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Uniformly closed replaced AKTT or AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

Abstract

The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results.

MSC:47H05, 47H09, 47H10.

1 Introduction and preliminaries

Let E be a real Banach space and C be a nonempty closed convex subset of E. A mapping T:CC is called nonexpansive if

TxTyxy,x,yC.

Let E be a real Banach space and C be a nonempty closed convex subset of E. A point pC is said to be an asymptotic fixed point of T if there exists a sequence { x n } n = 0 C such that x n p and lim n x n T x n =0. The set of asymptotic fixed point is denoted by F ˆ (T). We say that a mapping T is relatively nonexpansive (see [14]) if the following conditions are satisfied:

  1. (I)

    F(T);

  2. (II)

    ϕ(p,Tx)ϕ(p,x), xC, pF(T);

  3. (III)

    F(T)= F ˆ (T).

If T satisfies (I) and (II), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings.

Let E be a real Banach space. The modulus of smoothness of E is the function ρ E :[0,)[0,) defined by

ρ E (τ)=sup { 1 2 ( x + y + x y ) 1 : x 1 , y τ } .

E is uniformly smooth if and only if

lim τ 0 ρ E τ τ =0.

Let dimE2. The modulus of convexity of E is the function δ E (ϵ):=inf{1 x + y 2 :x=y=1;ϵ=xy}. E is uniformly convex if for any ϵ(0,2], there exists δ=δ(ϵ)>0 such that if x,yE with x1, y1 and xyϵ, then 1 2 (x+y)1δ. Equivalently, E is uniformly convex if and only if δ E (ϵ)>0 for all ϵ(0,2]. A normed space E is called strictly convex if for all x,yE, xy, x=y=1, we have λx+(1λ)y<1, λ(0,1).

Let E be the dual space of E. We denote by J the normalized duality mapping from E to 2 E defined by

J(x)= { f E : x , f = x 2 = f 2 } .

The following properties of J are well known (see [57] for more details):

  1. (1)

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

  2. (2)

    If E is reflexive, then J is a mapping from E onto E .

  3. (3)

    If E is smooth, then J is single valued.

Throughout this paper, we denote by ϕ the functional on E×E defined by

ϕ(x,y)= x 2 2 x , J ( y ) + y 2 ,x,yE.
(1.1)

Let E be a smooth, strictly convex, and reflexive real Banach space and let C be a nonempty closed convex subset of E. Following Alber [8], the generalized projection Π C from E onto C is defined by

Π C (x)= arg min y C ϕ(y,x),xE.

The existence and uniqueness of Π C follows from the property of the functional ϕ(x,y) and strict monotonicity of the mapping J. It is obvious that

( x y ) 2 ϕ(x,y) ( x + y ) 2 ,x,yE.
(1.2)

Next, we recall the notion of generalized f-projection operator and its properties. Let G:C× E R{+} be a functional defined as follows:

G(ξ,φ)= ξ 2 2ξ,φ+ φ 2 +2ρf(ξ),
(1.3)

where ξC, φ E , ρ is a positive number and f:CR{+} is proper, convex, and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties:

  1. (i)

    G(ξ,φ) is convex and continuous with respect to φ when ξ is fixed;

  2. (ii)

    G(ξ,φ) is convex and lower semi-continuous with respect to ξ when φ is fixed.

Definition 1.1 [9]

Let E be a real Banach space with its dual E . Let C be a nonempty, closed, and convex subset of E. We say that Π C f : E 2 C is a generalized f-projection operator if

Π C f φ= { u C : G ( u , φ ) = inf ξ C G ( ξ , φ ) } ,φ E .

For the generalized f-projection operator, Wu and Huang [9] proved in the following theorem some basic properties.

Lemma 1.2 [9]

Let E be a real reflexive Banach space with its dual E . Let C be a nonempty, closed, and convex subset of E. Then the following statements hold:

  1. (i)

    Π C f is a nonempty closed convex subset of C for all φ E .

