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J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
Research Article
Strong convergence of hybrid algorithms for fixed
point and bifunction equilibrium problems in
reflexive Banach spaces
Lingmin Zhanga,∗, Xinbin Lib
a
Institute of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei,
066004, China.
b
Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao, Hebei, 066004, China.
Communicated by C. Park
Abstract
Fixed point and bifunction equilibrium problems are studied via hybrid algorithms. Strong convergence
theorems are established in the framework of reflexive Banach spaces. The results presented in this paper
c
improve the corresponding results announced by many authors recently. 2016
All rights reserved.
Keywords: Asymptotically quasi-φ-nonexpansive mapping, equilibrium problem, fixed point, generalized
projection.
2010 MSC: 47J20, 47J25.
1. Introduction and Preliminaries
Bifunction equilibrium problem [4], which was introduced two decades ago, have emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, finance,
optimization, social, pure and applied sciences; see [3], [11], [12], [20], [22], [23] and the references therein.
Since bifunction equilibrium problem covers variational inequality problems, zero point problems, and variational inclusion problems, it has been investigated by many authors via fixed point algorithms; see, for
example, [5, 6, 8, 9, 10], [13]-[17], [24]-[28], [30] and the references therein.
Let E be a real Banach space and let E ∗ be the dual space of E. Let C be a nonempty subset of E
and let G be a bifunction from C × C to R, where R denotes the set of real numbers. Recall the following
equilibrium problem. Find x̄ ∈ C such that
∗
Corresponding author
Email address: zhanglm103@126.com (Lingmin Zhang)
Received 2015-09-01
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
G(x̄, y) ≥ 0,
1324
∀y ∈ C.
(1.1)
We use Sol(G) to denote the solution set of equilibrium problem (1.1). That is,
Sol(G) = {x ∈ C : 0 ≤ G(x, y)
∀y ∈ C}.
Given a mapping A : C → E ∗ , let
G(x, y) := hy − x, Axi
∀x, y ∈ C.
Then x̄ ∈ Sol(G) if x̄ is a solution of the following variational inequality. Find x̄ such that
hy − x̄, Ax̄i ≥ 0
∀y ∈ C.
(1.2)
In order to study the solution of problem (1.1), we assume that G satisfies the following conditions:
(C-1) G(x, x) = 0 ∀x ∈ C;
(C-2) 0 ≥ G(x, y) + G(y, x) ∀x, y ∈ C;
(C-3) G(x, y) ≥ lim supt↓0 G(tz + (1 − t)x, y) ∀x, y, z ∈ C;
(C-4) y 7→ G(x, y) is weakly lower semi-continuous and convex for each x ∈ C.
Recall that a Banach space E is said to be strictly convex iff k x+y
2 k < 1 for all x, y ∈ E with kxk = kyk = 1
and x 6= y. It is said to be uniformly convex if limn→∞ kxn − yn k = 0 for any two sequences {xn } and {yn }
in E such that
x n + yn
kxn k = kyn k = 1 and lim k
k = 1.
n→∞
2
Let UE = {x ∈ E : kxk = 1} be the unit sphere of E. Then the Banach space E is said to be smooth if
lim
t→0
kxk − kx + tyk
t
exists for each x, y ∈ UE . It is also said to be uniformly smooth if and only if the above limit is attained
uniformly for x, y ∈ UE . It is known that if E is uniformly smooth if and only if E ∗ is uniformly convex.
In this paper, we use → and * to denote the strong convergence and weak convergence, respectively.
Recall that a Banach space E has the Kadec-Klee property if for any sequence {xn } ⊂ E and x ∈ E with
xn * x and kxn k → kxk, then kxn − xk → 0 as n → ∞. It is well known that if E is a uniformly convex
Banach spaces, then E has the Kadec-Klee property.
∗
Recall that the normalized duality mapping J from E to 2E is defined by
Jx = {f ∗ ∈ E ∗ : kxk2 = hx, f ∗ i = kf ∗ k2 },
where h·, ·i denotes the generalized duality pairing.
It is known if E is uniformly smooth, then J is uniformly norm-to-norm continuous on every bounded
subset of E; if E is a smooth Banach space, then J is single-valued and demicontinuous, i.e., continuous
from the strong topology of E to the weak star topology of E; if E is smooth, strictly convex and reflexive
Banach space, then J is single-valued, one-to-one and onto.
Consider the functional defined by
φ(x, y) = kxk2 + kyk2 − 2hx, Jyi
∀x, y ∈ E.
Observe that, in a Hilbert space H, the equality is reduced to φ(x, y) = kx − yk2 for all x, y ∈ H. As
we all know that if C is a nonempty closed convex subset of a Hilbert space H and PC : H → C is the
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1325
