Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
Research Article
Strong convergence of hybrid Halpern processes in a
Banach space
Yuan Hecaia,∗, Zhaocui Minb
a
School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Henan, China.
b
School of Science, Hebei University of Engineering, Hebei, China.
Communicated by N. Hussain
Abstract
The purpose of this paper is to investigate convergence of a hybrid Halpern process for fixed point and
equilibrium problems. Strong convergence theorems of common solutions are established in a strictly convex
c
and uniformly smooth Banach space which also has the Kadec-Klee property. 2016
All rights reserved.
Keywords: Asymptotically nonexpansive mapping, fixed point, quasi-φ-nonexpansive mapping,
equilibrium problem, generalized projection.
2010 MSC: 65J15, 90C30.
1. Introduction and Preliminaries
Let E be a real Banach space and let E ∗ be the dual space of E. Let C be a nonempty subset of a E.
Let g be a bifunction from C × C to R, where R denotes the set of real numbers. Recall that the following
equilibrium problem [4]: Find x̄ ∈ C such that
g(x̄, y) ≥ 0
∀y ∈ C.
(1.1)
We use Sol(g) to denote the solution set of equilibrium problem (1.1). That is,
Sol(g) = {x ∈ C : g(x, y) ≥ 0
∀y ∈ C}.
Given a mapping A : C → E ∗ , let
G(x, y) = hAx, y − xi
∗
Corresponding author
Email address: hsyuanhc@yeah.net (Yuan Hecai)
Received 2015-09-12
∀x, y ∈ C.
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1777
Then x̄ ∈ Sol(g) iff x̄ is a solution of the following variational inequality. Find x̄ such that
hAx̄, y − x̄i ≥ 0
∀y ∈ C.
(1.2)
The following restrictions (R-a), (R-b), (R-c) and (R-d) imposed on g are essential in this paper.
(R-a) g(y, x) + g(x, y) ≤ 0 ∀x, y ∈ C;
(R-b) g(x, x) = 0 ∀x ∈ C;
(R-c) y 7→ g(x, y) is weakly lower semi-continuous and convex ∀x ∈ C;
(R-d) g(x, y) ≥ lim supt↓0 g(tz + (1 − t)x, y), ∀x, y, z ∈ C.
Equilibrium problem (1.1) is a bridge between nonlinear functional analysis and convex optimization
theory. Many problems arising in economics, medicine, engineering and physics can be studied via the
problem; see [3, 7, 8, 9, 10, 12, 14, 18, 19, 25] and the references therein.
∗
Recall that the normalized duality mapping J from E to 2E is defined by
Jx := {x∗ ∈ E ∗ : kx∗ k2 = hx, x∗ i = kxk2 }.
Let S E be the unit sphere of E. Recall that E is said to be a strictly convex space iff kx + yk < 2 for all
x, y ∈ S E and x 6= y. Recall that E is said to have a Gâteaux differentiable norm iff limt→∞ (ktx + yk − tkxk)
exists ∀x, y ∈ S E . In this case, we also say that E is smooth. E is said to have a uniformly Gâteaux
differentiable norm if for every y ∈ S E , the limit is attained uniformly for each x ∈ S E . E is also said to
have a uniformly Fréchet differentiable norm iff the above limit is attained uniformly for each x, y ∈ S E . In
this case, we say that E is uniformly smooth. 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 singlevalued and demicontinuous, i.e., continuous from the strong topology of E to the weak star topology of E;
if E is a strictly convex Banach space, then J is strictly monotone; if E is a reflexive and strictly convex
Banach space with a strictly convex dual E ∗ and J ∗ : E ∗ → E is the normalized duality mapping in E ∗ ,
then J −1 = J ∗ ; if E is a smooth, strictly convex and reflexive Banach space, then J is single-valued, oneto-one and onto; if E is a uniformly smooth, then it is smooth and reflexive. It is also known that E ∗ is
uniformly convex if and only if E is uniformly smooth. From now on, we use * and → to stand for the weak
convergence and strong convergence, respectively. Recall that E is said to have the Kadec-Klee property
if limn→∞ kxn − xk = 0 as n → ∞ for any sequence {xn } ⊂ E and x ∈ E with xn * x and kxn k → kxk
as n → ∞. It is well known that if E is a uniformly convex Banach spaces, then E has the Kadec-Klee
property; see [11] and the references therein.
Let f be a mapping on C. Recall that a point p is said to be a fixed point of f if and only if p = f p. p is
said to be an asymptotic fixed point of f if and only if C contains a sequence {xn }, where xn * p such that
g
xn − f xn → 0. From now on, We use F ix(f ) to stand for the fixed point set and F
ix(f ) to stand for the
asymptotic fixed point set. f is said to be closed if for any sequence {xn } ⊂ C such that limn→∞ xn = x0
and limn→∞ f xn = y 0 , then f x0 = y 0 .
Next, we assume that E is a smooth Banach space. Consider the functional defined on E by
φ(x, y) = kxk2 + kyk2 − 2hx, Jyi
∀x, y ∈ E.
Let C be a closed convex subset of a real Hilbert space H. For any x ∈ H, there exists a unique nearest
point in C, denoted by PC x, such that kx − PC xk ≤ kx − yk for all y ∈ C. The operator PC is called the
metric projection from H onto C. It is known that PC is firmly nonexpansive. In [2], Alber studied a new