  2. (ii)

    If E is smooth, then for all φ E , x Π C f φ if and only if

    xy,φJx+ρf(y)ρf(x)0,yC.
  3. (iii)

    If E is strictly convex and f:CR{+} is positive homogeneous (i.e., f(tx)=tf(x) for all t>0 such that txC where xC), then Π C f is a single-valued mapping.

Fan et al. [10] showed that the condition f is positive homogeneous which appeared in Lemma 1.2 can be removed.

Lemma 1.3 [10]

Let E be a real reflexive Banach space with its dual E and C a nonempty, closed, and convex subset of E. Then if E is strictly convex, then Π C f is a single-valued mapping.

Recall that J is a single-valued mapping when E is a smooth Banach space. There exists a unique element φ E such that φ=Jx for each xE. This substitution in (1.3) gives

G(ξ,Jx)= ξ 2 2ξ,Jx+ x 2 +2ρf(ξ).
(1.4)

Now, we consider the second generalized f-projection operator in a Banach space.

Definition 1.4 [11]

Let E be a real Banach space and C a nonempty, closed, and convex subset of E. We say that Π C f :E 2 C is a generalized f-projection operator if

Π C f x= { u C : G ( u , J x ) = inf ξ C G ( ξ , J x ) } ,xE.

Obviously, the definition of relatively quasi-nonexpansive mapping T is equivalent to

  1. (1)

    F(T);

  2. (2)

    G(p,JTx)G(p,Jx), xC, pF(T).

Lemma 1.5 [12]

Let E be a Banach space and f:ER{+} be a lower semi-continuous convex functional. Then there exist x E and αR such that

f(x) x , x +α,xE.

We know that the following lemmas hold for operator Π C f .

Lemma 1.6 [13]

Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:

  1. (i)

    Π C f is a nonempty, closed, and convex subset of C for all xE;

  2. (ii)

    for all xE, x ˆ Π C f x if and only if

    x ˆ y,JxJ x ˆ +ρf(y)ρf(x)0,yC;
  3. (iii)

    if E is strictly convex, then Π C f x is a single-valued mapping.

Lemma 1.7 [13]

Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let xE and x ˆ Π C f x. Then

ϕ(y, x ˆ )+G( x ˆ ,Jx)G(y,Jx),yC.

The fixed points set F(T) of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma.

Lemma 1.8 [14, 15]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasi-nonexpansive mapping of C into itself. Then F(T) is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 1.9 [16]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let { x n } n = 0 and { y n } n = 0 be sequences in E such that either { x n } n = 0 or { y n } n = 0 is bounded. If lim n ϕ( x n , y n )=0, then lim n x n y n =0.

Lemma 1.10 [17]

Let p>1 and r>0 be two fixed real numbers. Then a Banach space X is uniformly convex if and only if there is a continuous, strictly increasing and convex function g: R + R + , g(0)=0, such that

λ x + ( 1 λ ) y p λ x p +(1λ) y p W p (λ)g ( x y )

for all x,y B r and 0λ1, where W p (λ)=λ ( 1 λ ) p + λ p (1λ).

Remark We can see from the Lemma 1.10 that the function g has no relation with the selection of x, y and λ. However, the key point above, in the process of generalization and application about this lemma, has been ambiguous gradually. For instance, the following lemma states that the function g has something to do with λ, which always leads to failure in the proof.

Lemma (stated in [[11], Lemma 2.10])

Let E be a uniformly convex real Banach space. For arbitrary r>0, let B r (0):={xE:xr} and λ[0,1]. Then there exists a continuous strictly increasing convex function

g:[0,2r]R,g(0)=0

such that for every x,y B r (0), the following inequality holds:

λ x + ( 1 λ ) y 2 λ x 2 +(1λ) y 2 λ(1λ)g ( x y ) .