metric projection of H onto C, then PC 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 ΠC in a Banach space E which is an analogue of the metric
projection PC in Hilbert spaces.
Recall that generalized projection ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the
minimum point of the functional φ(x, y), that is, ΠC x = x̄, where x̄ is the solution to the minimization
problem
φ(x̄, x) = min φ(y, x).
y∈C
Existence and uniqueness of the operator ΠC follows from the properties of the functional φ(x, y) and strict
monotonicity of the mapping J. In Hilbert spaces, ΠC = PC . It is obvious from the definition of function φ
that
(kyk + kxk)2 ≥ φ(x, y) ≥ (kxk − kyk)2 ∀x, y ∈ E,
(1.3)
and
φ(x, y) − φ(x, z) = φ(z, y) + 2hx − z, Jz − Jyi
∀x, y, z ∈ E.
(1.4)
Let T : C → C be a mapping. In this paper, we use F ix(T ) to denote the fixed point set of T. T is said
to be closed if for any sequence {xn } ⊂ C such that limn→∞ xn = x0 and limn→∞ T xn = y0 , then T x0 = y0 .
Recall that a point p in C is said to be an asymptotic fixed point of T if C contains a sequence {xn }
which converges weakly to p such that limn→∞ kxn − T xn k = 0. The set of asymptotic fixed points of T
g
will be denoted by F
ix(T ).
Recall that a mapping T is said to be relatively nonexpansive if
g
F
ix(T ) = F ix(T ) 6= ∅,
φ(p, T x) ≤ φ(p, x)
∀x ∈ C, ∀p ∈ F ix(T ).
Recall that a mapping T is said to be relatively asymptotically nonexpansive if
g
F
ix(T ) = F ix(T ) 6= ∅,
φ(p, T n x) ≤ kn φ(p, x) ∀x ∈ C, ∀p ∈ F ix(T ), ∀n ≥ 1,
where {kn } ⊂ [1, ∞) is a sequence such that kn → 1 as n → ∞.
Remark 1.1. The class of relatively asymptotically nonexpansive mappings, which covers the class of relatively nonexpansive mappings, was first considered in [1]. See also [21] and the references therein.
Recall that a mapping T is said to be quasi-φ-nonexpansive if
F ix(T ) 6= ∅,
φ(p, T x) ≤ φ(p, x)
∀x ∈ C, ∀p ∈ F ix(T ).
Recall that a mapping T is said to be asymptotically quasi-φ-nonexpansive if there exists a sequence
{kn } ⊂ [1, ∞) with kn → 1 as n → ∞ such that
F ix(T ) 6= ∅,
φ(p, T n x) ≤ kn φ(p, x) ∀x ∈ C, ∀p ∈ F ix(T ), ∀n ≥ 1.
Remark 1.2. The class of asymptotically quasi-φ-nonexpansive mappings, which covers the class of quasi-φnonexpansive mappings [18], was considered in Qin, Cho and Kang [19] and Agarwal and Qin [17]; see also
Zhou, Gao and Tan [31].
Remark 1.3. 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
g
quasi-φ-nonexpansive do not require the restriction F ix(T ) = F
ix(T ).
In order to our main results, we also need the following lemmas.
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1326
Lemma 1.4 ([4]). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space
E. Let G be a bifunction from C × C to R satisfying (C-1), (C-2), (C-3) and (C-4). Let r > 0 and x ∈ E.
Then there exists z ∈ C such that
rG(z, y) + hz − y, Jx − Jzi ≥ 0
∀y ∈ C.
Lemma 1.5 ([2]). Let E be a reflexive, strictly convex, and smooth Banach space, C a nonempty, closed,
and convex subset of E and x ∈ E. Then
φ(y, x) ≥ φ(y, ΠC x) + φ(ΠC x, 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
ΠC x = x0 ,
if and only if
hx0 − y, Jx0 − Jxi ≤ 0
∀y ∈ C.
Lemma 1.7 ([19]). Let E be a uniformly convex and smooth Banach space and 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 ix(T ) is a closed convex subset of C.
Lemma 1.8 ([18]). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space
E. Let G be a bifunction from C × C to R satisfying (C-1), (C-2), (C-3) and (C-4). Let r > 0 and x ∈ E.
Define a mapping Sr : E → C by
Sr x = {z ∈ C : rG(z, y) + hy − z, Jz − Jxi ≥ 0
∀y ∈ C}.
Then the following conclusions hold:
(1) Sr is single-valued;
(2) Sr is a firmly nonexpansive-type mapping, i.e. for all x, y ∈ E,
hSr x − Sr y, Jx − Jyi ≥ hSr x − Sr y, JSr x − JSr yi;
(3) Sr is quasi-φ-nonexpansive;
(4) φ(q, x) ≥ φ(q, Sr x) + φ(Sr x, x) ∀q ∈ F ix(Sr );
(5) Sol(G) = F ix(Sr ) is convex and closed.
2. Main results
Theorem 2.1. Let E be a uniformly convex and smooth Banach space such that E ∗ has the Kadec-Klee
property and let C be a nonempty closed and convex subset of E. Let Λ be an index set and let Gi be a
bifunction from C × C to R satisfying (C-1), (C-2), (C-3) and (C-4). Let Ti : C → C be an asymptotically
quasi-φ-nonexpansive mapping for every i ∈ Λ.T Assume that Ti is closed and uniformly asymptotically
regular on C for every i ∈ Λ and ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) is nonempty and bounded. Let {xn } be a
sequence generated in the following manner:
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333


