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A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense

Abstract

In this paper, common solutions to an equilibrium problem and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense are discussed. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

MSC:47H09, 47J25.

1 Introduction

Equilibrium problems have been revealed as a very powerful and important tool in the study of nonlinear phenomena. They have turned out to be very useful in studying optimization problems, differential equations, and minimax theorems and in certain applications to mechanics and economic theory; see [127] and the references therein. Important practical situations motivate the study of a system of equilibrium problems. For instance, the flow of fluid through a fissured porous medium and certain models of plasticity led to such problems; see, for instance, [28]. Because of their importance, they have been extensively analyzed. The aim of this paper is to present an iterative method for solving solutions of an equilibrium problem, which is known as the Ky Fan inequality, and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is presented. A strong convergence theorem is established in a reflexive Banach space. Some results in Hilbert spaces are also discussed.

2 Preliminaries

Let E be a real Banach space. Recall that the normalized duality mapping J from E to 2 E is defined by

Jx= { f E : x , f = x 2 = f 2 } ,

where , denotes the generalized duality pairing. Recall that E is said to be strictly convex if x + y 2 <1 for all x,yE with x=y=1 and xy. It is said to be uniformly convex if lim n x n y n =0 for any two sequences { x n } and { y n } in E such that x n = y n =1 and lim n x n + y n 2 =1. Let U E ={xE:x=1} be the unit sphere of E. Then the Banach space E is said to be smooth if lim t 0 x + t y x t exists for each x,y U E . It is said to be uniformly smooth if the above limit is attained uniformly for x,y U E . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that if E is uniformly smooth if and only if E is uniformly convex.

Recall that E enjoys the Kadec-Klee property if for any sequence { x n }E, and xE with x n x, and x n x, then x n x0 as n. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Next, we assume that E is a smooth Banach space. Consider the functional defined by

ϕ(x,y)= x 2 2x,Jy+ y 2 ,x,yE.

Observe that in a Hilbert space H, the equality is reduced to ϕ(x,y)= x y 2 , x,yH. As we all know, if C is a nonempty closed convex subset of a Hilbert space H and P C :HC is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [29] recently introduced a generalized projection operator Π C in a Banach space E which is an analogue of the metric projection P C in Hilbert spaces. Recall that the generalized projection Π C :EC is a map that assigns to an arbitrary point xE the minimum point of the functional ϕ(x,y), that is, Π C x= x ¯ , where x ¯ is the solution to the minimization problem

ϕ( x ¯ ,x)= min y C ϕ(y,x).

Existence and uniqueness of the operator Π C follows from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J. In Hilbert spaces, Π C = P C . It is obvious from the definition of function ϕ that

( x y ) 2 ϕ(x,y) ( y + x ) 2 ,x,yE
(2.1)

and

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2xz,JzJy,x,y,zE.
(2.2)

Remark 2.1 If E is a reflexive, strictly convex and smooth Banach space, then ϕ(x,y)=0 if and only if x=y; for more details, see [29] and the reference therein.

Let f be a bifunction from C×C to , where denotes the set of real numbers and let A:C E be a mapping. Consider the following equilibrium problem. Find pC such that

f(p,q)+Ap,qp0,qC.
(2.3)

We use EP(f,A) to denote the solution set of inequality (2.3). That is,

EP(f)= { p C : f ( p , q ) + A p , q p 0 , q C } .

If A=0, then problem (2.3) is reduced to the following Ky Fan inequality. Find pC such that

f(p,q)0,qC.
(2.4)

We use EP(f) to denote the solution set of inequality (2.4). That is,

EP(f)= { p C : f ( p , q ) 0 , q C } .

If f=0, then problem (2.3) is reduced to the classical variational inequality. Find pC such that

Ap,qp0,qC.
(2.5)

We use VI(C,A) to denote the solution set of inequality (2.5). That is,

VI(C,A)= { p C : A p , q p 0 , q C } .

Recall that a mapping A:C E is said to be α-inverse-strongly monotone if there exists α>0 such that

AxAy,xyα A x A y 2 .