Let F be a bifunction of C×C into R. The equilibrium problem is to find x C such that F( x ,y)0, for all yC. We shall denote the solutions set of the equilibrium problem by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases.

For solving the equilibrium problem for a bifunction F:C×CR, let us assume that F satisfies the following conditions:

(A1) F(x,x)=0 for all xC;

(A2) F is monotone, i.e., F(x,y)+F(y,x)0 for all x,yC;

(A3) for each x,yC, lim t 0 F(tz+(1t)x,y)F(x,y);

(A4) for each xC, yF(x,y) is convex and lower semi-continuous.

Lemma 1.11 [18]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let F be a bifunction of C×C into R satisfying (A1)-(A4). Let r>0 and xE. Then there exists zC such that

F(z,y)+ 1 r yz,JzJx0,yK.

Lemma 1.12 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F:C×CR satisfies (A1)-(A4). For r>0 and xE, define a mapping T r F :EC as follows:

T r F (x)= { z C : F ( z , y ) + 1 r y z , J z J x 0 , y C }

for all zE. Then the following hold:

  1. (1)

    T r F is single valued;

  2. (2)

    T r F is a firmly nonexpansive-type mapping, i.e., for any x,yE,

    T r F x T r F y , J T r F x J T r F y T r F x T r F y , J x J y ;
  3. (3)

    F( T r F )=EP(F);

  4. (4)

    EP(F) is closed and convex.

Lemma 1.13 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that F:C×CR satisfies (A1)-(A4) and let r>0. Then for each xE and qF( T r F ),

ϕ ( q , T r F x ) +ϕ ( T r F x , x ) ϕ(q,x).

Let { T n } be a sequence of mappings from C into E, where C is a nonempty closed convex subset of a real Banach space E. For a subset B of C, we say that

  1. (i)

    ({ T n },B) satisfies condition AKTT (see [15]) if

    n = 1 sup { T n + 1 x T n x : x B } <;
  2. (ii)

    ({ T n },B) satisfies condition AKTT (see [15]) if

    n = 1 sup { J T n + 1 x J T n x : x B } <.

Recently, Shehu [11] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The author obtained the following theorem.

Theorem 1.14 [11]

Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex subset of E. For each k=1,2,,m, let F k be a bifunction from C×C satisfying (A1)-(A4) and let { T n } n = 1 be an infinite family of relatively quasi-nonexpansive mappings of C into itself such that F:=( n = 1 F( T n ))( k = 1 m EP( F k )). Let f:ER be a convex and lower semi-continuous mapping with Cint(D(f)) and suppose { x n } n = 0 is iteratively generated by x 0 C, C 1 =C, x 1 = Π C 1 f x 0 ,

{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n F m T r m 1 , n F m 1 T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w C n : G ( w , J u n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,
(1.5)

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in (0,1) such that lim inf n α n (1 α n )>0 { r k , n } n = 1 (0,) (k=1,2,,m) satisfying lim inf n r k , n >0 (k=1,2,,m). Suppose that for each bounded subset B of C, the ordered pair ({ T n },B) satisfies either condition AKTT or condition AKTT. Let T be the mapping from C into E defined by Tx= lim n T n x for all xC and suppose that T is closed and F(T)= n = 1 F( T n ). Then { x n } n = 0 converges strongly to Π F f x 0 .

In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

2 Main results

Now, we shall first introduce the notion of uniformly closed mappings and give an example which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

Definition 2.1 Let E be a Banach space, C be a nonempty closed convex subset of E. Let { T n } n = 1 :CE be a sequence of mappings of C into E such that n = 1 F( T n ) is nonempty. { T n } n = 1 is said to be uniformly closed, if p n = 1 F( T n ), whenever { x n }C converges strongly to p and x n T n x n 0 as n.

