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Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings

Abstract

We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.

MSC:65K10, 65K15, 90C25, 90C33.

1 Introduction

In this paper, we always assume that is a real Hilbert space with the inner product , and the induced norm . Let C be a nonempty closed convex subset of and the bifunction f:C×CR. Then f is called strongly monotone on C with β>0 iff

f(x,y)+f(y,x)β x y 2 x,yC;

monotone on C iff

f(x,y)+f(y,x)0x,yC;

pseudomonotone on C iff

f(x,y)0impliesf(y,x)0x,yC;

Lipschitz-type continuous on C in the sense of Mastroeni [1] iff there exist positive constants c 1 >0, c 2 >0 such that

f(x,y)+f(y,z)f(x,z) c 1 x y 2 c 2 y z 2 x,y,zC.

An equilibrium problem, shortly EP(f,C), is to find a point in

Sol(f,C)= { x C : f ( x , y ) 0 y C } .

Let a mapping T of C into itself. Then T is called contractive with constant δ(0,1) iff

T ( x ) T ( y ) δxyx,yC.

The mapping T is called strictly pseudocontractive iff there exists a constant k[0,1) such that

T ( x ) T ( y ) 2 x y 2 +k ( I T ) ( x ) ( I T ) ( y ) 2 .

In the case k=0, the mapping T is called nonexpansive on C. We denote by Fix(T) the set of fixed points of T.

Let T i :CC, iΓ, be a family of nonexpansive mappings where Γ stands for an index set. In this paper, we are interested in the problem of finding a common element of the solution set of problem EP(f,C) and the set of fixed points F= i Γ Fix( T i ), namely:

Find  x FSol(f,C),
(1.1)

where the function f and the mappings T i , iΓ, satisfy the following conditions:

(A1) f(x,x)=0 for all xC and f is pseudomonotone on C,

(A2) f is Lipschitz-type continuous on C with constants c 1 >0 and c 2 >0,

(A3) f is upper semicontinuous on C,

(A4) For each xC, f(x,) is convex and subdifferentiable on C,

(A5) FSol(f,C).

Under these assumptions, for each r>0 and xC, there exists a unique element zC such that

f(z,y)+ 1 r yz,zx0yC.
(1.2)

Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, equilibrium equilibriums, fixed point problems (see, e.g., [27]). Recently, it has become an attractive field for many researchers in both theory and its solution methods (see, e.g., [3, 4, 812] and the references therein). Most of these algorithms are based on inequality (1.2) for solving the underlying equilibrium problem when FSol(f,C). Motivated by this idea for finding a common point of Sol(f,C) and the fixed point set Fix(T) of a nonexpansive mapping T, Takahashi and Takahashi [13] first introduced an iterative scheme by the viscosity approximation method. The sequence { x n } is defined by

{ x 0 C , f ( u n , y ) + 1 r n y u n , u n x n 0 y C , x n + 1 = α n g ( x n ) + ( 1 α n ) T ( u n ) n 0 ,

where g:CC is contractive. Under certain conditions over the parameters { α n } and { r n }, they showed that the sequences { x n } and { u n } strongly converge to z= Pr Fix ( T ) Sol ( f , C ) g(z), where Pr C denotes the projection on C. At each iteration n in all of these algorithms, it requires to solve approximation auxiliary equilibrium problems for finding a common solution of an equilibrium problem and a fixed point problem. In order to avoid this requirement, Anh [14] recently proposed a hybrid extragradient algorithm for finding a common point of the set Fix(T)Sol(f,C). Starting with an arbitrary initial point x 0 C, iteration sequences are defined by

{ y k = argmin { λ k f ( x k , y ) + 1 2 y x k 2 : y C } , t k = argmin { λ k f ( y k , t ) + 1 2 t x k 2 : t C } , x k + 1 = α k x 0 + ( 1 α k ) T ( x k ) .
(1.3)

Under certain conditions onto parameters { λ k } and { α k }, he showed that the sequences { x k }, { y k } and { t k } weakly converge to the point xFix(T)Sol(f,C) in a real Hilbert space. At each main iteration n of the scheme, he only solved strongly convex problems on C, but the proof of convergence was still done under the assumptions that x n + 1 x n 0.

For finding a common point of a family of nonexpansive mappings T i (iΓ), as a corollary of Theorem 2.1 in [15], Zhou proposed the following iteration scheme:

{ x 0 H  chosen arbitrarily, C 1 , i = C , C 1 = i Γ C 1 , i , x 1 = Pr C 1 ( x 0 ) , y n , i = ( 1 α n , i ) x n + α n , i T i ( x n ) , C n + 1 , i = { z C n , i : α n , i ( 1 2 α n , i ) x n T i ( x n ) 2 x n z , y n , i T i ( y n , i ) } , C n + 1 = i Γ C n + 1 , i , x n + 1 = Pr C n + 1 ( x 0 ) .
(1.4)

Under the restrictions of the control sequences 0< lim inf n α n , i lim sup n α n , i a i < 1 2 , he showed that the sequence { x n } defined by (1.4) strongly converges to x = Pr F ( x 0 ) in a real Hilbert space , where F= i Γ Fix( T i ).