For solving problem (2.3), let us assume that the nonlinear mapping A:C E is α-inverse-strongly monotone and the bifunction f:C×CR satisfies the following conditions:

  1. (A1)

    F(x,x)=0, xC;

  2. (A2)

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

  3. (A3)
    lim sup t 0 F ( t z + ( 1 t ) x , y ) F(x,y),x,y,zC;
  4. (A4)

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

Let C be a nonempty subset of E and T:CC be a mapping. In this paper, we use F(T) to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C,

lim sup n { T n + 1 x T n x : x K } =0.

T is said to be closed if for any sequence { x n }C such that lim n x n = x 0 and lim n T x n = y 0 , then T x 0 = y 0 . In this paper, we use → and to denote the strong convergence and weak convergence, respectively.

Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence { x n } which converges weakly to p such that lim n x n T x n =0. The set of asymptotic fixed points of T will be denoted by F ˜ (T).

A mapping T is said to be relatively nonexpansive iff

F ˜ (T)=F(T),ϕ(p,Tx)ϕ(p,x),xC,pF(T).

A mapping T is said to be relatively asymptotically nonexpansive iff

F ˜ (T)=F(T),ϕ ( p , T n x ) (1+ μ n )ϕ(p,x),xC,pF(T),n1,

where { μ n }[0,) is a sequence such that μ n 0 as n.

Remark 2.2 The class of relatively asymptotically nonexpansive mappings was first considered in Agarwal et al. [30].

Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff

F(T),ϕ(p,Tx)ϕ(p,x),xC,pF(T).

Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n }[0,) with μ n 0 as n such that

F(T),ϕ ( p , T n x ) (1+ μ n )ϕ(p,x),xC,pF(T),n1.

Remark 2.3 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction F(T)= F ˜ (T); for more details, see [3133] the reference therein.

Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

Recall that T is said to be asymptotically quasi-ϕ-nonexpansive in the intermediate sense iff F(T) and the following inequality holds:

lim sup n sup p F ( T ) , x C ( ϕ ( p , T n x ) ϕ ( p , x ) ) 0.
(2.6)

Put

ξ n =max { 0 , sup p F ( T ) , x C ( ϕ ( p , T n x ) ϕ ( p , x ) ) } .

It follows that ξ n 0 as n. Then (2.6) is reduced to the following:

ϕ ( p , T n x ) ϕ(p,x)+ ξ n ,pF(T),xC.
(2.7)

Remark 2.5 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [34]; see also [35].

Remark 2.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [36] in the framework of Banach spaces.

In order to state our main results, we also need the following lemmas.

Lemma 2.7 [29]

Let C be a nonempty closed convex subset of a smooth Banach space E and xE. Then x 0 = Π C x if and only if

x 0 y,JxJ x 0 0,yC.

Lemma 2.8 [29]

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E and xE. Then

ϕ(y, Π C x)+ϕ( Π C x,x)ϕ(y,x),yC.

Lemma 2.9 [37]

Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let A:C E be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let r>0 be any given number, and let xE define a mapping K r :CC as follows: for any xC,

K r x= { p C : f ( p , q ) + A p , q p + 1 r q p , J p J x 0 } ,qC.

Then the following conclusions hold:

  1. (1)

    K r is single-valued;

  2. (2)

    K r is a firmly nonexpansive-type mapping, i.e., for all x,yE,

    K r x K r y,J K r xJ K r y S r x S r y,JxJy;
  3. (3)

    F( K r )=EP(f,A);

  4. (4)

    K r is quasi-ϕ-nonexpansive;

  5. (5)
    ϕ(q, K r x)+ϕ( K r x,x)ϕ(q,x),qF( K r );
  6. (6)

    EP(f,A) is closed and convex.

Lemma 2.10 [38]

Let E be a smooth and uniformly convex Banach space, and let r>0. Then there exists a strictly increasing, continuous, and convex function g:[0,2r]R such that g(0)=0 and

t x + ( 1 t ) y 2 t x 2 +(1t) y 2 t(1t)g ( x y )

for all x,y B r ={xE:xr} and t[0,1].

