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On generalized quasi-ϕ-nonexpansive mappings and their projection algorithms

Fixed Point Theory and Applications20132013:204

https://doi.org/10.1186/1687-1812-2013-204

Received: 29 April 2013

Accepted: 11 July 2013

Published: 29 July 2013

Abstract

A fixed point problem of a generalized asymptotically quasi-ϕ-nonexpansive mapping and an equilibrium problem are investigated. A strong convergence theorem for solutions of the fixed point problem and the equilibrium problem is established in a Banach space.

Keywords

asymptotically quasi-ϕ-nonexpansive mappingequilibrium problemfixed pointgeneralized asymptotically quasi-ϕ-nonexpansive mappinggeneralized projection

1 Introduction and preliminaries

Let E be a real Banach space, and 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 } ,
where , denotes the generalized duality pairing. A Banach space E is said to be strictly convex if x + y 2 < 1 for all x , y E with x = y = 1 and x y . 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 = { x E : x = 1 } be the unit sphere of E. Then the Banach space E is said to be smooth provided
lim t 0 x + t y x t

exists for each x , y U E . It is also 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 E is uniformly smooth if and only if E is uniformly convex.

Recall that a Banach space E enjoys the Kadec-Klee property if for any sequence { x n } E , and x E with x n x , and x n x , then x n x 0 as n . For more details on the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Let C be a nonempty subset of E. Let f be a bifunction from C × C to , where denotes the set of real numbers. In this paper, we investigate the following equilibrium problem. Find p C such that
f ( p , y ) 0 , y C .
(1.1)
We use EP ( f ) to denote the solution set of the equilibrium problem (1.1). That is,
EP ( f ) = { p C : f ( p , y ) 0 , y C } .
Given a mapping Q : C E , let
f ( x , y ) = Q x , y x , x , y C .
Then p EP ( f ) iff p is a solution of the following variational inequality. Find p such that
Q p , y p 0 , y C .
(1.2)
In order to study the solution problem of the equilibrium problem (1.1), we assume that f satisfies the following conditions:
  1. (A1)

    f ( x , x ) = 0 , x C ;

     
  2. (A2)

    f is monotone, i.e., f ( x , y ) + f ( y , x ) 0 , x , y C ;

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

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

     

As we all know, if C is a nonempty closed convex subset of a Hilbert space H and P C : H C 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 [2] 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.

Next, we assume that E is a smooth Banach space. Consider the functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
Observe that in a Hilbert space H, the equality is reduced to ϕ ( x , y ) = x y 2 , x , y H . The generalized projection Π C : E C is a map that assigns to an arbitrary point x E 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; see, for example, [1] and [2]. 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 , y E
(1.3)
and
ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x z , J z J y , x , y , z E .
(1.4)

Remark 1.1 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 [1] and [3].

Let T : C C 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 n sup x K T n + 1 x T n x = 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 , T x 0 = y 0 . In this paper, we use → and to denote the strong convergence and weak convergence, respectively.

A point p in C is said to be an asymptotic fixed point of T [3] 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 ) . T is said to be relatively nonexpansive [4, 5] iff F ˜ ( T ) = F ( T ) and ϕ ( p , T x ) ϕ ( p , x ) for all x C and p F ( T ) . T is said to be relatively asymptotically nonexpansive [6, 7] iff F ˜ ( T ) = F ( T ) and there exists a sequence { μ n } [ 1 , ) with μ n 1 as n such that ϕ ( p , T x ) μ n ϕ ( p , x ) for all x C , p F ( T ) and n 1 . T is said to be quasi-ϕ-nonexpansive [8, 9] iff F ( T ) and ϕ ( p , T x ) ϕ ( p , x ) for all x C and p F ( T ) . T is said to be asymptotically quasi-ϕ-nonexpansive [1012] iff F ( T ) and there exists a sequence { μ n } [ 1 , ) with μ n 1 as n such that ϕ ( p , T x ) μ n ϕ ( p , x ) for all x C , p F ( T ) and n 1 .

