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Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems

Abstract

In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.

MSC:47H09, 47J25.

1 Introduction

Let H be a real Hilbert space with the inner product , and the norm . Let C be a nonempty closed convex subset of H and F:C×CR be a bifunction, where R is the set of real numbers. The equilibrium problem (for short, EP) is to find x C such that

F ( x , y ) 0,yC.
(1.1)

The set of solutions of EP is denoted by EP(F). Given a mapping T:CC, let F(x,y)=Tx,yx for all x,yC. Then x EP(F) if and only if x C is a solution of the variational inequality Tx,yx0 for all yC, i.e., x is a solution of the variational inequality.

Let φ:CR{+} be a function. The mixed equilibrium problem (for short, MEP) is to find x C such that

F ( x , y ) +φ(y)φ ( x ) 0,yC.
(1.2)

The set of solutions of MEP is denoted by MEP(F,φ).

If φ=0, then mixed equilibrium problem (1.2) reduces to (1.1).

If F=0, then mixed equilibrium problem (1.2) reduces to the following convex minimization problem:

Find  x C such that φ(y)φ ( x ) ,yC.
(1.3)

The set of solutions of (1.3) is denoted by CMP(φ).

The mixed equilibrium problem (MEP) includes several important problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of abstract equilibrium problems (e.g., [1, 2]). Many authors have proposed some useful methods to solve the EP, MEP; see, for instance, [18] and the references therein.

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Following Kohsaka and Takahashi [911], a mapping T:CC is said to be nonspreading if

2 T x T y 2 T x y 2 + T y x 2 for all x,yC.

It is easy to see that the above inequality is equivalent to

T x T y 2 x y 2 +2xTx,yTyfor all x,yC.

In 1967, Browder and Petryshyn [12] introduced the concept of k-strictly pseudononspreading mapping.

Definition 1.1 [12]

Let H be a real Hilbert space. A mapping T:D(T)HH is said to be k-strictly pseudononspreading if there exists k[0,1) such that

T x T y 2 x y 2 +k x T x ( y T y ) 2 +2xTx,yTy,x,yD(T).

Clearly, every nonspreading mapping is k-strictly pseudononspreading.

In 2012, Osilike [13] introduced a class of nonspreading type mappings, which is more general than the mappings studied in [14] in Hilbert spaces, and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang [15] studied the multiple-set split feasibility problem for asymptotically strict pseudocontraction in the framework of infinite-dimensional Hilbert spaces.

Definition 1.2 [15]

Let H be a real Hilbert space. A mapping T:D(T)HH is said to be a k-asymptotically strict pseudocontraction if there exist a constant k[0,1) and a sequence { k n }[1,) with k n 1 (n) such that

T n x T n y 2 k n x y 2 +k x T n x ( y T n y ) 2

holds for all x,yD(T).

Definition 1.3 Let C be a nonempty subset of a real Hilbert space H. A mapping T:CC is said to be k-asymptotically strictly pseudononspreading if there exist a constant k[0,1) and a sequence { k n }[1,) with k n 1 (n) such that

T n x T n y 2 k n x y 2 + k x T n x ( y T n y ) 2 + 2 x T n x , y T n y , x , y C .
(1.4)

It is easy to see that the class of k-asymptotically strictly pseudononspreading mappings is more general than the classes of k-strictly pseudononspreading mappings and k-asymptotically strict pseudocontractions.

Example 1.4 Let X= l 2 with the norm defined by

x= i = 1 x i 2 ,x=( x 1 , x 2 ,, x n ,)X,

and C={x=( x 1 , x 2 ,, x n ,)| x i R 1 ,i=1,2,} be an orthogonal subspace of X (i.e., x,yC, we have x,y=0). It is obvious that C is a nonempty closed convex subset of X. For each x=( x 1 , x 2 ,, x n ,)C, we define the mapping T:CC by

Tx= { ( x 1 , x 2 , , x n , ) if  i = 1 x i < 0 ; ( x 1 , x 2 , , x n , ) if  i = 1 x i 0 .
(1.5)

Next we prove that T is a k-asymptotically strictly pseudononspreading mapping.

In fact, for any x,yC.

Case 1. If i = 1 x i <0 and i = 1 y i <0, then we have T n x=x, T n y=y, and so inequality (1.4) holds for any k[0,1).

Case 2. If i = 1 x i <0 and i = 1 y i 0, then we have that T n x=x, T n y= ( 1 ) n y. This implies that

{ T n x T n y 2 = x ( 1 ) n y 2 = x 2 + y 2 ; k n x y 2 = k n ( x 2 + y 2 ) ; x T n x ( y T n y ) 2 = [ 1 ( 1 ) n ] 2 y 2 ; 2 x T n x , y T n y = 0 .

Therefore inequality (1.4) holds for any k[0,1).

Case 3. If i = 1 x i 0 and i = 1 y i <0, then we have that T n x= ( 1 ) n x, T n y=y. Therefore we obtain

{ T n x T n y 2 = ( 1 ) n x y 2 = x 2 + y 2 ; k n x y 2 = k n ( x 2 + y 2 ) ; x T n x ( y T n y ) 2 = [ 1 ( 1 ) n ] 2 x 2 ; 2 x T n x , y T n y = 0 .