Example 1 Let E= l 2 , where

l 2 = { ξ = ( ξ 1 , ξ 2 , ξ 3 , , ξ n , ) : n = 1 | ξ n | 2 < } , ξ = ( n = 1 | ξ n | 2 ) 1 2 , ξ l 2 , ξ , η = n = 1 ξ n η n , ξ = ( ξ 1 , ξ 2 , ξ 3 , , ξ n , ) , η = ( η 1 , η 2 , η 3 , , η n , ) l 2 .

It is well known that l 2 is a Hilbert space, so that ( l 2 ) = l 2 . Let { x n }E be a sequence defined by

x 0 = ( 1 , 0 , 0 , 0 , ) , x 1 = ( 1 , 1 , 0 , 0 , ) , x 2 = ( 1 , 0 , 1 , 0 , 0 , ) , x 3 = ( 1 , 0 , 0 , 1 , 0 , 0 , ) , x n = ( ξ n , 1 , ξ n , 2 , ξ n , 3 , , ξ n , k , ) ,

where

ξ n , k ={ 1 , if  k = 1 , n + 1 , 0 , if  k 1 , k n + 1 ,

for all n1.

Define a countable family of mappings T n :EE as follows:

T n (x)={ n n + 1 x n , if  x = x n , x , if  x x n ,

for all n0.

Conclusion 2.2 { T n } n = 0 has a unique fixed point 0, that is, F( T n )={0}, n0.

Proof The conclusion is obvious. □

Let { T n } n = 1 be a countable family of quasi-relatively quasi-nonexpansive mappings, if

n = 0 F( T n )= F ˆ ( { T n } n = 0 ) ,

the { T n } n = 1 is said to be a countable family of relatively nonexpansive mappings in the sense of functional G, where

F ˆ ( { T n } n = 0 ) = { p C : x n p , x n T n x n 0 , x n C }

is said to be the asymptotic fixed point set of { T n } n = 1 .

Conclusion 2.3 { T n } n = 0 is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G.

Proof By Conclusion 2.2, we only need to show that G(0,J T n x)G(0,Jx), xE. Note that E= l 2 is a Hilbert space, for any n0 we can derive

G ( 0 , J T n x ) G ( 0 , J x ) x E ϕ ( 0 , T n x ) ϕ ( 0 , x ) 0 T n x 2 0 x 2 T n x 2 x 2 .

It is obvious that { x n } converges weakly to x 0 =(1,0,0,), and

x n T n x n = n n + 1 x n x n = 1 n + 1 x n 0,

as n, so x 0 is an asymptotic fixed point of { T n } n = 0 . Joining with Conclusion 2.2, we can obtain n = 0 F( T n ) F ˆ ( { T n } n = 0 ).

Thus, { T n } n = 0 is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. □

Conclusion 2.4 { T n } n = 0 is a countable family of uniformly closed relatively quasi-nonexpansive mappings in the sense of functional G.

Proof In fact, for any strong convergent sequence { z n }E such that z n z 0 and z n T n z n 0 as n, there exists a sufficiently large nature number N, such that z n x m for any n,m>N (since x n is not a Cauchy sequence it cannot converge to any element in E). Then T n z n = z n for n>N, it follows from z n T n z n 0 that 2 z n 0 and hence z n z 0 =0.

Therefore, { T n } n = 0 is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G. □

Now, we give an example which is a countable family of uniformly closed quasi-nonexpansive mappings but not satisfied condition AKTT and AKTT.

Example 2 Let X= 2 . For any complex number x=r e i θ X, define a countable family of quasi-nonexpansive mappings as follows:

T n :r e i θ r e i ( θ + n π 2 ) ,n=1,2,3,.

Proof It is easy to see that n = 1 F( T n )={0}. We first prove that { T n } is uniformly closed. In fact, for any strong convergent sequence { x n }X such that x n x 0 and x n T n x n 0 as n, there must be x 0 =0 n = 1 F( T n ). Otherwise, if x n x 0 0, and

x 4 n + 1 T 4 n + 1 x 4 n + 1 0,

since T 1 is continuous, we have

x 4 n + 1 T 4 n + 1 x 4 n + 1 = x 4 n + 1 T 1 x 4 n + 1 x 0 T 1 x 0 0 .