Remark 1.2 The class of asymptotically quasi-ϕ-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings which requires the restriction F ( T ) = F ˜ ( T ) .

Remark 1.3 The classes of asymptotically quasi-ϕ-nonexpansive mappings and quasi-ϕ-nonexpansive mappings are the generalizations of asymptotically quasi-nonexpansive mappings and quasi-nonexpansive mappings in Hilbert spaces.

Recently, Qin et al. [13] introduced a class of generalized asymptotically quasi-ϕ-nonexpansive mappings. Recall that a mapping T is said to be generalized asymptotically quasi-ϕ-nonexpansive iff F ( T ) and there exist a sequence { μ n } [ 1 , ) with μ n 1 as n and a sequence { ν n } [ 0 , ) with ν n 0 as n such that ϕ ( p , T x ) μ n ϕ ( p , x ) + ν n for all x C , p F ( T ) and n 1 .

Remark 1.4 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of generalized asymptotically quasi-nonexpansive mappings which was studied in [14].

Recently, fixed point and equilibrium problems (1.1) have been intensively investigated based on iterative methods; see [1528]. The projection method which grants strong convergence of the iterative sequences is one of efficient methods for the problems. In this paper, we investigate the equilibrium problem (1.1) and a fixed point problem of the generalized quasi-ϕ-nonexpansive mapping based on a projection method. A strong convergence theorem for solutions of the equilibrium and the fixed point problem is established in a Banach space.

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

Lemma 1.5 [2]

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and x E . Then
ϕ ( y , Π C x ) + ϕ ( Π C x , x ) ϕ ( y , x ) , y C .

Lemma 1.6 [2]

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

Lemma 1.7 [11]

Let E be a reflexive, strictly convex, and smooth Banach space such that both E and E have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let T : C C be a closed asymptotically quasi-ϕ-nonexpansive mapping. Then F ( T ) is a closed convex subset of C.

Lemma 1.8 [29, 30]

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from C × C to satisfying (A1)-(A4). Let r > 0 and x E . Then there exists z C such that f ( z , y ) + 1 r y z , J z J x 0 , y C . Define a mapping S r : E C by S r x = { z C : f ( z , y ) + 1 r y z , J z J x 0 , y C } . Then the following conclusions hold:
  1. (1)
    S r is a single-valued and firmly nonexpansive-type mapping, i.e., for all x , y E ,
    S r x S r y , J S r x J S r y S r x S r y , J x J y ;
     
  2. (2)

    F ( S r ) = EP ( f ) is closed and convex;

     
  3. (3)

    S r is quasi-ϕ-nonexpansive;

     
  4. (4)

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

     

Lemma 1.9 [31]

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 , 2 r ] R such that g ( 0 ) = 0 and
t x + ( 1 t ) y 2 t x 2 + ( 1 t ) y 2 t ( 1 t ) g ( x y )

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

2 Main results

Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. 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 T : C C be a closed generalized asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that F = F ( T ) EP ( f ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , 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 , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + ( μ n 1 ) M n + ν n } , x n + 1 = Π C n + 1 x 1 ,

where M n = sup { ϕ ( z , x n ) : z F } , { α n } is a real sequence in [ 0 , 1 ] such that lim inf n α n ( 1 α n ) > 0 , and { r n } is a real sequence in [ a , ) , where a is some positive real number. Then the sequence { x n } converges strongly to Π F x 1 .

