Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
Research Article
Some results on asymptotically quasi-φ-nonexpansive mappings in the intermediate sense and Ky Fan
inequalities
Hongwei Liang∗, Mingliang Zhang
School of Mathematics and Statistics, Henan University, Kaifeng 475000, China.
Communicated by X. Qin
Abstract
In this paper, we study asymptotically quasi-φ- nonexpansive mappings in the intermediate sense and Ky
Fan inequalities. A convergence theorem is established in a strictly convex and uniformly smooth Banach
space. The results presented in the paper improve and extend some recent results. c 2016 All rights reserved.
Keywords: Asymptotically nonexpansive mapping, quasi-φ-nonexpansive mapping, fixed point,
convergence theorem.
2010 MSC: 65J15, 90C33.
1. Introduction and Preliminaries
Let E be a real Banach space and let C be nonempty closed and convex subset of E. Let B : C × C → R
be a function. Recall the following equilibrium problem in the terminology of Blum and Oettli [4].
Find x̄ ∈ C such that B(x̄ y) ≥ 0, ∀y ∈ C.
In this paper, we use Sol(B) to denote the solution set of the equilibrium problem. That is, Sol(B) = {x ∈
C : B(x, y) ≥ 0, ∀y ∈ C}. The following restrictions on function B are essential in this paper.
(A-1) B(a, a) ≡ 0, ∀a ∈ C;
∗
Corresponding author
Email addresses: hdlianghw@yeah.net (Hongwei Liang), hdzhangml@yeah.net (Mingliang Zhang)
Received 2015-10-20
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1676
(A-2) 0 ≥ B(b, a) + B(a, b), ∀a, b ∈ C;
(A-3) b 7→ B(a, b) is convex and weakly lower semi-continuous, ∀a ∈ C;
(A-4) B(a, b) ≥ lim supt↓0 B(tc + (1 − t)a, b), ∀a, b, c ∈ C.
The equilibrium problem has been extensively studied based on iterative methods because of its applications in nonlinear analysis, optimization, economics, game theory, mechanics, medicine and so forth, see
[3], [7]-[11], [14], [17], [18], [25], [27]-[31] and the references therein.
Let E ∗ be the dual space of E. Let BE be the unit sphere of E. Recall that E is said to be uniformly
convex if for any a ∈ (0, 2] there exists b > 0 such that for any x, y ∈ BE ,
ky − xk ≥ a
implies
ky + xk ≤ 2 − 2b.
E is said to be a strictly convex space if and only if ky + xk < 2 for all x, y ∈ BE and x 6= y. It is known
that a uniformly convex Banach space is reflexive and strictly convex.
Recall that E is said to have a Gâteaux differentiable norm if and only if limt→0 kx+tyk−kxk
exists for each
t
x, y ∈ BE . In this case, we also say that E is smooth. E is said to have a uniformly Gâteaux differentiable
norm if for each y ∈ BE , limt→0 kx+tyk−kxk
is attained uniformly for all x ∈ BE . E is also said to have a
t
is attained uniformly for x, y ∈ BE . In this case,
uniformly Fréchet differentiable norm iff limt→0 kx+tyk−kxk
t
we say that E is uniformly smooth. It is known that a uniformly smooth Banach space is reflexive and
smooth. Recall that E is said to has the Kadec-Klee property if limm→∞ kxm − xk = 0, for any sequence
{xm } ⊂ E, and x ∈ E with {xn } converges weakly to x, and {kxn k} converges strongly to kxk. It is known
that every uniformly convex Banach space has the Kadec-Klee property; see [13] and the references therein.
∗
Recall that the normalized duality mapping J from E to 2E is defined by
Jx = {x∗ ∈ E ∗ : kxk2 = hx, x∗ i = kx∗ k2 }.
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.
Let T be a mapping on C. T is said to be closed if for any sequence {xm } ⊂ C such that limm→∞ xm = x0
and limm→∞ T xm = y 0 , then T x0 = y 0 . Let W be a bounded subset of C. Recall that T is said to be uniformly
asymptotically regular on C if and only if lim supn→∞ supx∈W {kT n x − T n+1 xk} = 0. From now on, we use
* and → to stand for the weak convergence and strong convergence, respectively and use F ix(T ) to denote
the fixed point set of mapping T .
Next, we assume that E is a smooth Banach space which means mapping J is single-valued. Study the
functional
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, that is, kPC x − PC yk2 ≤
hx − y, PC x − PC yi. In [2], Alber studied a new mapping P rojC in a Banach space E which is an analogue
of PC , the metric projection, in Hilbert spaces. Recall that the generalized projection P rojC : E → C is a
mapping that assigns to an arbitrary point x ∈ E the minimum point of φ(x, y).
Recall that T is said to be asymptotically quasi-φ-nonexpansive in the intermediate sense iff F ix(T ) 6= ∅
and
lim sup
sup
φ(p, T n x) − φ(p, x) ≤ 0.