So, inequality (1.4) holds for any k[0,1).

Case 4. If i = 1 x i 0 and i = 1 y i 0, then we have T n x= ( 1 ) n x, T n y= ( 1 ) n y. Hence we have

{ T n x T n y 2 = ( 1 ) n x ( 1 ) n y 2 = x y 2 = x 2 + y 2 ; k n x y 2 = k n ( x 2 + y 2 ) ; x T n x ( y T n y ) 2 = [ 1 ( 1 ) n ] 2 x y 2 = [ 1 ( 1 ) n ] 2 ( x 2 + y 2 ) ; 2 x T n x , y T n y = 0 .

Thus inequality (1.4) still holds for any k[0,1). Therefore the mapping defined by (1.5) is a k-asymptotically strictly pseudononspreading mapping.

A mapping T:CC is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that for all (x,y)H×H,

T n x T n y Lxy.
(1.6)

A Banach space E is said to satisfy Opial’s condition if, for any sequence { x n } in E, x n x implies that lim sup n x n x< lim sup n x n y for all yE with yx. It is well known that every Hilbert space satisfies Opial’s condition.

A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever { x n } is a sequence in D(T) such that { x n } converges weakly to x D(T) and {T x n } converges strongly to p, then T x =p.

T is said to be semi-compact if for any bounded sequence { x n }H with lim n x n T x n =0, there exists a subsequence { x n i } of { x n } such that { x n i } converges strongly to a point x H.

Recently, Zhao and Chang [16] proposed the following algorithm for solving k-strictly pseudononspreading mappings and equilibrium problem in Hilbert spaces.

{ F ( u n , y ) + 1 r n y u n , u n x n 0 , y C , x n + 1 = α 0 , n u n + i = 1 α i , n S i , β u n ,
(1.7)

where S i , β :=βI+(1β) S i , α i , n (0,1). Under some suitable conditions, they proved that the sequences { x n }, { y n } weakly and strongly converge to a solution of the problem x i = 1 F( S i )EP(F).

For finding a split feasibility problem for k-strictly pseudononspreading mappings in a Hilbert space, in [17], Quan and Chang presented the following iterative method:

{ x 1 H 1  chosen arbitrarily , u n = x n + γ A ( T n ( mod N ) I ) A x n , x n + 1 = ( 1 α n ) u n + α n S n ( mod N ) u n ,
(1.8)

where γ is a constant and γ(0, 1 κ λ ), λ is the spectral of the operator A A, κ=max{ κ 1 , κ 2 ,, κ N }, and { α n } is a sequence in (0,1ϱ] with ϱ=max{ ϱ 1 , ϱ 2 ,, ϱ N }. Under some suitable conditions, they proved that { x n } weakly and strongly converges to a split fixed point x Γ.

Inspired and motivated by the recent works of Zhao and Chang [16], Quan and Chang [17], etc., in this paper, we propose an iterative scheme to approximate a common element of the set of solutions of k-asymptotically strictly pseudononspreading mappings and mixed equilibrium problem in infinite-dimensional Hilbert spaces. Some weak and strong convergence theorems are proved. At the same time, the demiclosed principle of a k-asymptotically strictly pseudononspreading mapping is shown. The results presented in this paper improve and extend some recent corresponding results.

2 Preliminaries

Throughout this paper, we denote the strong convergence and weak convergence of a sequence { x n } to a point xX by x n x, x n x, respectively.

Let H be a Hilbert space with the inner product , and the norm , let C be a nonempty closed convex subset of H. For every point xH, there exists a unique nearest point of C, denoted by P C x, such that x P C xxy for all yC. Such a P C is called the metric projection from H onto C. It is well known that P C is a firmly nonexpansive mapping from H to C, i.e.,

P C x P C y 2 P C x P C y,xy,x,yH.

Further, for any xH and zC, z= P C x if and only if

xz,zy0,yC.
(2.1)

For solving mixed equilibrium problems, we assume that the bifunction F:C×CR satisfies the following conditions:

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

(A2) F(x,y)+F(y,x)0, x,yC;

(A3) For all x,y,zC, lim t 0 F(tz+(1t)x,y)F(x,y);

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

Lemma 2.1 [18]

Let C be a nonempty closed convex subset of a Hilbert space H. Let F be a bifunction from C×C to R satisfying (A1)-(A4), and let φ:CR{+} be a proper lower semi-continuous and convex function such that Cdomφ. For r>0 and xC, define a mapping T r :HC as follows:

T r (x)= { z C : F ( z , y ) + φ ( y ) φ ( z ) + 1 r y z , z x 0 , y C } ,xH.
(2.2)

Then

  1. (1)

    For each xH, T r (x);

  2. (2)

    T r is single-valued;

  3. (3)

    T r is firmly nonexpansive, that is, x,yH,

    T r x T r y 2 T r x T r y,xy;
  4. (4)

    F( T r )=MEP(F,φ);

  5. (5)

    MEP(F,φ) is closed and convex.