3 Main results

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C×C to satisfying (A1)-(A4), and let N be some positive integer. Let A:C E be a κ i -inverse-strongly monotone mapping. Let T i :CC be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense for every 1iN. Assume that T i is closed asymptotically regular on C and i = 1 N F( T i )EP(f,A) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 C chosen arbitrarily , y n = J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) , u n C  such that  f ( u n , y ) + A u n + y u n + 1 r n y u n , J u n J y n 0 , y C , H n = { z C : ϕ ( z , u n ) ϕ ( z , x n ) + N ξ n } , W n = { z C : x n z , J x 0 J x n 0 } , x n + 1 = Π H n W n x 0 ,

where ξ n =max{0, sup p i = 1 N F ( T i ) , x C (ϕ(p, T i n x)ϕ(p,x))}, { α n , i } is a real number sequence in (0,1) for every 1iN, { r n } is a real number sequence in [k,), where k is some positive real number. Assume that i = 0 N α n , i =1 and lim inf n α n , 0 α n , i >0 for every 1iN. Then the sequence { x n } converges strongly to Π i = 1 N F ( T i ) EP ( f , A ) x 0 , where Π i = 1 N F ( T i ) EP ( f , A ) is the generalized projection from E onto i = 1 N F( T i )EP(f,A).

Proof Since F i is closed and convex for every 1iN, we obtain from Lemma 2.9 that the common element set i = 1 N F( T i )EP(f,A) is closed and convex. Next, we show that both H n and W n are closed and convex. From the definition of H n and W n , it is obvious that H n is closed and W n is closed and convex. We show that H n is convex since ϕ(z, u n )ϕ(z, x n )+N ξ n is equivalent to

2z,J x n J u n x n 2 u n 2 +N ξ n .

It follows that H n is convex. This in turn shows that Π H n W n x 0 is well defined. Putting u n = k r n y n , from Lemma 2.9 we see that K r n is quasi-ϕ-nonexpansive. Now, we are in a position to prove that i = 1 N F( T i )EP(f,A) H n W n . Let w i = 1 N F( T i )EP(f,A),

ϕ ( w , u n ) = ϕ ( w , K r n y n ) ϕ ( w , y n ) = ϕ ( w , J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) ) = w 2 2 w , α h , 0 J x n + i = 1 N α n , i J T i n x n + α n , 0 J x n + i = 1 N α n , i J T i n x n 2 w 2 2 α n , 0 w , J x n 2 i = 1 N α n , i w , J T i n x n + α n , 0 x n 2 + i = 1 N α n , i T i n x n 2 = α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , T i n x n ) α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , x n ) + i = 1 N α n , i ξ n = ϕ ( w , x n ) + i = 1 N α n , i ξ n ϕ ( w , x n ) + N ξ n .
(3.1)

We have w H n . This implies that i = 1 N F( T i )EP(f,A) H n . On the other hand, we see that i = 1 N F( T i )EP(f,A) H 0 W 0 . Suppose that i = 1 N F( T i )EP(f,A) H m W m for some m. There exists an element x m + 1 H m W m such that x m + 1 = Π H m W m x 0 . In view of Lemma 2.7, we find that

x m + 1 w,J x 0 J x m + 1 0,w H m W m .

Since i = 1 N F( T i )EP(f,A) H m W m , we arrive at

x m + 1 w,J x 0 J x m + 1 0
(3.2)

for every w i = 1 N F( T i )EP(f,A). We therefore find that i = 1 N F( T i )EP(f,A) W m + 1 . It follows that i = 1 N F( T i )EP(f,A) H m + 1 W m + 1 . This shows that the sequence { x n } is well defined.

Next, we prove that the sequence { x n } is bounded. It follows from the definition of W n and Lemma 2.7 that x n = Π W n x 0 . In view of Lemma 2.8, we find that

ϕ( x n , x 0 )=ϕ( Π W n x 0 , x 0 )ϕ(w, x 0 )ϕ(w, x n )ϕ(w, x 0 )

for each w i = 1 N F( T i )EP(f,A) W n . This implies that {ϕ( x n , x 0 )} is bounded. It follows from (2.1) that { x n } is also bounded. Since x n + 1 = Π H n W n x 0 W n , we find from Lemma 2.7 that ϕ( x n , x 0 )ϕ( x n + 1 , x 0 ). Therefore, we obtain that {ϕ( x n , x 0 )} is nondecreasing. So there exists the limit of ϕ( x n , x 0 ). It follows from Lemma 2.8 that

ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π W n x 0 ) ϕ ( x n + 1 , x 0 ) ϕ ( Π W n x 0 , x 0 ) = ϕ ( x n + 1 , x 0 ) ϕ ( x n , x 0 ) .

