 t z t x n 2 ,
(3.7)
where
λ n ( t ) = V x n x n 2 + 2 z t x n V x n x n .
It follows from (3.7) that
z t f ( z t ) , j ( z t x n ) t 2 z t x n 2 + 1 2 t λ n ( t ) .
This implies that
lim sup n z t f ( z t ) , j ( z t x n ) t 2 z t x n 2 .
Since E is 2-uniformly smooth, j : E E is uniformly continuous on any bounded sets of E, which ensures that the lim sup n and lim sup t 0 are interchangeable, hence
lim sup n f ( x ) x , j ( x n x ) 0 .

This shows that (3.6) holds.

Finally, we show that x n x as n . Note that
x n + 1 x 2 = α n f ( x n ) x , j ( x n + 1 x ) + β n t n x , j ( x n + 1 x ) + γ n Q C e n x , j ( x n + 1 x ) α n f ( x n ) x , j ( x n + 1 q ) + β n t n x x n + 1 x + γ n Q C e n x x n + 1 x α n κ x n x x n + 1 q + α n f ( x ) x , j ( x n + 1 q ) + β n x n x x n + 1 x + γ n Q C e n x x n + 1 x α n κ + β n 2 ( x n x 2 + x n + 1 x 2 ) + α n f ( x ) x , j ( x n + 1 x ) + γ n 2 ( e n x 2 + x n + 1 x 2 ) α n κ + β n 2 x n x 2 + 1 α n ( 1 κ ) 2 x n + 1 x 2 + α n f ( x ) x , j ( x n + 1 x ) + γ n 2 e n x 2 .
It follows that
x n + 1 x 2 ( 1 α n ( 1 κ ) ) x n x 2 + 2 α n f ( x ) x , j ( x n + 1 x ) + γ n e n x 2 .
Using Lemma 2.3, we find from the restrictions (a) and (b) that
lim n x n x = 0 .

This completes the proof. □

Remark 3.2 The framework of the space in Theorem 3.1 can be applicable to L p , p 2 .

4 Applications

In this section, we always assume that E is a uniformly convex and 2-uniformly smooth Banach space. Let C be a nonempty, closed and convex subset of E.

First, we consider common fixed points of two strict pseudocontractions.

Theorem 4.1 Let E be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty closed convex subset of E. Let Q C be a sunny nonexpansive retraction from E onto C, and let T : C C be an α-strict pseudocontraction. Let S : C C be a λ-strict pseudocontraction. Assume that F : = F ( S ) F ( T ) . Let { α n } , { β n } and { γ n } be real number sequences in ( 0 , 1 ) . Suppose that x 1 = x C and that { x n } is given by
x n + 1 = α n f ( x n ) + ( 1 α n ) [ μ S t x n + ( 1 μ ) ( ( 1 α ) x n + α T x n ) ] , n 1 ,
where S t = ( 1 t ) x + t S x , t ( 0 , λ K 2 ] , f : C C is a κ-contractive mapping, { e n } is a bounded computational error in E, λ ( 0 , α / K 2 ] and μ ( 0 , 1 ) . Assume that the following restrictions are satisfied:
  1. (a)

    lim n α n = 0 , n = 1 α n = and n = 1 | α n + 1 α n | < ;

     
  2. (b)

    n = 1 γ n < .

     

Then { x n } converges strongly to x = Q F f ( x ) , where Q F is a sunny nonexpansive retraction from C onto F.

Proof Since ( I T ) is an α-inverse strongly accretive mapping, we find from Theorem 3.1 the desired conclusion. □

Closely related to the class of pseudocontractive mappings is the class of accretive mappings. Recall that an operator B with domain D ( B ) and range R ( B ) in E is accretive if for each x i D ( B ) and y i B x i ( i = 1 , 2 ),
y 2 y 1 , J ( x 2 x 1 ) 0 .
An accretive operator B is m-accretive if R ( I + r B ) = E for each r > 0 . Next, we assume that B is m-accretive and has a zero (i.e., the inclusion 0 B ( z ) is solvable). The set of zeros of B is denoted by Ω. Hence,
Ω = { z D ( B ) : 0 B ( z ) } = B 1 ( 0 ) .

For each r > 0 , we denote by J r the resolvent of B, i.e., J r = ( I + r B ) 1 . Note that if B is m-accretive, then J r : E E is nonexpansive and F ( J r ) = Ω for all r > 0 .

From the above, we have the following theorem.

Theorem 4.2 Let E be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant K, and let C be a nonempty, closed and convex subset of E. Let Q C be a sunny nonexpansive retraction from E onto C, and let A : C E be an α-inverse strongly accretive mapping. Let B : C C be an m-accretive operator. Assume that F : = A 1 ( 0 ) B 1 ( 0 ) . Let { α n } , { β n } and { γ n } be real number sequences in ( 0 , 1 ) . Suppose that x 1 = x C and that { x n } is given by
x n + 1 = α n f ( x n ) + β n [ μ J r x n + ( 1 μ ) ( x n λ A x n ) ] + γ n Q C e n , n 1 ,
where J r = ( I + r B ) 1 , f : C C is a κ-contractive mapping, { e n } is a bounded computational error in E, λ ( 0 , α / K 2 ] and μ ( 0 , 1 ) . Assume that the following restrictions are satisfied:
  1. (a)

    lim n α n = 0 , n = 1 α n = and n = 1 | α n + 1 α n | < ;

     
  2. (b)

    n = 1 γ n < .

     

Then { x n } converges strongly to x = Q F f ( x ) , where Q F is a sunny nonexpansive retraction from C onto F.

Declarations

Acknowledgements

The authors are grateful to the reviewers for useful suggestions which improved the contents of this paper.

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University
(2)
School of Mathematics and Sciences, Shijiazhuang University of Economics

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© Yu and Song; licensee Springer. 2014

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