Lemma 2.2 [13]

Let H be a real Hilbert space. Then the following results hold:

  1. (i)

    For all x,yH and for all t[0,1],

    t x + ( 1 t ) y 2 =t x 2 +(1t) y 2 t(1t) x y 2 .
  2. (ii)

    x + y 2 x 2 +2y,x+y.

  3. (iii)

    If { x n } n = 1 is a sequence in H which converges weakly to zH, then

    lim sup n x n y 2 = lim sup n x n z 2 + z y 2 ,yH.

The demiclosed principle and the closeness and convexity of the set of fixed points of a nonlinear mapping play very important roles in investigating many nonlinear problems. We now show the demiclosed principle of k-asymptotically strictly pseudononspreading mapping and the closeness and convexity of the set of fixed points of such a mapping, respectively.

Lemma 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let T:CC be a continuous k-asymptotically strictly pseudononspreading mapping. If F(T), then it is closed and convex.

Proof Let { x n } n = 1 F(T) be a sequence which converges to xC, we show that xF(T).

T n x x = T n x x n + x n x T n x T n x n + x n x.
(2.3)

Since T is k-asymptotically strictly pseudononspreading, we have

T n x T n x n 2 k n x x n 2 + k ( T n x n x n ) ( T n x x ) 2 + 2 x T n x , x n T n x n = k n x n x 2 + k T n x x 2 ( k n x n x + k T n x x ) 2 .
(2.4)

Using (2.4) in (2.3), we obtain

T n x x k n x n x+ k x T n x + x n x,

so

( 1 k ) T n x x ( k n + 1 ) x n x , T n x x k n + 1 1 k x n x .

Since k n 1 and x n x0 as n, we get that lim n T n xx=0. Since T is continuous, which implies that x= lim n T n x= lim n T( T n 1 x)=T( lim n T n 1 x)=Tx. Hence, xF(T).

Now, we show that F(T) is convex.

For x,yF(T) and t(0,1), put z=tx+(1t)y. We show that z=Tz. In fact, we have

z T n z 2 = z 2 2 z , T n z + T n z 2 = z 2 2 t x + ( 1 t ) y , T n z + T n z 2 = z 2 2 t x , T n z 2 ( 1 t ) y , T n z + T n z 2 = z 2 + t x 2 2 t x , T n z + t T n z 2 + ( 1 t ) y 2 2 ( 1 t ) y , T n z + ( 1 t ) T n z 2 t x 2 ( 1 t ) y 2 = z 2 + t x T n z 2 + ( 1 t ) y T n z 2 t x 2 ( 1 t ) y 2 z 2 + t [ k n x z 2 + k x T n x ( z T n z ) 2 + 2 x T n x , z T n z ] + ( 1 t ) [ k n y z 2 + k y T n y ( z T n z ) 2 + 2 y T n y , z T n z ] t x 2 ( 1 t ) y 2 z 2 + t k n x z 2 + t k z T n z 2 + ( 1 t ) k n y z 2 + ( 1 t ) k z T n z 2 t x 2 ( 1 t ) y 2 = z 2 + t k n x z 2 + ( 1 t ) k n y z 2 + k z T n z 2 t x 2 ( 1 t ) y 2 .

So,

( 1 k ) z T n z 2 z 2 + t k n x z , x z + ( 1 t ) k n y z , y z t x 2 ( 1 t ) y 2 ( k n 1 ) [ t x 2 + ( 1 t ) y 2 + z 2 ] .

Since k n 1 as n, we obtain that lim n z T n z 2 =0, which implies that lim n T n z=z, z= lim n T n z=T lim n ( T n 1 z)=Tz. Hence, zF(T), which means that F(T) is convex. □

Lemma 2.4 Let C be a nonempty closed convex subset of a real Hilbert space H, and let T:CC be a k-asymptotically strictly pseudononspreading and uniformly L-Lipschitzian mapping. Then, for any sequence { x n } in C converging weakly to a point p and { x n T x n } converging strongly to 0, we have p=Tp.

Proof Since lim n x n T x n =0, by induction we can prove that

lim n x n T m x n =0for each m1.

In fact, it is obvious that the conclusion is true for m=1. Suppose that the conclusion holds for m>1, now we prove that the conclusion is also true for m+1.

Indeed, since T is uniformly L-Lipschitzian, we have

x n T m + 1 x n x n T m x n + T m x n T m + 1 x n x n T m x n +L x n T x n .

So, lim n x n T m + 1 x n =0.

For each xH, define f:H[0,) by

f(x):= lim sup n x n x 2 .
(2.5)

Then from Lemma 2.2 we have

f(x)= lim sup n x n p 2 + p x 2 ,xH.
(2.6)

Thus, for any xH, f(x)=f(p)+ p x 2 and

f ( T m p ) =f(p)+ p T m p 2 ,
(2.7)
f ( T m p ) = lim sup n x n T m p 2 = lim sup n x n T m x n + T m x n T m p 2 = lim sup n T m x n T m p 2 lim sup n [ k m x n p 2 + k x n T m x n ( p T m p ) 2 + 2 x n T m x n , p T m p ] = lim sup n k m x n p 2 + k p T m p 2 = k m f ( p ) + k p T m p 2 .
(2.8)

It follows from (2.7) and (2.8) that

(1k) p T m p ( k m 1) f ( p ) .
(2.9)

That is,

lim m p T m p =0.
(2.10)

Hence we have

T p p T p T m p + T m p p L p T m 1 p + T m p p .