This is a contradiction. Therefore, { T n } is uniformly closed.

Besides, take any x=r e i θ 0. For any n by the definition of T n , we have

T n x T n + 1 x= r e π i 2 =r>0

and

J T n xJ T n + 1 x= r e π i 2 =r>0.

That is to say, { T n } does not satisfied condition AKTT and AKTT. □

Now we are in a position to present our main theorems.

Theorem 2.5 Let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself and other conditions are the same as Theorem 1.14 except for condition AKTT, AKTT and condition ‘Let T be the mapping from C into E defined by Tx= lim n T n x for all xC and suppose that T is closed and F(T)= n = 1 F( T n ) . Then the sequence { x n } n = 0 generated by (1.5) converges strongly to Π F f x 0 .

Proof We first show that C n , n1, is closed and convex. It is obvious that C 1 =C is closed and convex. Suppose that C n is closed convex for some n>1. From the definition of C n + 1 , we have z C n + 1 implies G(z,J u n )G(z,J x n ). This is equivalent to

2 ( z , J x n z , J u n ) x n 2 u n 2 .

This implies that C n + 1 is closed convex for the same n>1. Hence, C n is closed and convex for all n1. This shows that Π C n + 1 f x 0 is well defined for all n0.

By taking θ n k = T r k , n F k T r k 1 , n F k 1 T r 2 , n F 2 T r 1 , n F 1 , k=1,2,,m and θ n 0 =I for all n1, we obtain u n = θ n m y n .

We next show that F C n , n1. From Lemma 1.12, one sees that T r k , n F k , k=1,2,,m, is relatively nonexpansive mapping. For n=1, we have FC= C 1 . Now, assume that F C n for some n2. Then for each x F, we obtain

G ( x , J u n ) = G ( x , J θ n m y n ) G ( x , J y n ) = G ( x , ( α n J x n + ( 1 α n ) J T n x n ) ) = x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n 2 + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) = α n G ( x , J x n ) + ( 1 α n ) G ( x , J T n x n ) G ( x , J x n ) .
(2.1)

So, x C n . This implies that F C n , n1 and the sequence { x n } n = 0 generated by (1.5) is well defined.

We now show that lim n G( x n ,J x 0 ) exists. Since f:ER is a convex and lower semi-continuous, applying Lemma 1.5, we see that there exist u E and αR such that

f(y) y , u +α,yE.

It follows that

G ( x n , J x 0 ) = x n 2 2 x n , J x 0 + x 0 2 + 2 ρ f ( x n ) x n 2 2 x n , J x 0 + x 0 2 + 2 ρ x n , u + 2 ρ α = x n 2 2 x n , J x 0 ρ u + x 0 2 + 2 ρ α x n 2 2 x n J x 0 ρ u + x 0 2 + 2 ρ α = ( x n J x 0 ρ u ) 2 + x 0 2 J x 0 ρ u 2 + 2 ρ α .
(2.2)

Since x n = Π C n f x 0 , it follows from (2.2) that

G ( x , J x 0 ) G( x n ,J x 0 ) ( x n J x 0 ρ u ) 2 + x 0 2 J x 0 ρ u 2 +2ρα

for each x F(T). This implies that { x n } n = 1 is bounded and so is { G ( x n , J x 0 ) } n = 0 . By the construction of C n , we have C m C n and x m = Π C m f x 0 C n for any positive integer mn. It then follows from Lemma 1.7 that

ϕ( x m , x n )+G( x n ,J x 0 )G( x m ,J x 0 ).
(2.3)

It is obvious that

ϕ( x m , x n ) ( x m x n ) 2 0.