Proof In view of Lemma 1.7 and Lemma 1.8, we find that is closed and convex, so that Π F x is well defined for any x C . Next, we show that C n is closed and convex. It is obvious that C 1 = C is closed and convex. Suppose that C m is closed and convex for some m N . We now show that C m + 1 is also closed and convex. For z 1 , z 2 C m + 1 , we see that z 1 , z 2 C m . It follows that z = t z 1 + ( 1 t ) z 2 C m , where t ( 0 , 1 ) . Notice that
ϕ ( z 1 , u m ) ϕ ( z 1 , x m ) + ( μ m 1 ) M m + ν m
and
ϕ ( z 1 , u h ) ϕ ( z 1 , x m ) + ( μ m 1 ) M m + ν m .
The above inequalities are equivalent to
2 z 1 , J x m J u m x m 2 u m 2 + ( μ m 1 ) M m + ν m
(2.1)
and
2 z 2 , J x m J u m x m 2 u m 2 + ( μ m 1 ) M m + ν m .
(2.2)
Multiplying t and ( 1 t ) on both sides of (2.1) and (2.2), respectively, yields that
2 z , J x m J u m x m 2 y m 2 + ( μ m 1 ) M m + ν m .
That is,
ϕ ( z , u m ) ϕ ( z , x m ) + ( μ m 1 ) M m + ν m ,
(2.3)

where z C m . This gives that C m + 1 is closed and convex. Then C n is closed and convex. This shows that Π C n + 1 x 1 is well defined.

Next, we show that F C n . F C 1 = C is obvious. Suppose that F C m for some m N . Fix w F C m . It follows that
ϕ ( w , u m ) = ϕ ( w , S r m y m ) ϕ ( w , y m ) = ϕ ( w , J 1 ( α m J x m + ( 1 α m ) J T m x m ) ) = w 2 2 w , α m J x m + ( 1 α m ) J T m x m + α m J x m + ( 1 α m ) J T m x m 2 w 2 2 α m w , J x m 2 ( 1 α m ) w , J T m x m + α m x m 2 + ( 1 α m ) T m x m 2 = α m ϕ ( w , x m ) + ( 1 α m ) ϕ ( w , T m x m ) α m ϕ ( w , x m ) + ( 1 α m ) μ m ϕ ( w , x m ) + ( 1 α m ) ν m = ϕ ( w , x m ) ( 1 α m ) ϕ ( w , x m ) + ( 1 α m ) μ m ϕ ( w , x m ) + ( 1 α m ) ν m ϕ ( w , x m ) + ( 1 α m ) ( μ m 1 ) ϕ ( w , x m ) + ν m ϕ ( w , x m ) + ( μ m 1 ) M m + ν m ,
(2.4)
which shows that w C m + 1 . This implies that F C n for each n 1 . In view of x n = Π C n x 1 , from Lemma 1.6 we find that x n z , J x 1 J x n 0 for any z C n . It follows from F C n that
x n w , J x 1 J x n 0 , w F .
(2.5)
It follows from Lemma 1.5 that
ϕ ( x n , x 1 ) = ϕ ( Π C n x 1 , x 1 ) ϕ ( Π F x 1 , x 1 ) ϕ ( Π F x 1 , x n ) ϕ ( Π F x 1 , x 1 ) .
This implies that the sequence { ϕ ( x n , x 1 ) } is bounded. It follows from (1.3) that the sequence { x n } is also bounded. Since the space is reflexive, we may assume that x n x ¯ . Since C n is closed and convex, we find that x ¯ C n . On the other hand, we see from the weak lower semicontinuity of the norm that
ϕ ( x ¯ , x 1 ) = x ¯ 2 2 x ¯ , J x 1 + x 1 2 lim inf n ( x n 2 2 x n , J x 1 + x 1 2 ) = lim inf n ϕ ( x n , x 1 ) lim sup n ϕ ( x n , x 1 ) ϕ ( x ¯ , x 1 ) ,

which implies that lim n ϕ ( x n , x 1 ) = ϕ ( x ¯ , x 1 ) . 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 .