x0 ∈ E chosen arbitrarily,




C(1,i) = C,




C1 = ∩i∈Λ C(1,i) ,





x1 = ΠC1 x0 ,
y(n,i) = J −1 (α(n,i) Jx1 + (1 − α(n,i) )JTin xn ),



u(n,i) ∈ Cn such that r(n,i) Gi (u(n,i) , y) + hy − u(n,i) , Ju(n,i) − Jy(n,i) i ≥ 0





C(n+1,i) = {z ∈ C(n,i) : φ(z, xn ) + α(n,i) D ≥ φ(z, u(n,i) )},





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = ΠCn+1 x1 ,
where D := sup{φ(w, x1 ) : p ∈ ∩i∈Λ F ix(Ti )
T
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∀y ∈ Cn ,
∩i∈Λ Sol(Gi )}, {α(n,i) } is a real sequence in (0, 1) such that
limn→∞ α(n,i) = 0 and {r(n,i) } is a real sequence in [ai , ∞), where {ai } is a positive real number sequence
for every i ∈ Λ. Then the sequence {xn } converges strongly to Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 .
T
Proof. Using Lemma 1.7 and Lemma 1.8, we find that ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) is convex and closed so
that the generalization projection onto the set is well defined.
Next, we prove that Cn is convex and closed. To show Cn is convex and closed, it suffices to show that,
for each fixed but arbitrary i ∈ Λ, C(n,i) is convex and closed. This can be proved by induction on n. It is
obvious that C(1,i) = C is convex and closed. Assume that C(m,i) is convex and closed for some m ≥ 1. Let
For z1 , z2 ∈ C(m+1,i) , we see that z1 , z2 ∈ C(m,i) . It follows that z = tz1 + (1 − t)z2 ∈ C(m,i) , where t ∈ (0, 1).
Notice that
φ(z1 , xn ) + α(m,i) D ≥ φ(z1 , u(m,i) )
and
φ(z2 , xn ) + α(m,i) D ≥ φ(z2 , u(m,i) ).
The above inequalities are equivalent to
kxm k2 + α(m,i) D ≥ 2hz1 , Jxm − Ju(m,i) i + ku(m,i) k2
and
kxm k2 + α(m,i) D ≥ 2hz2 , Jxm − Ju(m,i) i + ku(m,i) k2 .
Multiplying t and (1 − t) on the both sides of the inequalities above, respectively yields that and
kxm k2 − ku(m,i) k2 + α(m,i) D ≥ 2hz, Jxm − Ju(m,i) i.
That is,
φ(z, xn ) + α(m,i) D ≥ φ(z, u(m,i) ),
where z ∈ C(m,i) . This finds that C(m+1,i) is convex and closed. We conclude that C(n,i) is convex and closed.
This in turn implies that Cn = ∩i∈Λ C(n,i) is convex T
and closed. This implies that ΠCn+1 x1 is well defined.
Now, we are in a position to prove ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) ⊂ Cn . Note that
\
C = C1 ⊇ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi )
T
is clear. Suppose
that C(m,i) ⊇ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) for some positive integer m. For any w ∈
T
∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) ⊂ C(m,i) , we see that
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1328
φ(w, xm ) + α(m,i) D ≥ α(m,i) φ(w, x1 ) + (1 − α(m,i) )φ(w, Tim xm )
= kwk2 − 2α(m,i) hw, Jx1 i − 2(1 − α(m,i) )hw, JTim xm i
+ α(m,i) kx1 k2 + (1 − α(m,i) )kTim xm k2
≥ kwk2 − 2hw, α(m,i) Jx1 + (1 − α(m,i) )JTim xm i
+ kα(m,i) Jx1 + (1 − α(m,i) )JTim xm k2
= φ(w, J
−1
(α(m,i) Jx1 + (1 −
(2.1)
α(m,i) )JTim xm ))
= φ(w, y(m,i) )
≥ φ(w, Sr(m,i) y(m,i) )
= φ(w, u(m,i) ),
where
D := sup{φ(w, x1 ) : w ∈ ∩i∈Λ F (Ti )
\
∩i∈Λ Sol(Gi )}.
T
This shows that
T w ∈ C(m+1,i) . This implies that C(n,i) ⊇ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(G
T i ). Hence, ∩i∈Λ C(n,i) ⊇
∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ). This completes the proof that Cn ⊇ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ). In the light