In particular,

ϕ( x n + 1 , x n )+G( x n ,J x 0 )G( x n + 1 ,J x 0 )

and

ϕ( x n + 1 , x n ) ( x n + 1 x n ) 2 0,

and so { G ( x n , J x 0 ) } n = 0 is nondecreasing. It follows that the limit of { G ( x n , J x 0 ) } n = 0 exists.

By the fact that C m C n and x m = Π C m f x 0 C n for any positive integer mn, we obtain

ϕ( x m , u n )ϕ( x m , x n ).

Now, (2.3) implies that

ϕ( x m , u n )ϕ( x m , x n )G( x m ,J x 0 )G( x n ,J x 0 ).
(2.4)

Taking the limit as m,n in (2.4), we obtain

lim n ϕ( x m , x n )=0.

It then follows from Lemma 1.9 that x m x n 0 as m,n. Hence, { x n } n = 0 is a Cauchy sequence. Since E is a Banach space and C is closed and convex, there exists pC such that x n p as n.

Now since ϕ( x m , x n )0 as m,n we have in particular that ϕ( x n + 1 , x n )0 as n and this further implies that lim n x n + 1 x n =0. Since x n + 1 = Π C n = 1 f x 0 C n + 1 we have

ϕ( x n + 1 , u n )ϕ( x n + 1 , x n ),n0.

Then we obtain

lim n ϕ( x n + 1 , u n )=0.

Since E is uniformly convex and smooth, we have from Lemma 1.9

lim n x n + 1 x n =0= lim n x n + 1 u n .

So,

x n u n x n + 1 x n + x n + 1 u n .

Hence,

lim n x n u n =0.
(2.5)

Since J is uniformly norm-to-norm continuous on bounded sets and lim n x n u n =0, we obtain

lim n J x n J u n =0.
(2.6)

Let r= sup n 1 { x n , T n x n }. Since E is uniformly smooth, we know that E is uniformly convex. Then from Lemma 1.10, we have

G ( x , J u n ) = G ( x , J θ n m y n ) G ( x , J y n ) = G ( x , ( α n J x n + ( 1 α n ) J T n x n ) ) = x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n + ( 1 α n ) J T n x n 2 + 2 ρ f ( x ) x 2 2 α n x , J x n 2 ( 1 α n ) x , J T n x n + α n J x n 2 + ( 1 α n ) J T n x n 2 α n ( 1 α n ) g ( J x n J T n x n ) + 2 ρ f ( x ) = α n G ( x , J x n ) + ( 1 α n ) G ( x , J T n x n ) α n ( 1 α n ) g ( J x n J T n x n ) G ( x , J x n ) α n ( 1 α n ) g ( J x n J T n x n ) .

It then follows that

α n (1 α n )g ( J x n J T n x n ) G ( x , J x n ) G ( x , J u n ) .

But

G ( x , J x n ) G ( x , J u n ) = x n 2 u n 2 2 x , J x n J u n x n 2 u n 2 + 2 | x , J x n J u n | | x n u n | ( x n + u n ) + 2 x J x n J u n x n u n ( x n + u n ) + 2 x J x n J u n .

From (2.5) and (2.6), we obtain

G ( x , J x n ) G ( x , J u n ) 0,n.

Using the condition lim inf n α n (1 α n )>0, we have

lim n g ( J x n J T n x n ) =0.

By the properties of g, we have lim n J x n J T n x n =0. Since J 1 is also uniformly norm-to-norm continuous on bounded sets, we have

lim n x n T n x n =0.

Since { T n } n = 1 are uniformly closed, and { x n } n = 1 is a Cauchy sequence. Then pF(T)= n = 1 F( T n ).