Now, we are in a position to prove that x ¯ F ( T ) . Since x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we find that ϕ ( x n , x 1 ) ϕ ( x n + 1 , x 1 ) . This shows that { ϕ ( x n , x 1 ) } is nondecreasing. It follows from the boundedness that lim n ϕ ( x n , x 1 ) exists. In view of the construction of x n + 1 = Π C n + 1 x 1 C n + 1 C n , we arrive at
ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π C n x 1 ) ϕ ( x n + 1 , x 1 ) ϕ ( Π C n x 1 , x 1 ) = ϕ ( x n + 1 , x 1 ) ϕ ( x n , x 1 ) .
This implies that
lim n ϕ ( x n + 1 , x n ) = 0 .
(2.6)
In light of x n + 1 = Π C n + 1 x 1 C n + 1 , we find that
ϕ ( x n + 1 , u n ) ϕ ( x n + 1 , x n ) + ( μ n 1 ) M n + ν n .
Thanks to (2.6), we find that
lim n ϕ ( x n + 1 , u n ) = 0 .
(2.7)
In view of (1.3), we see that lim n ( x n + 1 u n ) = 0 . It follows that lim n u n = x ¯ . This is equivalent to
lim n J u n = J x ¯ .
(2.8)
This implies that { J u n } is bounded. Note that both E and E are reflexive. We may assume 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 u E such that J u = 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 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 (2.8) that
lim n J u n = J x ¯ .
Since 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 .
(2.9)
This gives that
lim n J x n J u n = 0 .
(2.10)
Notice that
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 .
It follows from (2.9) and (2.10) that
lim n ( ϕ ( w , x n ) ϕ ( w , u n ) ) = 0 .
(2.11)
Since E is uniformly smooth, we know that E is uniformly convex. In view of Lemma 1.9, we see that
ϕ ( w , u n ) = ϕ ( w , S r n y n ) ϕ ( w , y n ) = ϕ ( w , J 1 [ α n J x n + ( 1 α n ) J T n x n ] ) = w 2 2 w , α n J x n + ( 1 α n ) J T n x n + α n J x n + ( 1 α n ) J T n x n 2 w 2 2 α n w , J x n 2 ( 1 α n ) w , J T n x n + α n x n 2 + ( 1 α n ) T n x n 2 α n ( 1 α n ) g ( J x n J ( T n x n ) ) = α n ϕ ( w , x n ) + ( 1 α n ) ϕ ( w , T n x n ) α n ( 1 α n ) g ( J x n J ( T n x n ) ) α n ϕ ( w , x n ) + ( 1 α n ) μ n ϕ ( w , x n ) + ν n α n ( 1 α n ) g ( J x n J ( T n x n ) ) ϕ ( w , x n ) + ( 1 α n ) ( μ n 1 ) ϕ ( w , x n ) + ν n α n ( 1 α n ) g ( J x n J ( T n x n ) ) .
This implies that
α n ( 1 α n ) g ( J x n J ( T n x n ) ) ϕ ( w , x n ) ϕ ( w , u n ) + ( 1 α n ) ( μ n 1 ) ϕ ( w , x n ) + ν n .
In view of the restrictions on the sequence { α n } , we find from (2.11) that
lim n J ( T n x n ) J x n = 0 .
(2.12)
Notice that
J ( T n x n ) J x ¯ J ( T n x n ) J x n + J x n J x ¯ .
It follows from (2.12) that
lim n J ( T n x n ) J x ¯ = 0 .
(2.13)
The demicontinuity of J 1 : E E implies that T n x n x ¯ . Note that
| T n x n x ¯ | = | J ( T n x n ) J x ¯ | J ( T n x n ) J x ¯ .
This implies from (2.13) that lim n T n x n = x ¯ . Since E has the Kadec-Klee property, we obtain that lim n T n x n x ¯ = 0 . Since
T n + 1 x n x ¯ T n + 1 x n T n x n + T n x n x ¯ ,

we find from the asymptotic regularity of T that lim n T n + 1 x n x ¯ = 0 , that is, T T n x n x ¯ 0 as n . It follows from the closedness of T that T x ¯ = x ¯ .