In this paper, motivated by Ceng et al. [16, 17], Wang and Guo [18], Zhou [15], Nadezhkina and Takahashi [10], Cho et al. [19], Takahashi and Takahashi [13], Anh [6, 12] and Anh et al. [20, 21], we introduce several modified hybrid extragradient schemes to modify the iteration schemes (1.3) and (1.4) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem EP(f,C) in the framework of a real Hilbert space .

To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.

Lemma 1.1 ([14], Lemma 3.1)

Let C be a nonempty closed convex subset of a real Hilbert space . Let f:C×CR be a pseudomonotone and Lipschitz-type continuous bifunction. For each xC, let f(x,) be convex and subdifferentiable on C. Suppose that the sequences { x n }, { y n }, { t n } are generated by scheme (1.3) and x Sol(f,C). Then

t n x 2 x n x 2 (12 λ n c 1 ) x n y n 2 (12 λ n c 2 ) y n t n 2 n0.

Lemma 1.2 Let C be a closed convex subset of a real Hilbert space , and let Pr C be the metric projection from on to C (i.e., for xH, Pr C is the only point in C such that x Pr C x=inf{xz:zC}). Given xH and zC. Then z= Pr C x if only if there holds the relation xz,yz0 for all yC.

Lemma 1.3 Let be a real Hilbert space. Then the following equations hold:

  1. (i)

    x y 2 = x 2 y 2 2xy,y for all x,yH.

  2. (ii)

    t x + ( 1 t ) y 2 =t x 2 +(1t) y 2 t(1t) x y 2 for all t[0,1] and x,yH.

2 Convergence theorems

Now, we prove the main convergence theorem.

Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space . Suppose that assumptions (A1)-(A5) are satisfied and { T i } i Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:

{ x 0 H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = i Γ C 1 , i , D 1 = i Γ D 1 , i , x 1 = Pr C 1 D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 y x n 2 : y C } , z n = argmin { λ n f ( y n , y ) + 1 2 z x n 2 : z C } , y n , i = ( 1 α n , i ) z n + α n , i T i z n , C n + 1 , i = { z C n , i : α n , i ( 1 2 α n , i ) z n T i z n 2 z n z , y n , i T i y n , i } , C n + 1 = i Γ C n + 1 , i , D n + 1 , i = { z D n , i : y n , i z x n z } , D n + 1 = i Γ D n + 1 , i , x n + 1 = Pr C n + 1 D n + 1 x 0 , 0 < lim inf α n , i lim sup α n , i < 1 , { λ n } [ a , b ]  for some  a , b ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n }, { y n } and { z n } strongly converge to the same point Pr F Sol ( f , C ) x 0 .

Proof The proof of this theorem is divided into several steps.

Step 1. Claim that C n and D n are closed and convex for all n0.

We have to show that for any fixed point but arbitrary iΓ, C n , i is closed and convex for every n0. This can be proved by induction on n. It is obvious that C 1 , i =C is closed and convex. Assume that C n , i is closed and convex for some n N ={1,2,}. We have that the set

A= { z C : α n , i ( 1 2 α n , i ) z n T i z n 2 z n z , y n , i T i y n , i }

is closed and convex, and C n + 1 , i = C n , i A, hence C n + 1 , i is closed and convex. Then C n is closed and convex for all n0. We can write D n + 1 , i under the form

D n + 1 , i = { z D n , i : y n , i x n 2 + 2 y n , i x n , x n z 0 } .

Then D n + 1 , i is closed and convex. Thus, D n is closed and convex.

Step 2. Claim that FSol(f,C) C n D n for all n N .

First, we show that F C n by induction on n. It suffices to show that F C n , i .

We have FC= C 1 , i is obvious. Suppose F C n , i for some nN. We have to show that F C n + 1 , i . Indeed, let wF, by inductive hypothesis, we have w C n , i and

z n T i z n 2 = z n T i z n , z n T i z n = 1 α n , i z n y n , i , z n T i z n = 1 α n , i z n y n , i , z n T i z n ( y n , i T i y n , i ) + 1 α n , i z n y n , i , y n , i T i y n , i = 1 α n , i z n y n , i , z n T i z n ( y n , i T i y n , i ) + 1 α n , i z n w + w y n , i , y n , i T i y n , i = 1 α n , i z n y n , i , z n y n , i + 1 α n , i z n y n , i , T i y n , i T i z n + 1 α n , i z n w , y n , i T i y n , i + 1 α n , i w y n , i , y n , i T i y n , i 2 α n , i z n y n , i 2 + 1 α n , i z n w , y n , i T i y n , i + 1 α n , i w y n , i , y n , i T i y n , i .
(2.1)

On the other hand, for all wF and y n , i C, we have

w y n , i 2 T i w T i y n , i , w y n , i = w T i y n , i , w y n , i = w y n , i + y n , i T i y n , i , w y n , i = w y n , i 2 + y n , i T i y n , i , w y n , i ,

and hence

w y n , i , y n , i T i y n , i 0.