This shows that lim n ϕ( x n + 1 , x n )=0. Since x n + 1 = Π H n W n x 0 H n , we find that lim n ϕ( x n + 1 , u n )=0. Since the space is reflexive, we may assume, without loss of generality, that x n x ¯ . Since W n is closed and convex, we find that x ¯ W n . This implies from x n = Π W n x 0 that ϕ( x n , x 0 )ϕ( x ¯ , x 0 ). On the other hand, we see from the weakly lower semicontinuity of that

ϕ ( x ¯ , x 0 ) = x ¯ 2 2 x ¯ , J x 0 + x 0 2 lim inf n ( x n 2 2 x n , J x 0 + x 0 2 ) = lim inf n ϕ ( x n , x 0 ) lim sup n ϕ ( x n , x 0 ) ϕ ( x ¯ , x 0 ) ,

which implies that lim n ϕ( x n , x 0 )=ϕ( x ¯ , x 0 ). Hence, we have lim n x n = x ¯ . In view of the Kadec-Klee property of E, we find that x n x ¯ as n. In view of (2.1), we see that lim n ( x n + 1 u n )=0. This implies that lim n u n = x ¯ . That is,

lim n J u n = lim n u n =J x ¯ .
(3.3)

This implies that {J u n } is bounded. Note that both E and E are reflexive. We may assume, without loss of generality, that J u n u E . In view of the reflexivity of E, we see that J(E)= E . This shows that there exists an element uE such that Ju= u . It follows that

ϕ ( x n + 1 , u n ) = x n + 1 2 2 x n + 1 , J u n + u n 2 = x n + 1 2 2 x n + 1 , J u n + J u n 2 .

Taking lim inf n on the both sides of the equality above yields that

0 x ¯ 2 2 x ¯ , u + u 2 = x ¯ 2 2 x ¯ , J u + J u 2 = x ¯ 2 2 x ¯ , J u + u 2 = ϕ ( x ¯ , u ) .

That is, x ¯ =u, which in turn implies that u =J x ¯ . It follows that J u n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain from (3.3) that lim n J u n =J x ¯ . Since J 1 : E E is demi-continuous and E enjoys the Kadec-Klee property, we obtain that u n x ¯ , as n. Note that x n u n x n x ¯ + x ¯ u n . It follows that

lim n x n u n =0.
(3.4)

Since J is uniformly norm-to-norm continuous on any bounded sets, we have

lim n J x n J u n =0.
(3.5)

On the other hand, we have

ϕ ( w , x n ) ϕ ( w , u n ) = x n 2 u n 2 2 w , J x n J u n x n u n ( x n + u n ) + 2 w J x n J u n .

We, therefore, find that

lim n ( ϕ ( w , x n ) ϕ ( w , u n ) ) =0.
(3.6)

Since E is uniformly smooth, we know that E is uniformly convex. In view of Lemma 2.10, we find that

ϕ ( w , u n ) = ϕ ( w , K r n y n ) ϕ ( w , y n ) = ϕ ( w , J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) ) = w 2 2 w , α n , 0 J x n + i = 1 N α n , i J T i n x n + α n , 0 J x n + i = 1 N α n , i J T i n x n 2 w 2 2 α n , 0 w , J x n 2 i = 1 N α n , i w , J T i n x n + α n , 0 x n 2 + i = 1 N α n , i T i n x n 2 α n , 0 α n , 1 g ( J x n J T 1 n x n ) = α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , T i n x n ) α n , 0 α n , 1 g ( J x n J T 1 n x n ) α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , x n ) + i = 1 N α n , i ξ h α n , 0 α n , 1 g ( J x n J T 1 n x n ) = ϕ ( w , x n ) + i = 1 N α n , i ξ n α n , 0 α n , 1 g ( J x n J T 1 n x n ) ϕ ( w , x n ) + N ξ n α n , 0 α n , 1 g ( J x n J T 1 n x n ) .