This is p=Tp, as desired. The proof is completed. □

Lemma 2.5 [19]

Let the number sequences { a n } and { α n } satisfy

a n + 1 (1+ α n ) a n ,n1,

where a n 0, α n 0 and n = 1 α n <. Then

  1. (1)

    lim n a n exists;

  2. (2)

    if lim inf n a n =0, then lim n a n =0.

3 Main results

Theorem 3.1 Let C be a nonempty and closed convex subset of a real Hilbert space H, let F be a bifunction from C×C to R satisfying (A1)-(A4), and let φ:CR{+} be a proper lower semi-continuous and convex function such that Cdomφ. Let T i :CC be a uniformly L i -Lipschitzian and τ i -asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S i :CC be a uniformly L ˜ i -Lipschitzian and l i -asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<, i=1,2,,N. Let { x n } be a sequence generated by

{ x 1 C , F ( u n , y ) + φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y C , y n = ( 1 β n ) u n + β n T n ( mod N ) n u n , x n + 1 = ( 1 α n ) y n + α n S n ( mod N ) n y n ,
(3.1)

where { α n } is a sequence in (0,1) with lim inf n α n >0, { β n } is a sequence in (0,1k) with lim inf n β n >0, k=max{ τ 1 , τ 2 ,, τ N }(0,1), and the sequence { r n }(0,) satisfies that lim inf n r n >0 and lim n | r n + 1 r n |=0. If Γ:= i = 1 N F( S i ) i = 1 N F( T i )MEP(F,φ), then the sequence { x n } converges weakly to a point x Γ.

Proof The proof is divided into four steps.

Step 1. Firstly, we prove that lim n x n p exists for any pΓ.

Taking pΓ and putting ρ=max{ l 1 , l 2 ,, l N }(0,1), it follows from Lemma 2.1 that u n = T r n x n , p= T r n p, we have

u n p T r n x n T r n p x n p,
(3.2)
x n + 1 p 2 = y n p + α n ( S n ( mod N ) n y n y n ) 2 = y n p 2 + 2 α n y n p , S n ( mod N ) n y n y n + α n 2 y n S n ( mod N ) n y n 2 .
(3.3)

Since

S n ( mod N ) n y n p 2 = S n ( mod N ) n y n S n ( mod N ) n p 2 ρ n y n p 2 + ρ ( I S n ( mod N ) n ) y n 2
(3.4)

and

S n ( mod N ) n y n p 2 = S n ( mod N ) n y n y n + y n p 2 = S n ( mod N ) n y n y n 2 + y n p 2 + 2 S n ( mod N ) n y n y n , y n p ,
(3.5)

from (3.4) and (3.5) we have

2 S n ( mod N ) n y n y n , y n p ( ρ n 1) y n p 2 +(ρ1) S n ( mod N ) n y n y n 2 .
(3.6)

Substituting (3.6) into (3.3) and simplifying, we have

x n + 1 p 2 y n p 2 + α n ( ρ n 1 ) y n p 2 + α n ( ρ 1 ) S n ( mod N ) n y n y n 2 + α n 2 y n S n ( mod N ) n y n 2 = [ 1 + α n ( ρ n 1 ) ] y n p 2 α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 .
(3.7)

On the other hand,

y n p 2 u n p 2 + 2 β n u n p , ( T n ( mod N ) n I ) u n + β n 2 T n ( mod N ) n I ) u n 2 = u n p 2 + 2 β n u n p , ( T n ( mod N ) n I ) u n + β n 2 T n ( mod N ) n I ) u n 2 .
(3.8)

Since T i is a τ i -asymptotically strictly pseudononspreading mapping, we have

T n ( mod N ) n u n p 2 k n u n p 2 +k u n T n ( mod N ) n u n 2 .
(3.9)

Again since

T n ( mod N ) n u n p 2 = T n ( mod N ) n u n u n 2 + u n p 2 + 2 T n ( mod N ) n u n u n , u n p ,
(3.10)

so we have

2 T n ( mod N ) n u n u n u n p ( k n 1 ) u n p 2 + ( k 1 ) u n T n ( mod N ) n u n 2 .
(3.11)

From (3.8) and (3.11), we have

y n p 2 u n p 2 + β n ( k n 1 ) u n p 2 + β n ( k 1 ) u n T n ( mod N ) n u n 2 + β n 2 T n ( mod N ) n I ) u n 2 [ 1 + β n ( k n 1 ) ] u n p 2 β n ( 1 k β n ) u n T n ( mod N ) n u n 2 .
(3.12)