mapping ΠC in a Banach space E which is an analogue of PC , the metric projection, in Hilbert spaces.
Recall that the generalized projection ΠC : E → C is a mapping that assigns to an arbitrary point x ∈ E
the minimum point of φ(x, y). It is obvious from the definition of function φ that
(kyk + kxk)2 ≥ φ(x, y) ≥ (kxk − kyk)2
∀x, y ∈ E,
(1.3)
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1778
and
φ(x, y) − 2hz − x, Jz − Jyi = φ(x, z) + φ(z, y)
∀x, y, z ∈ E.
(1.4)
Remark 1.1. If E is a strictly convex, reflexive and smooth Banach space, then φ(x, y) = 0 iff x = y.
Recall that a mapping f is said to be relatively nonexpansive ([5]) iff
φ(p, x) ≥ φ(p, f x)
g
∀x ∈ C, ∀p ∈ F
ix(f ) = F ix(f ) 6= ∅.
f is said to be relatively asymptotically nonexpansive ([1]) iff
g
φ(p, f n x) ≤ (1 + µn )φ(p, x) ∀x ∈ C, ∀p ∈ F
ix(f ) = F ix(f ) 6= ∅, ∀n ≥ 1,
where {µn } ⊂ [0, ∞) is a sequence such that µn → 0 as n → ∞.
Remark 1.2. The class of relatively asymptotically nonexpansive mappings, which include the class of relatively nonexpansive mappings ([5]) as a special case, were first considered in [1] and [26]; see the references
therein.
f is said to be quasi-φ-nonexpansive ([21]) iff
φ(p, x) ≥ φ(p, f x)
∀x ∈ C, ∀p ∈ F ix(f ) 6= ∅.
f is said to be asymptotically quasi-φ-nonexpansive ([22]) iff there exists a sequence {µn } ⊂ [0, ∞) with
µn → 0 as n → ∞ such that
φ(p, f n x) ≤ (1 + µn )φ(p, x)
∀x ∈ C, ∀p ∈ F ix(f ) 6= ∅, ∀n ≥ 1.
Remark 1.3. The class of asymptotically quasi-φ-nonexpansive mappings, which include the class of quasiφ-nonexpansive mappings ([21]) as a special case, were first considered in [20] and [22]; see the references
therein.
Remark 1.4. The class of asymptotically quasi-φ-nonexpansive mappings is more desirable than the class
of asymptotically relatively nonexpansive mappings. Quasi-φ-nonexpansive mappings and asymptotically
g
quasi-φ-nonexpansive do not require F ix(f ) = F
ix(f ).
Recently, Qin and Wang ([23]) introduced the asymptotically quasi-φ-nonexpansive mapping in the intermediate sense, which is a generalization of asymptotically quasi-nonexpansive mapping in the intermediate
sense in Banach spaces. Recall that f is said to be asymptotically quasi-φ-nonexpansive in the intermediate
sense iff F ix(f ) 6= ∅ and
lim sup
sup
φ(p, f n x) − φ(p, x) ≤ 0.
n→∞ p∈F ix(f ),x∈C
The so called convex feasibility problems which capture lots of applications in various subjects are to
find a special point in the intersection of convex (solution) sets. Recently, many author studied fixed points
of nonexpansive mappings and equilibrium (1.1); see [6], [13], [15]-[17], [24], [27]-[33] and the references
therein. The aim of this paper is to investigate convergence of a hybrid Halpern process for fixed point and
the equilibrium problem. Strong convergence theorems of common solutions are established in a strictly
convex and uniformly smooth Banach space which also has the Kadec-Klee property. In order to our main
results, we also need the following lemmas.
Lemma 1.5 ([2]). Let E be a strictly convex, reflexive and smooth Banach space and let C be a convex and
closed subset of E. Let x ∈ E. Then
φ(y, ΠC x) ≤ φ(y, x) − φ(ΠC x, x)
∀y ∈ C.
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1779
Lemma 1.6 ([4]). Let C be a convex and closed subset of a smooth Banach space E and let x ∈ E. Then
hy − x0 , Jx − Jx0 i ≤ 0 ∀y ∈ C iff x0 = ΠC x.
Lemma 1.7 ([4], [21]). 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 (R-a), (R-b), (R-c) and (R-d). Let r > 0 and
x ∈ E. Then
(a) There exists z ∈ C such that
hy − z, Jz − Jxi + rg(z, y) ≥ 0
∀y ∈ C.
(b) Define a mapping τr : E → C by
τr x = {z ∈ C : hy − z, Jz − Jxi + rg(z, y) ≥ 0
∀y ∈ C}.
Then the following conclusions hold:
(1) τr is single-valued;
(2) τr is a firmly nonexpansive-type mapping, i.e., for all x, y ∈ E,
hτr x − τr y, Jx − Jyi ≥ hτr x − τr y, Jτr x − Jτr yi;
(3)
(4)
(5)
(6)
F ix(τr ) = Sol(g);
τr is quasi-φ-nonexpansive;
φ(q, τr x) ≤ φ(q, x) − φ(τr x, x)
Sol(g) is convex and closed.
∀q ∈ F (τr );
Lemma 1.8 ([23]). 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 : C → C be a closed
asymptotically quasi-φ-nonexpansive mapping in the intermediate sense. Then F ix(f ) is a convex closed
subset of C.
2. Main results
Theorem 2.1. Let E be a strictly convex and uniformly smooth Banach space which also has the KadecKlee property. Let C be a convex and closed subset of E and let Λ be an index set. Let gi be a bifunction
from C × C to R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be an asymptotically quasi-φnonexpansive mapping in the intermediate sense for every i ∈ Λ.
T Assume that fi is continuous and uniformly
asymptotically regular on C for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩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,