It follows that α n , 0 α n , 1 g(J x n J T 1 n x n )ϕ(w, x n )ϕ(w, u n )+ ξ n . In view of the restriction on the sequences, we find from (3.6) that lim n g(J x n J T 1 n x n )=0. It follows that lim n J x n J T 1 n x n =0. In the same way, we obtain that lim n J x n J T i n x n =0, 2iN. Notice that J T i n x n J x ¯ J T i n x n J x n +J x n J x ¯ . It follows that lim n J T i n x n J x ¯ =0. The demicontinuity of J 1 : E E implies that T i n x n x ¯ . Note that

| T i n x n x ¯ | = | J T i n x n J x ¯ | J T i n x n J x ¯ .

This implies that lim n T i n x n = x ¯ . Since E has the Kadec-Klee property, we obtain that lim n T i n x n x ¯ =0. On the other hand, we have

T i n + 1 x n x ¯ T i n + 1 x n T i n x n + T i n x n x ¯ .

It follows from the asymptotic regularity of T i that lim n T i n + 1 x n x ¯ =0. That is, T i T i n x n x ¯ . From the closedness of T i , we find x ¯ = T i x ¯ for every 1iN. This proves x ¯ i = 1 N F( T i ). Now, we state that x ¯ EP(f,A). In view of Lemma 2.9, we find that

ϕ( u n , y n )ϕ(w, y n )ϕ(w, u n )ϕ(w, x n )+N ξ n ϕ(w, u n ).

It follows from (3.6) that lim n ϕ( u n , y n )=0. This implies that lim n ( u n y n )=0. It follows from (3.4) that lim n y n = x ¯ . It follows that

lim n J y n = lim n y n = x ¯ =J x ¯ .

This shows that {J y n } is bounded. Since E is reflexive, we may assume that J y n y E . In view of J(E)= E , we see that there exists yE such that Jy= y . It follows that ϕ( u n , y n )= u n 2 2 u n ,J y n + J y n 2 . Taking lim inf n on the both sides of the equality above yields that

0 x ¯ 2 2 x ¯ , y + y 2 = x ¯ 2 2 x ¯ , J y + J y 2 = x ¯ 2 2 x ¯ , J y + y 2 = ϕ ( x ¯ , y ) .

That is, x ¯ =y, which in turn implies that y =J x ¯ . It follows that J y n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain that J y n J x ¯ 0 as n. Note that J 1 : E E is demi-continuous. It follows that y n x ¯ . Since E enjoys the Kadec-Klee property, we obtain that y n x ¯ as n. Note that u n y n u n x ¯ + x ¯ y n . This implies that lim n u n y n =0. Since J is uniformly norm-to-norm continuous on any bounded sets, we have lim n J u n J y n =0. In view of the assumption r n k, we see that

lim n J u n J y n r n =0.
(3.7)

Since u n = K r n y n , we find that

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

where F( u n ,y)=f( u n ,y)+A u n +y u n for every yC. It follows from (A2) that

y u n J u n J y n r n 1 r n y u n ,J u n J y n F(y, u n ),yC.

It follows from (3.7) that

F(y, x ¯ )0,yC.

For 0<t<1 and yC, define y t =ty+(1t) x ¯ . It follows that y t C, which yields that F( y t , x ¯ )0. It follows from (A1) and (A4) that

0=F( y t , y t )tF( y t ,y)+(1t)F( y t , x ¯ )tF( y t ,y).

That is,

F( y t ,y)0.

Letting t0, we obtain from (A3) that F( x ¯ ,y)0, yC. That is, f( u n ,y)+A u n +y u n 0 for every yC. This implies that x ¯ EP(f,A). This completes the proof that x ¯ i = 1 N F( T i )EP(f,A). Since x n + 1 = Π H n W n x 0 . In view of Lemma 2.7, we find that

x n + 1 w,J x 0 J x n + 1 0,w H n W n .

Since i = 1 N F( T i )EP(f,A) H n W n , we arrive at

x n + 1 w,J x 0 J x n + 1 0.