n→∞ p∈F ix(T ),x∈C
Putting ξn = max{0, supp∈F ix(T ),x∈C φ(p, T n x) − φ(p, x) }, we see ξn → 0 as n → ∞. Hence, we have
φ(p, T n x) ≤ φ(p, x) + ξn ,
∀x ∈ C, ∀p ∈ F ix(T ).
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1677
T is said to be asymptotically quasi-φ-nonexpansive iff F ix(T ) 6= ∅ and
φ(p, T n x) ≤ (1 + un )φ(p, x),
∀x ∈ C, ∀p ∈ F ix(T ), ∀n ≥ 1,
where {un } is a sequence {un } ⊂ [0, ∞) with un → 0 as n → ∞.
T is said to be quasi-φ-nonexpansive iff F ix(T ) 6= ∅ and
φ(p, T x) ≤ φ(p, x),
∀x ∈ C, ∀p ∈ F ix(T ).
Recall that p is said to be an asymptotic fixed point of T if and only if C contains a sequence {xn },
g
where xn * p such that xn − T xn → 0. Here, we use F
ix(T ) to denote the asymptotic fixed point set of T .
g
T is said to be asymptotically relatively quasi-φ-nonexpansive iff F ix(T ) = F
ix(T ) 6= ∅ and
φ(p, T n x) ≤ (1 + un )φ(p, x),
g
∀x ∈ C, ∀p ∈ F ix(T ) = F
ix(T ), ∀n ≥ 1,
where {un } is a sequence {un } ⊂ [0, ∞) with un → 0 as n → ∞.
g
T is said to be relatively nonexpansive iff F ix(T ) = F
ix(T ) 6= ∅ and
φ(p, T x) ≤ φ(p, x),
g
∀x ∈ C, ∀p ∈ F ix(T ) = F
ix(T ).
Remark 1.1. The class of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense [24] is
reduced to the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was
considered in [5] as a non-Lipschitz continuous mappings, in the framework of Hilbert spaces.
Remark 1.2. The class of quasi-φ-nonexpansive mappings [21] is a generalization of relatively nonexpansive
mappings [6]. The class of quasi-φ-nonexpansive mappings do not require the strong restriction that the
fixed point set equals the asymptotic fixed point set.
Remark 1.3. The class of asymptotically quasi-φ-nonexpansive mappings [22] is more desirable than the class
of asymptotically relatively nonexpansive [1] mappings. Asymptotically quasi-φ-nonexpansive mappings are
reduced to asymptotically quasi-nonexpansive mappings in the framework of Hilbert spaces.
In this paper, we study the equilibrium problem in the terminology of Blum and Oettli [4] and a
finite family of asymptotically quasi-φ-nonexpansive mappings in the intermediate sense. With the aid of
generalization projections, we establish a strong theorem in a strictly convex and uniformly smooth Banach
space. The results obtained in this paper mainly improve the corresponding results in [15], [16], [19], [20],
[23], [30]. In order to prove our main results, we also need the following lemmas.
Lemma 1.4 ([2]). Let E be a strictly convex, reflexive, and smooth Banach space and let C be a nonempty,
closed, and convex subset of E. Let x ∈ E. Then
φ(y, x) − φ(ΠC x, x) ≥ φ(y, ΠC x),
∀y ∈ C,
0 ≥ hy − x0 , Jx − Jx0 i, ∀y ∈ C if and only if x0 = ΠC x.
Lemma 1.5 ([26]). Let r be a positive real number and let E be uniformly convex. Then there exists a
convex, strictly increasing and continuous function cog : [0, 2r] → R such that cog(0) = 0 and
tkak2 + (1 − t)kbk2 ≥ k(1 − t)b + tak2 + t(1 − t)cog(kb − ak)
for all a, b ∈ B r := {a ∈ E : kak ≤ r} and t ∈ [0, 1].
Lemma 1.6 ([4], [21]). Let E be a strictly convex, smooth, and reflexive Banach space and let C be a
closedconvex subset of E. Let B be a function with restrictions (A-1), (A-2), (A-3) and (A-4), from C × C
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1678
to R. Let x ∈ E and let r > 0. Then there exists z ∈ C such that hz − y, Jz − Jxi + rB(z, y) ≤ 0, ∀y ∈ C
Define a mapping K B,r by
K B,r x = {z ∈ C : 0 ≤ hy − z, Jz − Jxi + rB(z, y),
∀y ∈ C}.
The following conclusions hold:
(1) K B,r is single-valued quasi-φ-nonexpansive;
(2) Sol(B) = F ix(K B,r ) is convex and closed.
Lemma 1.7 ([24]). 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 T be an asymptotically quasi-φ-nonexpansive
mapping in the intermediate sense on C. F ix(T ) is convex.
2. Main results
Theorem 2.1. Let E be a strictly convex and uniformly smooth Banach space which also has the Kadec-Klee
property. Let C be a convex and closed subset of E and let B be a function with restrictions (A-1), (A-2),
(A-3) and (A-4). Let {Tm }N
m=1 , where N is some positive integer, be a sequence of asymptotically quasi-φnonexpansive mappings in the intermediate
sense on C. Assume that every Tm is uniformly asymptotically
T N
regular and closed and Sol(B) ∩m=1 F ix(Tm ) is nonempty. Let {α(n,0) }, {α(n,1) }, · · · , {α(n,N ) } be real
P
sequences in (0,1) such that N
m=0 α(n,m) = 1 and lim inf n→∞ α(n,0) α(n,m) > 0 for any 1 ≤ m ≤ N . Let
{xn } be a sequence generated by