x1 = ΠC1 :=∩i∈Λ C(1,i) x0 ,

y(n,i) = J −1 ((1 − α(n,i) )Jfin z(n,i) + α(n,i) Jx1 ),



C(n+1,i) = {z ∈ C(n,i) : φ(z, xn ) + α(n,i) D + (1 − α(n,i) )ξ(n,i) ≥ φ(z, y(n,i) )},





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = ΠCn+1 x1 ,
T
where ξ(n,i) = max{0, supp∈F ix(fi ),x∈C φ(p, fin x) − φ(p, x) , D = sup{φ(w, x1 ) : w ∈ ∩i∈Λ F ix(fi )
∩i∈Λ Sol(gi )}, z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥ hz(n,i) − y, Jz(n,i) − Jxn i ∀y ∈ Cn , {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 [ri , ∞), where
{ri } is a positive real number sequence for every i ∈ Λ. Then the sequence {xn } converges strongly to
Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1780
Proof. We divide the proof into six steps.
T
Step 1. We prove that ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) is convex and closed.
In the light of Lemma 1.7Tand Lemma 1.8, we easily find the conclusion. This shows that the generalized
projection onto ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) is well defined.
Step 2. We prove that Cn is convex and closed.
C(1,i) = C is convex and closed. Next, we assume that C(k,i) is convex and closed for some k ≥
1. For q1 , q2 ∈ C(k+1,i) ⊂ C(k,i) , we have q = tq1 + (1 − t)q2 ∈ C(k,i) , where t ∈ (0, 1). Notice that
φ(q1 , xk ) + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ φ(q1 , y(k,i) ) and φ(q2 , xk ) + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ φ(q2 , y(k,i) ).
The above inequalities are equivalent to
kxk k2 − ky(k,i) k2 + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ 2hq1 , Jxk − Jy(k,i) i
and
kxk k2 − ky(k,i) k2 + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ 2hq2 , Jxk − Jy(k,i) i.
Using the above inequalities, we find that
kxk k2 − ky(k,i) k2 + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ 2hq, Jxk − Jy(k,i) i.
That is,
φ(q, xk ) + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ φ(q, y(k,i) ),
where q ∈ C(k,i) . This finds that C(k+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)
T is convex and closed. Hence, ΠCn+1 x1 is well defined.
Step 3. We prove
that ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) ⊂ Cn .
T
T
∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) ⊂ C1 = C is clear.
T Suppose that ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) ⊂ C(k,i) for some
positive integer k. For any w ∈ ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) ⊂ C(k,i) , we see that
φ(w, xk ) + α(k,i) D + (1 − α(k,i) )ξ(k,i) ≥ φ(w, xk ) + α(k,i) φ(w, x1 ) − α(k,i) φ(w, xk ) + (1 − α(k,i) )ξ(k,i)
≥ α(k,i) φ(w, x1 ) + (1 − α(k,i) )φ(w, τ(k,i) xk ) + (1 − α(k,i) )ξ(k,i)
= α(k,i) φ(w, x1 ) + (1 − α(k,i) )φ(w, fik z(k,i) )
≥ kwk2 + α(k,i) kx1 k2 + (1 − α(k,i) )kfik z(k,i) k2
− 2(1 − α(k,i) )hw, Jfik z(k,i) i − 2α(k,i) hw, Jx1 i
2
= kwk + kα(k,i) Jx1 + (1 − α(k,i) )Jfik z(k,i) )k2
− 2hw, α(k,i) Jx1 + (1 − α(k,i) )Jfik z(k,i) )i
= φ(w, J −1 (α(k,i) Jx1 + (1 − α(k,i) )Jfik z(k,i) ))
= φ(w, y(k,i) ),