By using (3.7) and (3.12), we have

x n + 1 p 2 [ 1 + α n ( ρ n 1 ) ] { [ 1 + β n ( k n 1 ) ] u n p 2 β n ( 1 k β n ) u n T n ( mod N ) n u n 2 } α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 = [ 1 + α n ( ρ n 1 ) ] [ 1 + β n ( k n 1 ) ] u n p 2 β n [ 1 + α n ( ρ n 1 ) ] ( 1 k β n ) u n T n ( mod N ) n u n 2 α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 { 1 + β n ( k n 1 ) + α n ( ρ n 1 ) [ 1 + β n ( k n 1 ) ] } x n p 2 β n [ 1 + α n ( ρ n 1 ) ] ( 1 k β n ) u n T n ( mod N ) n u n 2 α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 .
(3.13)

Let M n := β n ( k n 1)+ α n ( ρ n 1)[1+ β n ( k n 1)]. Since ( ρ n 1)< and ( k n 1)<, so M n <, we have

x n + 1 p 2 ( 1 + M n ) x n p 2 α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 β n [ 1 + α n ( ρ n 1 ) ] ( 1 k β n ) u n T n ( mod N ) n u n 2 ( 1 + M n ) x n p 2 .
(3.14)

Using Lemma 2.5, we show that lim n x n p exists. Further, it follows from (3.2) and (3.12) that { y n } and { u n } are bounded.

On the other hand, from (3.14) we have

β n [ 1 + α n ( ρ n 1 ) ] ( 1 k β n ) u n T n ( mod N ) n u n 2 + α n ( 1 ρ α n ) S n ( mod N ) n y n y n 2 ( 1 + M n ) x n p 2 x n + 1 p 2 .
(3.15)

Since lim n x n p exists and by the fact that M n 0, taking limit on both sides of inequality (3.15), we get

lim n ( T n ( mod N ) n I ) u n =0,
(3.16)
lim n y n S n ( mod N ) n y n =0.
(3.17)

Step 2. Now, we prove that lim n x n + 1 x n =0, lim n y n + 1 y n =0 and lim n x n u n =0.

It follows from Lemma 2.1 that u n = T r n x n , p= T r n p, so

u n p 2 = T r n x n T r n p 2 x n p , u n p = 1 2 ( x n p 2 + u n p 2 x n u n 2 ) .
(3.18)

This shows that

u n p 2 x n p 2 x n u n 2 .
(3.19)

By (3.13) and (3.19), we obtain

x n + 1 p 2 x n p 2 x n u n 2 + M n x n p 2 .
(3.20)

So,

x n u n 2 x n q 2 x n + 1 q 2 + M n x n p 2 .
(3.21)

Thus, we obtain

lim n x n u n =0.
(3.22)

In fact, it follows from (3.1) that

x n + 1 x n = ( 1 α n ) y n + α n S n ( mod N ) n y n x n = ( 1 α n ) ( u n + β n ( T n ( mod N ) n I ) u n ) + α n S n ( mod N ) n y n x n = ( 1 α n ) β n ( T n ( mod N ) n I ) u n + α n ( S n ( mod N ) n y n u n ) + ( u n x n ) = ( 1 α n ) β n ( T n ( mod N ) n I ) u n + α n ( S n ( mod N ) n y n y n ) + α n ( y n u n ) + ( u n x n ) = ( 1 α n ) β n ( T n ( mod N ) n I ) u n + α n ( S n ( mod N ) n y n y n ) + α n β n ( T n ( mod N ) n I ) u n + ( u n x n ) = β n ( T n ( mod N ) n I ) u n + α n ( S n ( mod N ) n y n y n ) + u n x n β n ( T n ( mod N ) n I ) u n + α n S n ( mod N ) n y n y n + u n x n .
(3.23)

From (3.16), (3.17) and (3.22) we have

lim n x n + 1 x n =0.
(3.24)

Similarly, it follows from (3.1) that

y n + 1 y n = u n + 1 + β n + 1 ( T n + 1 ( mod N ) n + 1 I ) u n + 1 u n + β n ( T n ( mod N ) n I ) u n u n + 1 u n + β n + 1 ( T n + 1 ( mod N ) n + 1 I ) u n + 1 + β n ( T n ( mod N ) n I ) u n ,
(3.25)

where

u n + 1 u n = T r n + 1 x n + 1 T r n x n T r n + 1 x n + 1 T r n + 1 x n + T r n + 1 x n T r n x n x n + 1 x n + T r n + 1 x n T r n x n .
(3.26)

On the other hand, it follows from Lemma 2.1 that u n = T r n x n and u n + 1 = T r n + 1 x n + 1 . We have

F( u n + 1 ,y)+φ(y)φ( u n + 1 )+ 1 r n + 1 y u n + 1 , u n + 1 x n + 1 0,yC,

and

F( u n ,y)+φ(y)φ( u n )+ 1 r n y u n , u n x n 0,yC.

Particularly, we have

F( u n + 1 , u n )+φ( u n )φ( u n + 1 )+ 1 r n + 1 u n u n + 1 , u n + 1 x n + 1 0,yC,
(3.27)

and

F( u n , u n + 1 )+φ( u n + 1 )φ( u n )+ 1 r n u n + 1 u n , u n x n 0,yC.
(3.28)

Summing up (3.27) and (3.28) and using (A2), we obtain

1 r n + 1 u n u n + 1 , u n + 1 x n + 1 + 1 r n u n + 1 u n , u n x n 0.