x0 ∈ E chosen arbitrarily,





C1 = C,






x1 = P rojC1 x0 ,
rn B(un, u) ≥ hun − u, Jun − Jxn i, ∀u ∈ C
n ,

PN

n
Jyn =

m=1 α(n,m) JTm xn + α(n,0) Jun ,



C

n+1 = {z ∈ Cn : φ(z, yn ) ≤ (1 − α(n,0) )ξn + φ(z, xn )},



x
n+1 = P rojCn+1 x1 ,
n x) − φ(p, x) , 0} : 1 ≤ m ≤ N , and {r } is a real sequence
where ξn = max max{supp∈F ix(Tm ),x∈C φ(p, Tm
n
such that lim inf n→∞ rn > 0. Then {xn } converges strongly to P rojSol(B) T ∩N F ix(Tm ) x1 .
m=1
Proof. The proof is split into seven steps.
T
Step 1. Prove that Sol(B) ∩N
m=1 F ix(Tm ) is convex and closed.
Using Lemmas 1.6 and 1.7, we find that F ix(Tm ) is convex and Sol(B) is convex and closed. Since Tm
is closed, we find that F ix(Tm ) is also closed. So, P rojSol(B)∩∩N F ix(Tm ) x is well defined, for any element
m=1
x in E.
Step 2. Prove that Cn is convex and closed.
It is obvious that C1 = C is convex and closed. Assume that Ci is convex and closed for some i ≥ 1. Let
p1 , p2 ∈ Ci+1 . It follows that p = sp1 + (1 − s)p2 ∈ Ci , where s ∈ (0, 1). Since
(1 − α(i,0) )ξi + φ(p1 , xi ) ≥ φ(p1 , yi ),
and
(1 − α(i,0) )ξi + φ(p2 , xi ) ≥ φ(p2 , yi ),
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1679
one has
(1 − α(i,0) )ξi ≥ 2hp1 , Jxi − Jyi i − kxi k2 + kyi k2 ,
and
(1 − α(i,0) )ξi ≥ 2hp2 , Jxi − Jyi i − kxi k2 + kyi k2 .
Using the above two inequalities, one has
φ(p, yi ) − φ(p, xi ) ≤ (1 − α(i,0) )ξi .
This shows that Ci+1 is closed and convex. Hence, Cn is a convex and closed set.
Step 3. Prove ∩N
m=1 F ix(Tm ) ∩ Sol(B) ⊂ Cn .
It is obvious
∩N
m=1 F ix(Tm ) ∩ Sol(B) ⊂ C1 = C.
N
Suppose that ∩N
m=1 F ix(Tm )∩Sol(B) ⊂ Ci for some positive integer i. For any z ∈ ∩m=1 F ix(Tm )∩Sol(B) ⊂
Ci , we see that
φ(z, xi ) + (1 − α(i,0) )ξi
N
X
≥
i
α(i,m) φ(z, Tm
xi ) + α(i,0) φ(z, ui )
m=1
≥ kzk2 +
N
X
i
α(i,m) kTm
xi k2 + α(i,0) kJui k2
m=1
− 2α(i,0) hz, Jui i − 2
N
X
i
α(i,m) hz, JTm
xi i
m=1
N
X
≥ kzk2 + k
i
α(i,m) JTm
xi + α(i,0) Jui k2
m=1
− 2hz,
N
X
i
α(i,m) JTm
xi + α(i,0) Jui i
m=1
= φ(z, yi ),
where
ξi = max max{
i
φ(p, Tm
x) − φ(p, x) , 0} : 1 ≤ m ≤ N .
sup
p∈F ix(Tm ),x∈C
This shows that z ∈ Ci+1 . This implies that ∩N
m=1 F ix(Tm ) ∩ Sol(B) ⊂ Cn .
Step 4. Prove that {xn } is bounded.
Now, we have hxn − z, Jx1 − Jxn i ≥ 0, for any z ∈ Cn . It follows that
0 ≤ hxn − z, Jx1 − Jxn i,
∀z ∈ ∩N
m=1 F ix(Tm ) ∩ Sol(B) ⊂ Cn .