Next, we show that x ¯ EP ( f ) . In view of Lemma 1.8, we find from (2.4) that
ϕ ( u n , y n ) ϕ ( w , y n ) ϕ ( w , u n ) ϕ ( w , x n ) + ( μ n 1 ) M n + ν n ϕ ( w , u n ) = ϕ ( w , x n ) ϕ ( w , u n ) + ( k n 1 ) M n .
(2.14)
It follows from (2.11) that lim n ϕ ( u n , y n ) = 0 . This implies that lim n ( u n y n ) = 0 . In view of (2.9), we see that u n x ¯ as n . This implies that y n x ¯ 0 as n . It follows that lim n J y n = J x ¯ . Since E is reflexive, we may assume that J y n r E . In view of J ( E ) = E , we see that there exists r E such that J r = r . It follows that
ϕ ( u n , y n ) = u n 2 2 u n , J y n + y n 2 = u n 2 2 u n , J y n + J y n 2 .
Taking lim inf n on both sides of the equality above yields that
0 x ¯ 2 2 x ¯ , r + r 2 = x ¯ 2 2 x ¯ , J r + J r 2 = x ¯ 2 2 x ¯ , J r + r 2 = ϕ ( x ¯ , r ) .
That is, x ¯ = r , which in turn implies that r = 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 demicontinuous. 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 .
(2.15)
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
lim n J u n J y n = 0 .
From the assumption r n a , we see that
lim n J u n J y n r n = 0 .
(2.16)
In view of u n = S r n y n , we see that
f ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C .
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 ( u n , y ) f ( y , u n ) , y C .
By taking the limit as n in the above inequality, from (A4) and (2.16) we obtain that
f ( y , x ¯ ) 0 , y C .
For 0 < t < 1 and y C , define y t = t y + ( 1 t ) 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 ) t f ( y t , y ) + ( 1 t ) f ( y t , x ¯ ) t f ( y t , y ) .
That is,
f ( y t , y ) 0 .

Letting t 0 , we obtain from (A3) that f ( x ¯ , y ) 0 , y C . This implies that x ¯ EP ( f ) . This shows that x ¯ F = EP ( f ) F ( T ) .

Finally, we prove that x ¯ = Π F x 1 . Letting n in (2.5), we see that
x ¯ w , J x 1 J x ¯ 0 , w F .

In view of Lemma 1.6, we find that x ¯ = Π F x 1 . This completes the proof. □

If T is asymptotically quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.

Corollary 2.2 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. 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 T : C C be a closed asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and that F = F ( T ) EP ( f ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , 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 , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + ( μ n 1 ) M n } , x n + 1 = Π C n + 1 x 1 ,

where M n = sup { ϕ ( z , x n ) : z F } , { α n } is a real sequence in [ 0 , 1 ] such that lim inf n α n ( 1 α n ) > 0 , and { r n } is a real sequence in [ a , ) , where a is some positive real number. Then the sequence { x n } converges strongly to Π F x 1 .

If T is quasi-ϕ-nonexpansive, then Theorem 2.1 is reduced to the following.

Corollary 2.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. 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 T : C C be a closed quasi-ϕ-nonexpansive mapping. Assume that F = F ( T ) EP ( f ) is nonempty. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , 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 , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) } , x n + 1 = Π C n + 1 x 1 ,

where { α n } is a real sequence in [ 0 , 1 ] such that lim inf n α n ( 1 α n ) > 0 , and { r n } is a real sequence in [ a , ) , where a is some positive real number. Then the sequence { x n } converges strongly to Π F x 1 .

If T is the identity, then Theorem 2.1 is reduced to the following.

Corollary 2.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4). Assume that EP ( f ) is nonempty. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , u n C such that f ( u n , y ) + 1 r n y u n , J u n J x n 0 , y C , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + ( μ n 1 ) M n + ν n } , x n + 1 = Π C n + 1 x 1 ,

where M n = sup { ϕ ( z , x n ) : z F } , { α n } is a real sequence in [ 0 , 1 ] such that lim inf n α n ( 1 α n ) > 0 , and { r n } is a real sequence in [ a , ) , where a is some positive real number. Then the sequence { x n } converges strongly to Π EP ( f ) x 1 .

Declarations

Acknowledgements

The study was supported by the Natural Science Foundation of Zhejiang Province (Y6110270).

Authors’ Affiliations

(1)
School of Mathematics Physics and Information Science, Zhejiang Ocean University, Zhoushan, China

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