Next, we show that p k = 1 m EP( F k ). From (2.1), we obtain

ϕ ( x , u n ) = ϕ ( x , θ n m y n ) = ϕ ( x , T r m , n F m θ n m 1 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , x n ) .
(2.7)

Since x EP( F m )=F( T r m , n F m ) for all n1, it follows from (2.7) and Lemma 1.13 that

ϕ ( u n , θ n m 1 y n ) = ϕ ( T r m , n F m θ n m 1 y n , θ n m 1 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , u n ) ϕ ( x , x n ) ϕ ( x , u n ) .

From (2.5) and (2.6), we obtain lim n ϕ( θ n m y n , θ n m 1 y n )= lim n ϕ( u n , θ n m 1 y n )=0. From Lemma 1.9, we have

lim n θ n m y n θ n m 1 y n = lim n u n θ n m 1 y n =0.
(2.8)

Hence, we have from (2.8) that

lim n J θ n m y n J θ n m 1 y n =0.
(2.9)

Again, since x EP( F m 1 )=F( T r m 1 , n F m 1 ) for all n1, it follows from (2.7) and Lemma 1.13 that

ϕ ( θ n m 1 y n , θ n m 2 y n ) = ϕ ( T r m 1 , n F m 1 θ n m 2 y n , θ n m 2 y n ) ϕ ( x , θ n m 2 y n ) ϕ ( x , θ n m 1 y n ) ϕ ( x , x n ) ϕ ( x , u n ) .

Again, from (2.5) and (2.6), we obtain lim n ϕ( θ n m 1 y n , θ n m 2 y n )=0. From Lemma 1.9, we have

lim n θ n m 1 y n θ n m 2 y n =0
(2.10)

and hence,

lim n J θ n m 1 y n J θ n m 2 y n =0.
(2.11)

In a similar way, we can verify that

lim n θ n m 2 y n θ n m 3 y n == lim n θ n 1 y n y n =0.
(2.12)

From (2.8), (2.10), and (2.12), we can conclude that

lim n θ n k y n θ n k 1 y n =0,k=1,2,,m.
(2.13)

Since x n p, n, we obtain from (2.5) that u n p, n. Again, from (2.8), (2.10), (2.12), and u n p, n, we have that θ n k y n p, n for each k=1,2,,m. Also, using (2.13), we obtain

lim n J θ n k y n J θ n k 1 y n =0,k=1,2,,m.

Since lim inf n r k , n >0, k=1,2,,m,

lim n J θ n k y n J θ n k 1 y n r k , n =0.
(2.14)

By Lemma 1.12, we have for each k=1,2,,m

F k ( θ n k y n , y ) + 1 r k , n y θ n k y n , J θ n k y n J θ n k 1 y n 0,yC.

Furthermore, using (A2) we obtain

1 r k , n y θ n k y n , J θ n k y n J θ n k 1 y n F k ( y , θ n k y n ) .
(2.15)

By (A4), (2.14), and θ n k y n p, we have for each k=1,2,,m

F k (y,p)0,yC.

For fixed yC, let z t =ty+(1t)p for all t(0,1]. This implies that z t C. This yields F k ( z t ,p)0. It follows from (A1) and (A4) that

0= F k ( z t , z t )t F k ( z t ,y)+(1t) F k ( z t ,p)t F k ( z t ,y)

and hence

0 F k ( z t ,y).

From condition (A3), we obtain

F k (p,y)0,yC.

This implies that pEP( F k ), k=1,2,,m. Thus, p k = 1 m EP( F k ). Hence, we have pF= k = 1 m EP( F k )( n = 1 F( T n )).

Finally, we show that p= Π F f x 0 . Since F= k = 1 m EP( F k )( n = 1 F( T n )) is a closed and convex set, from Lemma 1.6, we know that Π F f x 0 is single valued and denote w= Π F f x 0 . Since x n = Π c n f x 0 and wF C n , we have

G( x n ,J x 0 )G(w,J x 0 ),n0.

We know that G(ξ,Jφ) is convex and lower semi-continuous with respect to ξ when φ is fixed. This implies that

G(p,J x 0 ) lim inf n G( x n ,J x 0 ) lim sup n G( x n ,J x 0 )G(w,J x 0 ).