Thus,

u n + 1 u n , u n x n r n u n + 1 x n + 1 r n + 1 0,

which implies that

0 u n + 1 u n , u n x n r n r n + 1 ( u n + 1 x n + 1 ) = u n + 1 u n , u n u n + 1 + u n + 1 x n r n r n + 1 ( u n + 1 x n + 1 ) .

Therefore,

u n + 1 u n 2 u n + 1 u n , x n + 1 x n + ( 1 r n r n + 1 ) ( u n + 1 x n + 1 ) u n + 1 u n [ x n + 1 x n + | 1 r n r n + 1 | u n + 1 x n + 1 ] .

Thus, we have

u n + 1 u n x n + 1 x n + | 1 r n r n + 1 | u n + 1 x n + 1 .
(3.29)

It follows from (3.16), (3.17), (3.24), (3.25) and (3.29) that

lim n u n + 1 u n =0
(3.30)

and

lim n y n + 1 y n =0.
(3.31)

Put L={ L 1 , L 2 ,, L N , L ˜ 1 , L ˜ 2 ,, L ˜ N }. Since

u n T n ( mod N ) u n u n T n ( mod N ) n u n + T n ( mod N ) n u n T n ( mod N ) u n u n T n ( mod N ) n u n + L T n ( mod N ) n 1 u n u n u n T n ( mod N ) n u n + L [ T n 1 ( mod N ) n 1 u n T n 1 ( mod N ) n 1 u n 1 + T n 1 ( mod N ) n 1 u n 1 u n 1 + u n 1 u n ] u n T n ( mod N ) n u n + L 2 u n u n 1 + L T n 1 ( mod N ) n 1 u n 1 u n 1 + L u n 1 u n ,
(3.32)

from (3.16), (3.22), (3.30) and (3.32) we get

lim n T n ( mod N ) u n u n =0.
(3.33)

Similarly, we have

y n S n ( mod N ) y n y n S n ( mod N ) n y n + S n ( mod N ) n y n y n S n ( mod N ) y n y n S n ( mod N ) n y n + L [ S n 1 ( mod N ) n 1 y n S n 1 ( mod N ) n 1 y n 1 + S n 1 ( mod N ) n 1 y n 1 y n 1 + y n 1 y n ] y n T n ( mod N ) n y n + L 2 y n y n 1 + L T n 1 ( mod N ) n 1 y n 1 y n 1 + L y n 1 y n .

This implies that

lim n y n S n ( mod N ) y n =0.
(3.34)

Since x n + 1 y n = α n y n S n ( mod N ) n y n , so

lim n x n + 1 y n =0.
(3.35)

By (3.24) and (3.35), we have

lim n x n y n =0.
(3.36)

It follows from (3.22) and (3.36) that

lim n y n u n =0.
(3.37)

Since lim n x n p exists for any pΓ and x n p x n y n y n p x n p+ x n y n , it follows from (3.36) that lim n y n p= lim n x n p holds. Similarly, lim n u n p= lim n x n p holds for any pΓ.

Step 3. We show that x Γ:= i = 1 N F( S i ) i = 1 N F( T i )MEP(F,φ).

Firstly, we show that x i = 1 N F( S i ) i = 1 N F( T i ).

In fact, since { y n } is bounded, there exists a subsequence { y n i }{ y n } such that { y n i } x C. Hence, for any positive integer j=1,2,,N, there exists a subsequence { n i (j)}{ n i } with n i (j)(modN)=j such that { y n i ( j ) } x . Again, by (3.34) we know that y i N + j S j u i N + j 0 as i, therefore we have that lim n i ( j ) y n i ( j ) S j y n i ( j ) =0.

Since S j is demiclosed at zero, it follows from Lemma 2.4 that x F( S j ). By the arbitrariness of j=1,2,,N, we have

x i = 1 N F( S i ).

On the other hand, since lim n y n u n =0, we know that u n i x , too. Similarly, it follows from (3.33) and Lemma 2.4 that x F( T j ). By the arbitrariness of j=1,2,,N, we have

x i = 1 N F( T i ).

Now, we show that x MEP(F,φ).

By Lemma 2.1, since u n = T r n x n , we have

F( u n ,y)+φ(y)φ( u n )+ 1 r n y u n , u n x n 0,yK.
(3.38)

From (A2), we obtain

φ(y)φ( u n )+ 1 r n y u n , u n x n F( u n ,y)F(y, u n ),
(3.39)

and hence

φ(y)φ( u n i )+ 1 r n i y u n i , u n i x n i F(y, u n i ).
(3.40)

By lim inf n r n >0, we have lim i u n i x n i r n i =0. Since u n i x , it follows from (A4) and the weak lower semicontinuity of φ that

F ( y , x ) φ(y)+φ ( x ) 0.
(3.41)

Put z t =ty+(1t) x for all t(0,1] and yC. Consequently, we get z t C. Hence

F( z t ,p)φ( z t )+φ ( x ) 0.
(3.42)

From (A1) and (A4), and the convexity of φ, we have

0 = F ( z t , z t ) φ ( z t ) + φ ( z t ) t F ( z t , y ) + ( 1 t ) φ ( z t , x ) + t φ ( y ) + ( 1 t ) φ ( x ) φ ( z t ) t [ F ( z t , y ) + φ ( y ) φ ( z t ) ] .