Letting n in the above inequality, we obtain that

x ¯ w,J x 0 J x ¯ 0,w i = 1 N F( T i )EP(f,A).

It follows from Lemma 2.7 that x ¯ = Π i = 1 N F ( T i ) EP ( f , A ) x 0 . This completes the proof. □

Remark 3.2 Theorem 3.1 includes the corresponding results in the literature as special cases. Since every uniformly convex Banach space enjoys the Kadec-Klee property, the framework of the space can be applicable to L p , p1.

Next, we state a result on Ky Fan inequality (2.4) and a single asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense.

Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C×C to satisfying (A1)-(A4). Let T:CC be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and F(T)EP(f) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 C chosen arbitrarily , y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n C  such that  f ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C , H n = { z C : ϕ ( z , u n ) ϕ ( z , x n ) + N ξ n } , W n = { z C : x n z , J x 0 J x n 0 } , x n + 1 = Π H n W n x 0 ,

where ξ n =max{0, sup p F ( T ) , x C (ϕ(p, T n x)ϕ(p,x))}, { α n } is a real number sequence in (0,1), { r n } is a real number sequence in [k,), where k is some positive real number. Assume that lim inf n α n (1 α n )>0. Then the sequence { x n } converges strongly to Π F ( T ) EP ( f ) x 0 , where Π F ( T ) EP ( f ) is the generalized projection from E onto F(T)EP(f).

If the mapping T is quasi-ϕ-nonexpansive, we find from Corollary 3.3 the following.

Corollary 3.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C×C to satisfying (A1)-(A4). Let T:CC be a quasi-ϕ-nonexpansive mapping. Assume that F(T)EP(f) is nonempty. Let { x n } be a sequence generated in the following manner:

{ x 0 C chosen arbitrarily , y n = J 1 ( α n J x n + ( 1 α n ) J T x n ) , u n C  such that  f ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C , H n = { z C : ϕ ( z , u n ) ϕ ( z , x n ) } , W n = { z C : x n z , J x 0 J x n 0 } , x n + 1 = Π H n W n x 0 ,

where { α n } is a real number sequence in (0,1), { r n } is a real number sequence in [k,), where k is some positive real number. Assume that lim inf n α n (1 α n )>0. Then the sequence { x n } converges strongly to Π F ( T ) EP ( f ) x 0 , where Π F ( T ) EP ( f ) is the generalized projection from E onto F(T)EP(f).

Finally, we give a result in the framework of Hilbert spaces based on Theorem 3.1.

Corollary 3.5 Let E be a Hilbert space, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C×C to satisfying (A1)-(A4), and let N be some positive integer. Let A:CE be a κ i -inverse-strongly monotone mapping. Let T i :CC be an asymptotically quasi-nonexpansive mapping in the intermediate sense for every 1iN. Assume that T i is closed asymptotically regular on C and i = 1 N F( T i )EP(f,A) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:

{ x 0 C chosen arbitrarily , y n = α n , 0 x n + i = 1 N α n , i T i n x n , u n C  such that  f ( u n , y ) + A u n + y u n + 1 r n y u n , u n y n 0 , y C , H n = { z C : z u n 2 z x n 2 + N ξ n } , W n = { z C : x n z , x 0 x n 0 } , x n + 1 = Proj H n W n x 0 ,

where ξ n =max{0, sup p i = 1 N F ( T i ) , x C ( p T i n x 2 p x 2 )}, { α n , i } is a real number sequence in (0,1) for every 1iN, { r n } is a real number sequence in [k,), where k is some positive real number. Assume that i = 0 N α n , i =1 and lim inf n α n , 0 α n , i >0 for every 1iN. Then the sequence { x n } converges strongly to Proj i = 1 N F ( T i ) EP ( f , A ) x 0 , where Proj i = 1 N F ( T i ) EP ( f , A ) is the metric projection from E onto i = 1 N F( T i )EP(f,A).

Proof Since ϕ(x,y)= x y 2 and J=I in the framework of Hilbert spaces, we draw the desired conclusion immediately. □

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Qing, Y., Lv, S. A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2013, 305 (2013). https://doi.org/10.1186/1687-1812-2013-305

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