where D := supw∈∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) φ(w, x1 ). This proves w ∈ C(k+1,i) . Hence, we have
∩i∈Λ F ix(fi )
This in turn implies that ∩i∈Λ F ix(fi )
T
\
∩i∈Λ Sol(gi ) ⊂ C(n,i) .
∩i∈Λ Sol(gi ) ⊂ ∩i∈Λ C(n,i) . It follows that
∩i∈Λ F ix(fi )
\
∩i∈Λ Sol(gi ) ⊂ Cn .
Step 4. We prove that {xn } is a bounded sequence.
Using Lemma 1.6, we see
hxn − z, Jx1 − Jxn i ≥ 0 ∀z ∈ Cn .
(2.1)
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
Since ∩i∈Λ F ix(fi )
T
1781
∩i∈Λ Sol(gi ) is subset of Cn , we find that
hxn − w, Jx1 − Jxn i ≥ 0 ∀w ∈ ∩i∈Λ F (Ti )
\
∩i∈Λ EF (fi ).
(2.2)
Using Lemma 1.5, we get
φ(Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 , xn ) + φ(xn , x1 ) ≤ φ(Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 , x1 ).
Hence, we have
φ(xn , x1 ) ≤ φ(Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 , x1 ).
This implies that {φ(xn , x1 )} is a bounded sequence. It follows from (1.3) that sequence {xn } is also a
bounded sequence.
T
Step 5. Since the space is reflexive, we may assume that xn * x̄. We prove x̄ ∈ ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ).
Since Cn is convex and closed, we have x̄ ∈ Cn . Hence, φ(xn , x1 ) ≤ φ(x̄, 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 (kxn k2 + kx1 k2 − 2hxn , Jx1 i)
n→∞
2
= kx̄k + kx1 k2 − 2hx̄, Jx1 i
= φ(x̄, x1 ).
This implies that φ(xn , x1 ) → φ(x̄, x1 ) as n → ∞. Hence, we have limn→∞ kxn k = kx̄k. In view of KadecKlee property of E, we find that xn → x̄ as n → ∞. Since xn+1 ∈ Cn , one has φ(xn , x1 ) ≤ φ(xn+1 , x1 ). So,
{φ(xn , x1 )} is a nondecreasing sequence. Since φ(xn , x1 ) ≤ φ(x̄, x1 ), one see that limn→∞ φ(xn , x1 ) exists.
This implies that limn→∞ φ(xn+1 , xn ) = 0. Since xn+1 ∈ Cn+1 , we find that
φ(xn+1 , xn ) + α(n,i) D + (1 − α(n,i) )ξ(n,i) ≥ φ(xn+1 , y(n,i) ) ≥ 0.
Using restriction imposed on {α(n,i) }, on has limn→∞ φ(xn+1 , y(n,i) ) = 0. Using (1.3), we see that
lim (ky(n,i) k − kxn+1 k) = 0,
n→∞
which in turn finds
lim ky(n,i) k = kx̄k.
n→∞
That is,
lim kJy(n,i) k = kJ x̄k = lim ky(n,i) k = kx̄k.
n→∞
n→∞
E∗
Since both
and E are reflexive spaces, we may assume that Jy(n,i) * y (∗,i) ∈ 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 = kxn+1 k2 + ky(n,i) k2 − 2hxn+1 , Jy(n,i) i
= φ(xn+1 , y(n,i) ).
Taking lim inf n→∞ on the both sides of the equality above yields that
0 ≤ φ(x̄, y i ) = kx̄k2 − 2hx̄, Jy i i + ky i k2
= kx̄k2 − 2hx̄, Jy i i + kJy i k2
≤ kx̄k2 − 2hx̄, y (∗,i) i + ky (∗,i) k2
≤ 0.
This implies y i = x̄. Hence, we have y (∗,i) = J x̄. It follows that Jy(n,i) * J x̄ ∈ E ∗ . Since limn→∞ α(n,i) = 0
for every i ∈ Λ, we find limn→∞ kJy(n,i) − Jfin z(n,i) k = 0. Using the fact
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1782
kJ x̄ − Jfin z(n,i) k ≤ kJy(n,i) − J x̄k + kJy(n,i) − Jfin z(n,i) k,
one has Jfin z(n,i) → J x̄ as n → ∞ for every i ∈ λ. Since J −1 is demicontinuous, we have fin z(n,i) * x̄ for
every i ∈ ∆. Since |kfin z(n,i) k − kx̄k| ≤ kJ(fin z(n,i) ) − J x̄k, one has kfin z(n,i) k → kx̄k, as n → ∞ for every
i ∈ Λ. Since E has the Kadec-Klee property, one obtains