of the construction ΠCn x1 = xn , we find from Lemma 1.6 that hxn − z, Jx1 − Jxn i ≥ 0 for any z ∈ Cn . Since
\
Cn ⊇ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ),
we find that
0 ≥ hw − xn , Jx1 − Jxn i,
∀w ∈ ∩i∈Λ F ix(Ti )
\
∩i∈Λ Sol(Gi ).
(2.2)
Using Lemma 1.5, one sees that
φ(Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 , x1 )
≥ φ(Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 , x1 ) − φ(Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 , xn )
≥ φ(xn , x1 ).
This implies that {φ(xn , x1 )} is bounded. Hence, {xn } is also bounded. Since the space is reflexive, we may
assume that xn * x̄. Since Cn is convex closed, we find Cn 3 x̄. This implies that φ(x̄, x1 ) ≥ φ(xn , x1 ). On
the other hand, we see from the weakly lower semicontinuity of the norm that
φ(x̄, x1 ) ≥ lim sup φ(xn , x1 )
n→∞
≥ lim inf φ(xn , x1 )
n→∞
= lim inf (kxn k2 − 2hxn , Jx1 i + kx1 k2 )
n→∞
2
≥ kx̄k − 2hx̄, Jx1 i + kx1 k2
= φ(x̄, x1 ),
which implies that limn→∞ φ(xn , x1 ) = φ(x̄, x1 ). Hence, we have limn→∞ kxn k = kx̄k. Using the Kadec-Klee
property of E, one has xn → x̄ as n → ∞. Since xn = ΠCn x1 , and
xn+1 = ΠCn+1 x1 ∈ Cn+1 ⊂ Cn ,
one sees φ(xn+1 , x1 ) ≥ φ(xn , x1 ). This shows that {φ(xn , x1 )} is nondecreasing. Since it is bounded, we find
that limn→∞ φ(xn , x1 ) exists. It follows that
φ(xn+1 , x1 ) − φ(xn , x1 ) ≥ φ(xn+1 , xn ).
This implies that
lim φ(xn+1 , xn ) = 0.
n→∞
(2.3)
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1329
In the light of Cn+1 3 ΠCn+1 x1 = xn+1 , we find that
φ(xn+1 , xn ) + α(n,i) D ≥ φ(xn+1 , u(n,i) ).
This implies from (2.3) that
lim φ(xn+1 , u(n,i) ) = 0.
n→∞
(2.4)
Using inequality (1.3), we see
lim (kxn+1 k − ku(n,i) k) = 0.
n→∞
This implies that
lim ku(n,i) k = kx̄k.
n→∞
On the other hand, we have
lim ku(n,i) k = kx̄k = lim kJu(n,i) k = kJ x̄k.
n→∞
n→∞
(2.5)
This implies that {Ju(n,i) } is bounded. Since both E and E ∗ are reflexive, we may assume that Ju(n,i) *
u(∗,i) ∈ E ∗ . Since E is reflexive, we see J(E) = E ∗ . This shows that there exists an element ui ∈ E such
that Jui = u(∗,i) . It follows that
φ(xn+1 , u(n,i) ) = kJu(n,i) k2 + kxn+1 k2 − 2hxn+1 , Ju(n,i) i.
Therefore, one has
0 ≥ kx̄k2 + ku(∗,i) k2 − 2hx̄, u(∗,i) i = kx̄k2 + kui k2 − 2hx̄, Jui i = φ(x̄, ui ) ≥ 0.
That is, x̄ = ui , which in turn implies that u(∗,i) = J x̄. It follows that Ju(n,i) * J x̄ ∈ E ∗ . Since E ∗ has the
Kadec-Klee property, we obtain from (2.5) that
lim Ju(n,i) = J x̄.
n→∞
Since J −1 : E ∗ → E is demi-continuous and E has the Kadec-Klee property, we obtain u(n,i) → x̄, as
n → ∞. Using (2.1) and (2.3), one has
lim φ(xn+1 , y(n,i) ) = 0.
n→∞
(2.6)
Using inequality (1.3), we see
lim (kxn+1 k − ky(n,i) k) = 0.
n→∞
This implies that
lim ky(n,i) k = kx̄k.
n→∞
On the other hand, we have
kx̄k = kJ x̄k = lim kJy(n,i) k = lim ky(n,i) k.
n→∞
n→∞
(2.7)
This implies that {Jy(n,i) } is bounded. Since both E and E ∗ are reflexive, we may assume that Jy(n,i) *
y (∗,i) ∈ E ∗ . Since E is reflexive, we see E ∗ = J(E). This shows that there exists an element y i ∈ E such
that y (∗,i) = Jy i . It follows that
kxn+1 k2 + kJy(n,i) k2 − 2hxn+1 , Jy(n,i) i = φ(xn+1 , y(n,i) ).