Therefore

F( z t ,y)+φ(y)φ( z t )0.

Letting t0, and from the weak lower semicontinuity of φ, we have

F ( x , y ) +φ(y)φ ( x ) 0.

This implies that x MEP(F,φ). Hence x Γ.

Step 4. Finally, we prove that x n x and u n x , x Γ.

Due to u n i x , we know that x n i x from (3.37). Suppose that there exists another subsequence { x n j } of { x n } such that { x n j } y Γ with y x . Using the same proof method as in Step 3, we know that y Γ. Consequently, lim n x n y exists. By using Opial’s property of a Hilbert space, we have

lim inf n i x n i x < lim inf n i x n i y = lim inf n x n y = lim inf n j x n j y < lim inf n j x n j x = lim inf n x n x = lim inf n i x n i x .

This is a contradiction. Therefore x n x . By (3.1) and (3.22), we have u n x . Therefore, the conclusion follows.

This completes the proof of Theorem 3.1. □

Taking φ=0, N=1 in Theorem 3.1, we have the following result.

Corollary 3.2 Let C be a nonempty and closed convex subset of a real Hilbert space H, let F be a bifunction from C×C to R satisfying (A1)-(A4), and let T:CC be a uniformly L-Lipschitzian and k-asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S:CC be a uniformly L ˜ -Lipschitzian and ρ-asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<. Let { x n } be a sequence generated by

{ x 1 C , F ( u n , y ) + 1 r n y u n , u n x n 0 , y C , y n = ( 1 β n ) u n + β n T n u n , x n + 1 = ( 1 α n ) y n + α n S n y n ,
(3.43)

where k(0,1), ρ(0,1), { α n } is a sequence in (0,1) with lim inf n α n >0, { β n } is a sequence in (0,1k) with lim inf n β n >0 and the sequence { r n }(0,) with lim inf n r n >0 and lim n | r n + 1 r n |=0. If Γ:=F(S)F(T)EP(F), then the sequence { x n } converges weakly to a point x Γ.

Corollary 3.3 Let C be a nonempty and closed convex subset of a real Hilbert space H, let F be a bifunction from C×C to R satisfying (A1)-(A4), and let φ:CR{+} be a proper lower semi-continuous and convex function such that Cdomφ. Let T i :CC be a uniformly L i -Lipschitzian and τ i -asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S i :CC be a uniformly L ˜ i -Lipschitzian and l i -asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<, i=1,2,,N. Let { x n } be a sequence generated by

{ x 1 C , F ( u n , y ) + φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y C , y n = ( 1 β n ) u n + β n T n ( mod N ) n u n , x n + 1 = ( 1 α n ) y n + α n S n ( mod N ) y n ,
(3.44)

where k=max{ τ 1 , τ 2 ,, τ N }(0,1), ρ=max{ l 1 , l 2 ,, l N }(0,1), { α n } is a sequence in (0,1) with lim inf n α n >0, { β n } is a sequence in (0,1k) with lim inf n β n >0 and the sequence { r n }(0,) with lim inf n r n >0 and lim n | r n + 1 r n |=0. If Γ:= i = 1 F( S i ) i = 1 F( T i )MEP(F,φ), and there exists a positive integer j such that S j is semi-compact, then the sequence { x n } converges strongly to a point x Γ.

Proof Without loss of generality, we can assume that S 1 is semi-compact. It follows from (3.34) that

y n i ( 1 ) S 1 y n i ( 1 ) 0, n i ( 1 ) .

Therefore, there exists a subsequence of { y n i ( 1 ) } (for the sake of convenience we still denote it by { y n i ( 1 ) }) such that y n i ( 1 ) y H 1 . Since y n i ( 1 ) y , x = y , and so y n i ( 1 ) x Γ. By virtue of the fact that lim n y n p exists, we know that

lim n y n x = lim n u n x = lim n x n x =0.

That is, { x n }, { u n } and { y n } converge strongly to the point x Γ. This completes the proof. □

4 Applications

4.1 Application to a convex minimization problem

It is well known that mixed equilibrium problem (1.2) reduces to the convex minimization problem as F=0. Therefore, Theorem 3.1 can be used to solve convex minimization problem (1.3), and the following result can be directly deduced from Theorem 3.1.