On the other hand, we find from Lemma 1.4,
φ(P roj∩N
m=1 F ix(Tm )∩Sol(B)
≥ φ(P roj∩N
x1 , x1 )
m=1 F ix(Tm )∩Sol(B)
x1 , x1 ) − φ(P roj∩N
m=1 F ix(Tm )∩Sol(B)
x1 , xn )
≥ φ(xn , x1 ),
which shows that {φ(xn , x1 )} is bounded. Hence, {xn } is also bounded. Without loss of generality, we
assume xn * x̄. Since every Cn is convex and closed. So x̄ ∈ Cn .
Step 5. Prove x̄ ∈ ∩N
m=1 F ix(Tm ).
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1680
Since x̄ ∈ Cn , one has φ(xn , x1 ) ≤ φ(x̄, x1 ). This implies that
φ(x̄, x1 ) ≤ lim inf (kxn k2 + kx1 k2 − 2hxn , Jx1 i) = lim sup φ(xn , x1 ) ≤ φ(x̄, x1 ).
n→∞
n→∞
Hence, one has
lim φ(xn , x1 ) = φ(x̄, x1 ).
n→∞
It follows that
lim kxn k = kx̄k.
n→∞
Using the Kadec-Klee property, one obtains that {xn } converges strongly to x̄ as n → ∞. Since xn+1 ∈
Cn+1 ⊂ Cn , we find that
φ(xn+1 , x1 ) ≥ φ(xn , x1 ),
which shows that {φ(xn , x1 )} is nondecreasing. It follows that limn→∞ φ(xn , x1 ) exists. Since
φ(xn+1 , x1 ) − φ(xn , x1 ) ≥ φ(xn+1 , xn ) ≥ 0,
one has limn→∞ φ(xn+1 , xn ) = 0. Using the fact xn+1 ∈ Cn+1 , one sees
φ(xn+1 , yn ) − φ(xn+1 , xn ) ≤ (1 − α(n,0) )ξn .
Since
lim φ(xn+1 , xn ) = lim ξn = 0,
n→∞
n→∞
one has
lim φ(xn+1 , yn ) = 0.
n→∞
Therefore, one has
lim (kyn k − kxn+1 k) = 0.
n→∞
This implies that
lim kJyn k = lim kyn k = kx̄k = kJ x̄k.
n→∞
n→∞
This implies that {Jyn } is bounded. Without loss of generality, we assume that {Jyn } converges weakly to
y ∗ ∈ E ∗ . In view of the reflexivity of E, we see that J(E) = E ∗ . This shows that there exists an element
y ∈ E such that Jy = y ∗ . It follows that
φ(xn+1 , yn ) + 2hxn+1 , Jyn i = kxn+1 k2 + kJyn k2 .
Taking lim inf n→∞ , one has 0 ≥ kx̄k2 − 2hx̄, y ∗ i + ky ∗ k2 = kx̄k2 + kJyk2 − 2hx̄, Jyi = φ(x̄, y) ≥ 0. That is,
x̄ = y, which in turn implies that J x̄ = y ∗ . Hence, Jyn * J x̄ ∈ E ∗ . Since E is uniformly smooth. Hence,
E ∗ is uniformly convex and it has the Kadec-Klee property, we obtain
lim Jyn = J x̄.
n→∞
Since J −1 : E ∗ → E is demi-continuous and E has the Kadec-Klee property, one gets that yn → x̄, as
n → ∞. Using the fact
(kxn k + kyn k)kyn − xn k + 2hz, Jyn − Jxn i ≥ φ(z, xn ) − φ(z, yn ),
we find
lim φ(z, xn ) − φ(z, yn ) = 0.
n→∞
It follows from Lemma 1.5, that
(2.1)
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1681
n
φ(z, xn ) + (1 − α(n,0) )ξn − α(n,0) α(n,m) g(kJTm
xn − Jun k)
≥
≥
N
X