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1330
Therefore, one has
0 ≥ kx̄k2 + ky (∗,i) k2 − 2hx̄, y (∗,i) i
= kx̄k2 + ky i k2 − 2hx̄, Jy i i
= φ(x̄, y i ).
That is, x̄ = y i , which in turn implies that y (∗,i) = J x̄. It follows that Jy(n,i) * J x̄ ∈ E ∗ . Since E ∗ has the
Kadec-Klee property, we obtain from (2.7) that
lim Jy(n,i) = J x̄.
n→∞
Since J −1 : E ∗ → E is demi-continuous and E has the Kadec-Klee property, we obtain y(n,i) → x̄, as n → ∞.
In view of Sr(n,i) y(n,i) = u(n,i) , we see that
hy − u(n,i) , Ju(n,i) − Jy(n,i) i + r(n,i) Gi (u(n,i) , y) ≥ 0 ∀y ∈ Cn .
It follows from (C-2) that
ky − u(n,i) kkJu(n,i) − Jy(n,i) k ≥ r(n,i) Gi (y, u(n,i) )
∀y ∈ Cn .
In view of (C-4), one has Gi (y, x̄) ≤ 0 ∀y ∈ Cn . For 0 < ti < 1 and y ∈ C, define
y(t,i) = ti y + (1 − ti )x̄.
It follows that y(t,i) ∈ C, which yields that Gi (y(t,i) , x̄) ≤ 0. It follows from the (C-1) and (C-4) that
ti Gi (y(t,i) , y) ≥ ti Gi (y(t,i) , y) + (1 − ti )Gi (y(t,i) , x̄)
≥ Gi (y(t,i) , y(t,i) )
= 0.
That is, Gi (y(t,i) , y) ≥ 0. Letting ti ↓ 0, we obtain from (C-3) that 0 ≤ Gi (x̄, y), ∀y ∈ C. This implies that
x̄ ∈ Sol(Gi ) for every i ∈ Λ. This shows that x̄ ∈ ∩i∈Λ Sol(Gi ).
Next, we prove x̄ ∈ ∩i∈Λ F ix(Ti ). Using the condition imposed on {α(n,i) }, one has
lim kJy(n,i) − JTin xn k = 0.
n→∞
Using the fact
kJ x̄ − JTin xn k ≤ kJy(n,i) − J x̄k + kJy(n,i) − JTin xn k,
one has JTin xn → J x̄ as n → ∞, for every i ∈ Λ. Since J −1 is demi-continuous, we have Tin xn * x̄ for
every i ∈ Λ. Since
kJ(Tin xn ) − J x̄k ≥ |kTin xn k − kx̄k|,
one has kTin xn k → kx̄k, as n → ∞, for every i ∈ Λ. Since E has the Kadec-Klee property, one obtains
lim kTin xn − x̄k = 0.
n→∞
On the other hand, we have
kTin+1 xn − x̄k ≤ kTin+1 xn − Tin xn k + kTin xn − x̄k.
In view of the uniformly asymptotic regularity of Ti , one has
lim kTin+1 xn − x̄k = 0,
n→∞
that is, Ti Tin xn − x̄ → 0 as n → ∞. Since every Ti is closed, we find that Ti x̄ = x̄ for every i ∈ Λ.
Finally, we prove x̄ = Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 . Letting n → ∞ in (2.2), we see that
\
hx̄ − w, Jx1 − J x̄i ≥ 0, ∀w ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ).
In view of Lemma 1.6, we find that that x̄ = Π∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 . This completes the proof.
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1331
Remark 2.2. The algorithm is efficient for an uncountable infinite family of operators and bifunctions. The
space does not require the uniform smoothness. Theorem 2.1, which mainly improve the corresponding
results in [7], [19] and [29], unify the corresponding results announced recently.
From Theorem 2.1, the following result is not hard to derive.
Corollary 2.3. Let E be a uniformly convex and smooth Banach space such that E ∗ has the Kadec-Klee
property and let C be a nonempty, convex and closed subset of E. Let G be a bifunction from C × C to
R satisfying (C-1), (C-2), (C-3) and (C-4). Let T : C → C be an asymptotically quasi-φ-nonexpansive
mapping. Assume that T is closed and uniformly asymptotically regular on C and F ix(T ) ∩ Sol(G) is
nonempty and bounded. Let {xn } be a sequence generated in the following manner:


x0 ∈ E chosen arbitrarily,





C1 = C,




x1 = ΠC1 x0 ,

yn = J −1 ((1 − αn )JT n xn + αn Jx1 ),



un ∈ Cn such that rn G(un , y) + hy − un , Jun − Jyn i ≥ 0 ∀y ∈ Cn ,





Cn+1 = {z ∈ Cn : φ(z, xn ) + αn D ≥ φ(z, un )},




xn+1 = ΠCn+1 x1 ,
where D := sup{φ(w, x1 ) : p ∈ F ix(T ) ∩ Sol(G)}, {αn } is a real sequence in (0, 1) such that limn→∞ αn = 0,
and {rn } is a real sequence in [a, ∞), where a is a positive real number. Then the sequence {xn } converges
strongly to ΠF ix(T )∩Sol(G) x1 .
In Hilbert spaces, Theorem 2.1 is reduced to the following.
Corollary 2.4. Let E be a Hilbert space and let C be a nonempty closed and convex subset of E. Let Λ
be an index set and let Gi be a bifunction from C × C to R satisfying (C-1), (C-2), (C-3) and (C-4). Let
Ti : C → C be an asymptotically quasi-nonexpansive mapping for every i T
∈ Λ. Assume that Ti is closed
and uniformly asymptotically regular on C for every i ∈ Λ and ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi ) is nonempty and
bounded. Let {xn } be a sequence generated in the following manner:


x0 ∈ E chosen arbitrarily,





C(1,i) = C,





C1 = ∩i∈Λ C(1,i) ,





x1 = PC1 x0 ,
y(n,i) = α(n,i) x1 + (1 − α(n,i) )Tin xn ,



u(n,i) ∈ Cn such that r(n,i) Gi (u(n,i) , y) + hy − u(n,i) , u(n,i) − y(n,i) i ≥ 0 ∀y ∈ Cn ,





C(n+1,i) = {z ∈ C(n,i) : kz − xn k2 + α(n,i) D ≥ kz − u(n,i) k2 },





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = PCn+1 x1 ,
T
where D := sup{kw − x1 k2 : p ∈ ∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(Gi )}, {α(n,i) } is a real sequence in (0, 1) such that
limn→∞ α(n,i) = 0 and {r(n,i) } is a real sequence in [ai , ∞), where {ai } is a positive real number sequence
for every i ∈ Λ. Then the sequence {xn } converges strongly to P∩i∈Λ F ix(Ti ) T ∩i∈Λ Sol(Gi ) x1 .
Proof. In the framework of Hilbert space, the class of asymptotically quasi-φ-nonexpansive mappings is
reduced to the class of asymptotically quasi-nonexpansive mappings and the φ(x, y) = kx − yk2 . Using
Theorem 2.1, we can conclude the desired result immediately.
L. Zhang, X. Li, J. Nonlinear Sci. Appl. 9 (2016), 1323–1333
1332
Acknowledgement
The authors are grateful to the reviewers for the useful suggestions which improve the contents of this
article.
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