Theorem 4.1 Let C be a nonempty and closed convex subset of a real Hilbert space H, let φ:CR{+} be a proper lower semi-continuous and convex function such that Cdomφ. Let T i :CC be a uniformly L i -Lipschitzian and τ i -asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S i :CC be a uniformly L ˜ i -Lipschitzian and l i -asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<, i=1,2,,N. Let { x n } be a sequence generated by

{ x 1 C , φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y C , y n = ( 1 β n ) u n + β n T n ( mod N ) n u n , x n + 1 = ( 1 α n ) y n + α n S n ( mod N ) n y n ,
(4.1)

where k=max{ τ 1 , τ 2 ,, τ N }(0,1), ρ=max{ l 1 , l 2 ,, l N }(0,1), { α n } is a sequence in (0,1) with lim inf n α n >0, { β n } is a sequence in (0,1k) with lim inf n β n >0, and the sequence { r n }(0,) with lim inf n r n >0 and lim n | r n + 1 r n |=0. If i = 1 N F( S i ) i = 1 N F( T i )CMP(φ), then the sequence { x n } converges weakly to a point x i = 1 N F( S i ) i = 1 N F( T i )CMP(φ).

4.2 Application to a convex feasibility problem

The so-called convex feasibility problem for a family of mappings { T i } i = 1 ω (where ω may be a finite positive integer or +∞) is to find a point of the nonempty intersection i = 1 ω C i , where C i is the fixed point set of mapping T i , i=1,2,,ω.

In Theorem 3.1 if F=0, φ=0, then the condition ‘ u n C such that yC, y u n , u n x n 0’ is equivalent to u n = P C ( x n ). Therefore, the following result can be directly obtained from Theorem 3.1.

Theorem 4.2 Let C be a nonempty and closed convex subset of a real Hilbert space H, let T i :CC be a uniformly L i -Lipschitzian and τ i -asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S i :CC be a uniformly L ˜ i -Lipschitzian and l i -asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<, i=1,2,,N. Let { x n } be a sequence generated by

{ x 1 C , u n = P C ( x n ) , y n = ( 1 β n ) u n + β n T n ( mod N ) n u n , x n + 1 = ( 1 α n ) y n + α n S n ( mod N ) n y n ,
(4.2)

where { α n } is a sequence in (0,1) with lim inf n α n >0 and { β n } is a sequence in (0,1k) with lim inf n β n >0, k=max{ τ 1 , τ 2 ,, τ N }(0,1). If Γ:= i = 1 N F( S i ) i = 1 N F( T i ), then the sequence { x n } converges weakly to a point x Γ, which is a solution of the convex feasibility problem for mappings { T i } i = 1 N and { S i } i = 1 N .

4.3 Application to the mixed variational inequality problem of Browder type

A variational inequality problem (VIP) is formulated as a problem of finding a point x with property x C, A x ,z x 0, zC. We will denote the solution set of VIP by VI(A,C). We know that given a mapping T:CC, let F(x,y)=Tx,yx for all x,yC. Then x EP(F) if and only if x C is a solution of the variational inequality Tx,yx0 for all yC, i.e., x is a solution of the variational inequality.

In [20], the mixed variational inequality of Browder type (VI) is shown to be equivalent to finding a point uC such that

Au,yu+φ(y)φ(u)0,yC.

We will denote the solution set of a mixed variational inequality of Browder type by VI(A,C,φ).

A mapping A:CH is said to be an α-inverse-strongly monotone mapping if there exists a constant α>0 such that AxAy,xyα A x A y 2 for any x,yC. Setting F(x,y)=Ax,yx, it is easy to show that F satisfies conditions (A1)-(A4) as A is an α-inverse-strongly monotone mapping. Then it follows from Theorem 3.1 that the following result holds.

Theorem 4.3 Let C be a nonempty and closed convex subset of a real Hilbert space H, let A:CH be an α-inverse-strongly monotone mapping, and let φ:CR{+} be a proper lower semi-continuous and convex function such that Cdomφ. Let T i :CC be a uniformly L i -Lipschitzian and τ i -asymptotically strictly pseudononspreading mapping with the sequence { k n }[1,+) such that n = 1 ( k n 1)<, let S i :CC be a uniformly L ˜ i -Lipschitzian and l i -asymptotically strictly pseudononspreading mapping with the sequence { ρ n }[1,+) such that n = 1 ( ρ n 1)<, i=1,2,,N. Let { x n } be a sequence generated by

{ x 1 C , A u n , y u n + φ ( y ) φ ( u n ) + 1 r n y u n , u n x n 0 , y C , y n = ( 1 β n ) u n + β n T n ( mod N ) n u n , x n + 1 = ( 1 α n ) y n + α n S n ( mod N ) n y n ,
(4.3)

where { α n } is a sequence in (0,1) with lim inf n α n >0, { β n } is a sequence in (0,1k) with lim inf n β n >0, k=max{ τ 1 , τ 2 ,, τ N }(0,1), and the sequence { r n }(0,) satisfies lim inf n r n >0 and lim n | r n + 1 r n |=0. If Γ:= i = 1 N F( S i ) i = 1 N F( T i )VI(A,C,φ), then the sequence { x n } converges weakly to a point x Γ.

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Acknowledgements

The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

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Correspondence to Lin Wang.

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Both authors contributed equally to this work. Both authors read and approved the final manuscript.

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Keywords

  • k-asymptotically strictly pseudononspreading mapping
  • mixed equilibrium problem
  • weak and strong convergence
  • demiclosed principle