Combining this with (2.1), we obtain

z n T i z n 2 2 α n , i z n y n , i 2 + 1 α n , i z n w , y n , i T i y n , i 2 α n , i z n T i z n 2 + 1 α n , i z n w , y n , i T i y n , i .

This follows that

α n , i (12 α n , i ) z n T i z n 2 z n w , y n , i T i y n , i .

By the definition of C n + 1 , i , we have w C n + 1 , i , and so F C n + 1 , i for all iΓ, which deduces that F C n . This shows that FSol(f,C) C n for all n N .

Next, we will prove FSol(f,C) D n by induction on n N . It suffices to show that FSol(f,C) D n , i . Indeed, FC= D 1 , i so FSol(f,C) D 1 , i . Suppose that FSol(f,C) D n , i . Let x FSol(f,C), then x D n , i . Using Lemma 1.1, we get

y n , i x 2 = ( 1 α n , i ) z n + α n , i T i z n x 2 ( 1 α n , i ) z n x 2 + α n , i T i z n T i x 2 z n x 2 x n x 2 ( 1 2 λ n c 1 ) x n y n 2 ( 1 2 λ n c 2 ) y n z n 2 x n x 2 .
(2.2)

Then we have x D n + 1 , i and hence FSol(f,C) D n + 1 , i . This shows that FSol(f,C) D n , which yields that FSol(f,C) C n D n for all n N .

Step 3. Claim that the sequence { x n } is bounded and there exists the limit lim n x n x 0 =c.

From x n = Pr C n D n x 0 , it follows that

x 0 x n , x n y 0y C n D n .
(2.3)

Then, using Step 2, we have FSol(f,C) C n D n and

x 0 x n , x n w 0wFSol(f,C).
(2.4)

Combining this and assumption (A5), the projection Pr F Sol ( f , C ) x 0 is well defined and there exits a unique point p such that p= Pr F Sol ( f , C ) x 0 . So, we have

0 x 0 x n , x n p = x 0 x n , x n x 0 + x 0 p x 0 x n 2 + x 0 x n x 0 p ,

and hence

x 0 x n x 0 p .

Then the sequence { x n } is bounded. So, the sequences { y n }, { z n }, { y n , i }, { T i y n , i } also are bounded. Since x n + 1 C n + 1 D n + 1 C n D n and (2.3), we have

0 x 0 x n , x n x n + 1 = x 0 x n , x n x 0 + x 0 x n + 1 x 0 x n 2 + x 0 x n x 0 x n + 1 ,

and hence x 0 x n x 0 x n + 1 . This together with the boundedness of { x n } implies that the limit lim n x n x 0 =c exists.

Step 4. We claim that lim n x n =qC.

Since C m D m C n D n , x m = Pr C m D m x 0 C n D n for any positive integer mn and (2.3), we have

x 0 x n , x n x n + m 0.

Then

x n x n + m 2 = x n x 0 + x 0 x n + m 2 = x n x 0 2 + x 0 x n + m 2 2 x 0 x n , x 0 x n + m x 0 x n + m 2 x n x 0 2 2 x 0 x n , x n x n + m x 0 x n + m 2 x n x 0 2 .
(2.5)

Passing the limit in (2.5) as n, we get lim n x n x n + m =0 m N . Hence, { x n } is a Cauchy sequence in a real Hilbert space and so lim n x n =qC.

Step 5. We claim that q= Pr F Sol ( f , C ) x 0 , where q= lim n x n .

First we show that qFSol(f,C). Since x n + 1 = Pr C n + 1 D n + 1 x 0 , we have x n + 1 D n + 1 . Then x n + 1 D n + 1 , i and

y n , i x n + 1 x n x n + 1 ,

which yields that

x n y n , i x n x n + 1 + x n + 1 y n , i 2 x n x n + 1 .

Combining this and lim n x n x m =0 for all m N , we get

lim n x n y n , i =0.
(2.6)

For each x Sol(f,C)F, by (2.2) we have

( 1 2 b c 1 ) x n y n 2 ( 1 2 λ n c 1 ) x n y n 2 x n x 2 y n , i x 2 = ( x n x + y n , i x ) ( x n x y n , i x ) ( x n x + y n , i x ) ( x n y n , i ) .