lim kfin z(n,i) − x̄k = 0.
n→∞
On the other hand, we have
kfin+1 z(n,i) − x̄k ≤ kfin+1 z(n,i) − fin z(n,i) k + kfin z(n,i) − x̄k.
In view of the uniformly asymptotic regularity of fi , one has
lim kfin+1 z(n,i) − x̄k = 0,
n→∞
that is, fi fin z(n,i) − x̄ → 0 as n → ∞. Since every fi is a continuous, we find that fi x̄ = x̄ for every i ∈ Λ.
Next, we prove x̄ ∈ ∩i∈Λ Sol(gi ).
Since fi is continuous, using (2.1), we find that limn→∞ φ(xn+1 , z(n,i) ) = 0. Using (1.3), we see that
limn→∞ (kz(n,i) k − kxn+1 k) = 0, which in turn finds limn→∞ kz(n,i) k = kx̄k. That is,
lim kJz(n,i) k = kJ x̄k = lim kz(n,i) k = kx̄k.
n→∞
n→∞
Since both E ∗ and E are reflexive, we may assume that Jz(n,i) * z (∗,i) ∈ E ∗ . This shows that there exists
an element z i ∈ E such that z (∗,i) = Jz i . It follows that
kxn+1 k2 + kJz(n,i) k2 − 2hxn+1 , Jz(n,i) i = kxn+1 k2 + kz(n,i) k2 − 2hxn+1 , Jz(n,i) i
= φ(xn+1 , z(n,i) ).
Taking lim inf n→∞ on the both sides of the equality above yields that
φ(x̄, z i ) = kx̄k2 − 2hx̄, Jz i i + kz i k2
= kx̄k2 − 2hx̄, Jz i i + kJz i k2
≤ kx̄k2 − 2hx̄, z (∗,i) i + kz (∗,i) k2
≤ 0.
This implies z i = x̄. Hence, we have z (∗,i) = J x̄. It follows that Jz(n,i) * J x̄ ∈ E ∗ . Using the Kadec-Klee
property we find that Jz(n,i) → J x̄ ∈ E ∗ . Since J −1 is demicontinuous, we have z(n,i) * x̄. Using the fact
that
kJy(n,i) − Jxn k ≤ kJy(n,i) − J x̄k + kJxn − J x̄k,
we see that lim kJy(n,i) − Jxn k = 0. In view of z(n,i) = τr(n,i) xn , we see that
n→∞
ky − z(n,i) kkJz(n,i) − Jxn k ≥ r(n,i) gi (y, z(n,i) ) ∀y ∈ Cn .
It follows that gi (y, x̄) ≤ 0 ∀y ∈ Cn . For 0 < ti < 1 and y ∈ Cn , define y(t,i) = ti y + (1 − ti )x̄. It follows that
y(t,i) ∈ Cn , which yields that gi (y(t,i) , x̄) ≤ 0. Hence, we have
0 = gi (y(t,i) , y(t,i) ) ≤ ti gi (y(t,i) , y) + (1 − ti )gi (y(t,i) , x̄) ≤ ti gi (y(t,i) , y).
That is, gi (y(t,i) , y) ≥ 0. Letting ti ↓ 0, we obtain from (R − d) that gi (x̄, y) ≥ 0, ∀y ∈ C. This implies
that x̄ ∈ Sol(g
T i ) for every i ∈ Λ. This shows that x̄ ∈ ∩i∈Λ Sol(gi ). This completes the proof that x̄ ∈
∩i∈Λ F ix(Ti ) ∩i∈Λ Sol(gi ).
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1783
Step 6. Prove x̄ = Π∩i∈Λ F ix(fi ) 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(fi )
\
∩i∈Λ Sol(gi ).
In view of Lemma 1.6, we find that that x̄ = Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 . This completes the proof.
If f is a asymptotically quasi-φ-nonexpansive mapping, we find from Theorem 2.1 the following.
Corollary 2.2. Let E be a strictly convex and uniformly smooth Banach space which also has the KadecKlee property. Let C be a convex and closed subset of E and let Λ be an index set. Let gi be a bifunction
from C × C to R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be an asymptotically quasi-φnonexpansive mapping for every i ∈ Λ. Assume
that fi is continuous and uniformly asymptotically regular
T
on C for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩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,