From the definition of Π F f x 0 and pF, we see that p=w. This completes the proof. □

Corollary 2.6 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each k=1,2,,m, let F k be a bifunction from C×C satisfying (A1)-(A4) and let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F:=( n = 1 F( T n ))( k = 1 m EP( F k )). Suppose { x n } n = 0 is iteratively generated by x 0 C, C 1 =C, x 1 = Π C 1 f x 0 ,

{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n F m T r m 1 , n F m 1 T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w C n : ϕ ( w , u n ) ϕ ( w , x n ) } , x n + 1 = Π C n + 1 x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in (0,1) such that lim inf n α n (1 α n )>0, and { r k , n } n = 1 (0,) (k=1,2,,m) satisfying lim inf n r k , n >0 (k=1,2,,m). Then { x n } n = 0 converges strongly to Π F x 0 .

Proof Take f(x)=0 for all xE in Theorem 2.5, then G(ξ,Jx)=ϕ(ξ,x) and Π C f x 0 = Π C x 0 . Then Corollary 2.6 holds. □

Take F k 0 (k=1,2,,m), it is obvious that the following holds.

Corollary 2.7 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. Let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F=( n = 1 F( T n )). Let f:ER be a convex and lower semi-continuous mapping with Cint(D(f)) and suppose { x n } n = 0 is iteratively generated by x 0 C, C 1 =C, x 1 = Π C 1 f x 0 ,

{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , C n + 1 = { w C n : G ( w , J y n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in (0,1) such that lim inf n α n (1 α n )>0, and { r k , n } n = 1 (0,) (k=1,2,,m) satisfying lim inf n r k , n >0 (k=1,2,,m). Then { x n } n = 0 converges strongly to Π F x 0 .

3 Applications

Let φ:CR be a real-valued function. The convex minimization problem is to find x C such that

φ ( x ) φ(y),
(3.1)

yC. The set of solutions of (3.1) is denoted by CMP(φ). For each r>0 and xE, define the mapping

T r φ (x)= { z C : φ ( y ) + 1 r y z , J z J x φ ( z ) , y C } .

Theorem 3.1 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each k=1,2,,m, let φ k be a bifunction from C×C satisfying (A1)-(A4) and let { T n } n = 1 be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that F:=( n = 1 F( T n ))( k = 1 m CMP( φ k )). Let f:ER be a convex and lower semi-continuous mapping with Cint(D(f)) and suppose { x n } n = 0 is iteratively generated by x 0 C, C 1 =C, x 1 = Π C 1 f x 0 ,

{ y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n = T r m , n φ m T r m 1 , n φ m 1 T r 2 , n φ 2 T r 1 , n φ 1 y n , C n + 1 = { w C n : G ( w , J u n ) G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n 1 ,

where J is the duality mapping on E. Suppose { α n } n = 1 is a sequence in (0,1) such that lim inf n α n (1 α n )>0 and { r k , n } n = 1 (0,) (k=1,2,,m) satisfying lim inf n r k , n >0 (k=1,2,,m). Then { x n } n = 0 converges strongly to Π F f x 0 .

Proof Define F k (x,y)= φ k (y) φ k (x), x,yC and k=1,2,,m. Then F( T r k F k )=EP( F k )=CMP( φ k )=F( T r k φ k ) for each k=1,2,,m, and therefore { F k } k = 1 m satisfies conditions (A1) and (A2). Furthermore, one can easily show that { F k } k = 1 m satisfies (A3) and (A4). Therefore, from Theorem 2.5, we obtain Theorem 3.1. □

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Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

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Correspondence to Yongfu Su.

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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Keywords

  • relatively quasi-nonexpansive mapping
  • equilibrium problems
  • generalized f-projection operator
  • hybrid algorithm
  • uniformly closed mappings