m=1
N
X
n
n
α(n,m) φ(z, Tm
xn ) + α(n,0) φ(z, un ) − α(n,0) α(n,m) g(kJTm
xn − Jun k)
2
α(n,m) kzk +
m=0
N
X
n
α(n,m) kTm
xn k2 + α(n,0) kJun k2
m=1
− 2α(n,0) hz, Jun i − 2
N
X
n
α(n,m) hz, JTm
xn i
m=1
n
− α(n,0) α(n,m) g(kJTm
xn − Jun k)
≥ φ(z, yn ).
This implies
n
0 ≤ α(n,0) α(n,m) g(kJTm
xn − Jun k) ≤ φ(z, xn ) − φ(z, yn ) + (1 − α(n,0) )ξn .
Since lim inf n→∞ α(n,0) α(n,m) > 0, one sees from 2.1
n
xn k = 0
lim kJun − JTm
n→∞
for any 1 ≤ m ≤ N. Using the fact
N
X
n
α(n,m) (JTm
xn − Jun ) = Jyn − Jun ,
m=1
n x → J x̄ as n → ∞. Since J −1 : E ∗ → E is
one has {Jun } converges strongly to J x̄. It follows that JTm
n
n
demi-continuous, one has Tm xn * x̄. Using the fact
n
n
n
|kTm
xn k − kx̄k| = |kJTm
xn k − kJ x̄k| ≤ kJTm
xn − J x̄k,
n x k → kx̄k as n → ∞. Since E has the Kadec-Klee property, one has
one has kTm
n
n
lim kx̄ − Tm
xn k = 0.
n→∞
Since Tm is also uniformly asymptotically regular, one has
n+1
lim kx̄ − Tm
xn k = 0.
n→∞
n x ) → x̄. Using the closedness of T , we find T x̄ = x̄. This proves x̄ ∈ F ix(T ), that is,
That is, Tm (Tm
n
m
m
m
N
x̄ ∈ ∩m=1 F ix(Tm ).
Step 6. Prove x̄ ∈ Sol(B).
Since B is a monotone bifunction, one has
rn B(u, un ) ≤ ku − un kkJun − Jxn k.
Since lim inf n→∞ rn > 0, we may assume there exists λ > 0 such that rn ≥ λ. It follows that
B(u, un ) ≤ ku − un k
kJun − Jxn k
.
λ
Hence, one has B(u, x̄) ≤ 0. For 0 < s < 1, define us = (1 − s)x̄ + su. This implies that 0 ≥ B(us , x̄). Hence,
we have
sB(us , u) ≥ B(us , us ) = 0.
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1682
It follows that B(x̄, u) ≥ 0, ∀u ∈ C. This implies that x̄ ∈ Sol(B).
Step 7. Prove x̄ = P roj∩N
m=1 F ix(Tm )∩Sol(B)
x1 .
Using Lemma 1.5, we find
0 ≤ hxn − z, Jx1 − Jxn i, ∀z ∈ ∩N
m=1 F ix(Tm ) ∩ Sol(B).
Let n → ∞, one has
0 ≤ hx̄ − z, Jx1 − J x̄i.
It follows that x̄ = P roj∩N
m=1 F ix(Tm )∩Sol(B)
x1 . This completes the proof.
If N = 1, we have the following result.
Corollary 2.2. 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 B be a bifunction with (A-1), (A-2), (A-3) and (A-4).
Let T be an asymptotically quasi-φ-nonexpansive mapping inTthe intermediate sense on C. Assume that
T is uniformly asymptotically regular and closed and Sol(B) F ix(T ) is nonempty. Let {α(n,0) } be a real
sequence in (0,1) such that lim inf n→∞ α(n,0) (1 − α(n,0) ) > 0. Let {xn } be a sequence generated by