x1 = ΠC1 :=∩i∈Λ C(1,i) x0 ,
y(n,i) = J −1 ((1 − α(n,i) )Jfin z(n,i) + α(n,i) Jx1 ),



C(n+1,i) = {z ∈ C(n,i) : φ(z, xn ) + α(n,i) D ≥ φ(z, y(n,i) )},





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = ΠCn+1 x1 ,
T
where D = sup{φ(w, x1 ) : w ∈ ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi )}, z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥ hz(n,i) −
y, Jz(n,i) − Jxn i ∀y ∈ Cn , {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 [ri , ∞), where {ri } is a positive real number sequence for every i ∈ Λ. Then the sequence
{xn } converges strongly to Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
If T is a quasi-φ-nonexpansive mapping, we find from Theorem 2.1 the following.
Corollary 2.3. Let E be a strictly convex and uniformly smooth Banach space which also has the KKP.
Let C be a convex and closed subset of E and let Λ be an index set. Let gi be a bifunction from C × C to
R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be a quasi-φ-nonexpansive
mapping for every
T
i ∈ Λ. Assume that fi is continuous for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) is nonempty. Let {xn }
be a sequence generated in the following manner:


x0 ∈ E chosen arbitrarily,





C(1,i) = C,





x1 = ΠC1 :=∩i∈Λ C(1,i) x0 ,
y(n,i) = J −1 ((1 − α(n,i) )Jfi z(n,i) + α(n,i) Jx1 ),