Using this, the boundedness of sequences { x n }, { y n , i } and (2.6), we obtain

lim n x n y n =0.
(2.7)

By a similar way, we also have lim n z n y n =0. Then it follows from the inequality

x n z n x n y n + y n z n

that

lim n x n z n =0.
(2.8)

On the other hand, we have

y n , i z n y n , i x n + x n z n .

Combining this, (2.6) and (2.8), we obtain lim n y n , i z n =0. By the definition of the sequence { y n , i }, we have

y n , i z n = α n , i T i z n z n ,

and hence

lim n T i z n z n =0,

which yields that

T i x n x n T i x n T i z n + T i z n z n + x n z n 2 x n z n + T i z n z n 0 as  n

and

lim n T i x n x n =0.

It follows from Step 4 that lim n T i x n =q. Hence qF.

Now we show that qSol(f,C). By Step 5, we have y n q as n.

Since y n is the unique solution of the strongly convex problem

min { 1 2 y x n 2 + λ n f ( x n , y ) : y C } ,

we get

0 2 ( λ n f ( x n , y ) + 1 2 y x n 2 ) ( y n ) + N C ( y n ) .

From this it follows that

0= λ n w+ y n x n + w ¯ ,

where w 2 f( x n ,)( y n ) and w ¯ N C ( y n ). By the definition of the normal cone N C , we have

y n x n , y y n λ n w , y n y yC.
(2.9)

On the other hand, since f( x n ,) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists w 2 f( x n ,)( y n ) such that

f ( x n , y ) f ( x n , y n ) w , y y n yC.

Combining this with (2.9), we have

λ n ( f ( x n , y ) f ( x n , y n ) ) y n x n , y n y yC.

Then, using { λ n }[a,b](0, 1 L ), (2.7), x n q, y n q as n and the upper semicontinuity of f, we have

f(q,y)0qC.

This means that qSol(f,C). By taking the limit in (2.4), we have

x 0 q , q w 0wFSol(f,C),

which implies that q= Pr F Sol ( f , C ) x 0 . Thus, the subsequences { x n }, { y n }, { z n } strongly converge to the same point q= Pr F Sol ( f , C ) x 0 . This completes the proof. □

Now, notice that wF

z n T i z n 2 = z n w + w T i z n 2 = z n w 2 + w T i z n 2 + 2 z n w , w T i z n 2 z n w 2 + 2 z n w , w z n + z n T i z n = 2 z n w 2 2 z n w 2 + 2 z n w , z n T i z n = 2 z n w , z n T i z n .

Hence

y n , i w 2 = ( 1 α n , i ) ( z n w ) + α n , i ( T i z n w ) 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n z n + z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 = ( 1 α n , i ) z n w 2 + α n , i T i z n z n 2 + α n , i z n w 2 + 2 α n , i T i z n z n , z n w α n , i ( 1 α n , i ) T i z n z n 2 z n w 2 + 2 α n , i z n w , z n T i z n + 2 α n , i T i z n z n , z n w α n , i ( 1 α n , i ) T i z n z n 2 = z n w 2 α n , i ( 1 α n , i ) T i z n z n 2 .
(2.10)

From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.

Theorem 2.2 Let C be a nonempty closed convex subset of a real Hilbert space . Suppose that assumptions (A1)-(A5) are satisfied and { T i } i Γ is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let { x n } be a sequence generated by the following scheme:

{ x 0 H  chosen arbitrarily , C 1 , i = D 1 , i = C , C 1 = i Γ C 1 , i , D 1 = i Γ D 1 , i , x 1 = Pr C 1 D 1 x 0 , y n = argmin { λ n f ( x n , y ) + 1 2 y x n 2 : y C } , z n = argmin { λ n f ( y n , y ) + 1 2 z x n 2 : z C } , y n , i = ( 1 α n , i ) z n + α n , i T i z n , C n + 1 , i = { z C n , i : y n , i z 2 z n z 2 α n , i ( 1 α n , i ) z n T i z n 2 } , C n + 1 = i Γ C n + 1 , i , D n + 1 , i = { z D n , i : y n , i z x n z } , D n + 1 = i Γ D n + 1 , i , x n + 1 = Pr C n + 1 D n + 1 x 0 , 0 < lim inf α n , i lim sup α n , i < 1 , { λ n } [ a , b ]  for some  a , b ( 0 , 1 L ) ,  where  L = max { 2 c 1 , 2 c 2 } .

Then the sequences { x n }, { y n } and { z n } converge strongly to the same point Pr F Sol ( f , C ) x 0 .

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Thanh, D.D. Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings. Fixed Point Theory Appl 2014, 200 (2014). https://doi.org/10.1186/1687-1812-2014-200

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Keywords

  • equilibrium problems
  • fixed point
  • pseudomonotone
  • Lipschitz-type continuous
  • extragradient method
  • nonexpansive mappings