x0 ∈ E chosen arbitrarily,





C1 = C, x1 = P rojC1 x0 ,



rn B(un , u) ≥ hun − u, Jun − Jxn i, ∀u ∈ Cn ,

yn = J −1 (1 − α(n,0) )JT n xn + α(n,0) Jun ,





Cn+1 = {z ∈ Cn : φ(z, yn ) ≤ (1 − α(n,0) )ξn + φ(z, xn )},




xn+1 = P rojCn+1 x1 ,
where ξn = max{supp∈F ix(T ),x∈C φ(p, T n x)−φ(p, x) , 0}, and {rn } is a real sequence such that lim inf n→∞ rn
> 0. Then {xn } converges strongly to P rojSol(B)∩F ix(T ) x1 .
If T is the identity mapping, we have the following results on the equilibrium problem.
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 B be a bifunction with (A-1), (A-2), (A-3) and (A-4). Let
N ≥ 1 be some positive integer and assume Sol(B) 6= ∅. Let {α(n,0) }, {α(n,1) }, · · · , {α(n,N ) } be real sequences
P
in (0,1) such that N
m=0 α(n,m) = 1 and lim inf n→∞ α(n,0) α(n,m) > 0 for any 1 ≤ m ≤ N . Let {xn } be a
sequence generated by


x0 ∈ E chosen arbitrarily,





C1 = C, x1 = P rojC1 x0 ,



rn B(un , u) ≥ hun − u, Jun − Jxn i, ∀u ∈ Cn ,
P
N
−1

y
=
J
α
Jx
+
α
Ju
n
n
n ,
(n,0)

m=1 (n,m)




Cn+1 = {z ∈ Cn : φ(z, yn ) ≤ φ(z, xn )},




xn+1 = P rojCn+1 x1 ,
where {rn } is a real sequence such that lim inf n→∞ rn > 0. Then {xn } converges strongly to P rojSol(B) x1 .
p
In the framework of Hilbert spaces, φ(x, y) = kx−yk, ∀x, y ∈ E. The generalized projection is reduced
to the metric projection and the class of asymptotically-φ-nonexpansive mappings in the intermediate sense
is reduced to the class of asymptotically quasi-nonexpansive mappings in the intermediate sense.
H. Liang, M. Zhang, J. Nonlinear Sci. Appl. 9 (2016), 1675–1684
1683
Corollary 2.4. Let E be a Hilbert space. Let C be a convex and closed subset of E and let B be a
function with (A-1), (A-2), (A-3) and (A-4). Let {Tm }N
m=1 , where N is some positive integer, be a
sequence of asymptotically quasi-nonexpansive mappings in the intermediate
T N sense on C. Assume that
every Tm is uniformly asymptotically regular and closed and Sol(B)
∩m=1 F ix(Tm ) is nonempty. Let
P
{α(n,0) }, {α(n,1) }, · · · , {α(n,N ) } be real sequences in (0,1) such that N
m=0 α(n,m) = 1 and
lim inf α(n,0) α(n,m) > 0
n→∞
for any 1 ≤ m ≤ N . Let {xn } be a sequence generated by


x0 ∈ E chosen arbitrarily,





C1 = C, x1 = PC1 x0 ,



r B(u , u) ≥ hu − u, u − x i, ∀u ∈ C ,
n
n
n
n
n
n
PN
n

yn = m=1 α(n,m) Tm xn + α(n,0) un ,





Cn+1 = {z ∈ Cn : kz − yn k2 ≤ (1 − α(n,0) )ξn + kz − xn k2 },



x
n+1 = P rojCn+1 x1 ,
n xk2 − kp − xk2 , 0} : 1 ≤ m ≤ N , and {r } is a real
where ξn = max max{supp∈F ix(Tm ),x∈C kp − Tm
n
sequence such that lim inf n→∞ rn > 0. Then {xn } converges strongly to PSol(B) T ∩N F ix(Tm ) x1 .
m=1
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