C(n+1,i) = {z ∈ C(n,i) : φ(z, xn ) ≥ φ(z, y(n,i) )},





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = ΠCn+1 x1 ,
where z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥ hz(n,i) − y, Jz(n,i) − Jxn i ∀y ∈ Cn , {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 [ri , ∞), where {ri } is a positive real
number sequence for every i ∈ Λ. Then the sequence {xn } converges strongly to Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
In the Hilbert spaces, we have the following deduced results.
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1784
Corollary 2.4. Let E be a Hilbert space. Let C be a convex and closed subset of E and let Λ be an index
set. Let gi be a bifunction from C × C to R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be
an asymptotically quasi-nonexpansive mapping in the intermediate sense for every i ∈ Λ. TAssume that fi
is continuous and uniformly asymptotically regular on C for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩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,




x1 = PC1 :=∩i∈Λ C(1,i) x0 ,

y(n,i) = (1 − α(n,i) )fin z(n,i) + α(n,i) x1 ,



C(n+1,i) = {z ∈ C(n,i) : kz − xn k2 + α(n,i) D + (1 − α(n,i) )ξ(n,i) ≥ kz − y(n,i) k2 },





Cn+1 = ∩i∈Λ C(n+1,i) ,




xn+1 = PCn+1 x1 ,
where
ξ(n,i) = max{0,
sup
\
kp − fin xk2 − kp − xk2 , D = sup{kw − x1 k2 : w ∈ ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi )},
p∈F ix(fi ),x∈C
z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥ hz(n,i) − y, z(n,i) − xn i ∀y ∈ Cn , {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 [ri , ∞), where {ri } is a positive real number
sequence for every i ∈ Λ. Then the sequence {xn } converges strongly to Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
If T f is an asymptotically quasi-nonexpansive mapping, we find from Theorem 2.1 the following.
Corollary 2.5. Let E be a Hilbert space. Let C be a convex and closed subset of E and let Λ be an index
set. Let gi be a bifunction from C × C to R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be an
asymptotically quasi-nonexpansive mapping for every i ∈ Λ. T
Assume that fi is continuous and uniformly
asymptotically regular on C for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩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, x1 = PC1 :=∩i∈Λ C(1,i) x0 ,
y(n,i) = (1 − α(n,i) )fin z(n,i) + α(n,i) x1 ,



C(n+1,i) = {z ∈ C(n,i) : kz − xn k2 + α(n,i) D ≥ kz − y(n,i) k2 },



Cn+1 = ∩i∈Λ C(n+1,i) , xn+1 = PCn+1 x1 ,
T
where D = sup{kw − x1 k2 : w ∈ ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi )}, z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥
hz(n,i) − y, z(n,i) − xn i ∀y ∈ Cn , {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 [ri , ∞), where {ri } is a positive real number sequence for every i ∈ Λ. Then the sequence
{xn } converges strongly to Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
If f is a closed quasi-nonexpansive mapping, we find from Theorem 2.1 the following.
Corollary 2.6. Let E be a Hilbert space. Let C be a convex and closed subset of E and let Λ be an index
set. Let gi be a bifunction from C × C to R satisfying (R-a), (R-b), (R-c), (R-d) and let fi : C → C be
a quasi-nonexpansive mapping for every i ∈ Λ.T Assume that fi is continuous and uniformly asymptotically
regular on C for every i ∈ Λ and ∩i∈Λ F ix(fi ) ∩i∈Λ Sol(gi ) is nonempty. Let {xn } be a sequence generated
in the following manner:
Y. Hecai, Z. Min, J. Nonlinear Sci. Appl. 9 (2016), 1776–1786
1785


x0 ∈ E chosen arbitrarily,




C(1,i) = C, x1 = PC1 :=∩i∈Λ C(1,i) x0 ,

y(n,i) = (1 − α(n,i) )fi z(n,i) + α(n,i) x1 ,



C(n+1,i) = {z ∈ C(n,i) : kz − xn k2 ≥ kz − y(n,i) k2 },




Cn+1 = ∩i∈Λ C(n+1,i) , xn+1 = PCn+1 x1 ,
where z(n,i) ∈ Cn such that r(n,i) gi (z(n,i) , y) ≥ hz(n,i) − y, z(n,i) − xn i ∀y ∈ Cn , {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 [ri , ∞), where {ri } is a positive real
number sequence for every i ∈ Λ. Then the sequence {xn } converges strongly to Π∩i∈Λ F ix(fi ) T ∩i∈Λ Sol(gi ) x1 .
Acknowledgements
The authors are grateful to the reviewers for useful suggestions which improve the contents of this article.
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