Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 131890, 23 pages
doi:10.1155/2011/131890
Research Article
A New Hybrid Iterative Scheme for Countable
Families of Relatively Quasi-Nonexpansive
Mappings and System of Equilibrium Problems
Yekini Shehu1, 2
1
2
Mathematics Institute, African University of Science and Technology, Abuja, Nigeria
Department of Mathematics, University of Nigeria, Nsukka, Nigeria
Correspondence should be addressed to Yekini Shehu, deltanougt2006@yahoo.com
Received 14 February 2011; Accepted 14 April 2011
Academic Editor: Yonghong Yao
Copyright q 2011 Yekini Shehu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We construct a new iterative scheme by hybrid methods and prove strong convergence theorem
for approximation of a common fixed point of two countable families of closed relatively quasinonexpansive mappings which is also a solution to a system of equilibrium problems in a
uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the
properties of generalized f -projection operator. Using this result, we discuss strong convergence
theorem concerning variational inequality and convex minimization problems in Banach spaces.
Our results extend many known recent results in the literature.
1. Introduction
Let E be a real Banach space with dual E∗ and C a nonempty, closed, and convex subset of E.
A mapping T : C → C is called nonexpansive if
Tx − Ty ≤ x − y,
∀x, y ∈ C.
1.1
A point x ∈ C is called a fixed point of T if Tx x. The set of fixed points of T is denoted by
FT : {x ∈ C : Tx x}.
∗
We denote by J the normalized duality mapping from E to 2E defined by
2 Jx f ∈ E∗ : x, f x2 f .
1.2
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The following properties of J are well known the reader can consult 1–3 for more details.
1 If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each
bounded subset of E.
2 Jx /
∅, x ∈ E.
3 If E is reflexive, then J is a mapping from E onto E∗ .
4 If E is smooth, then J is single valued.
Throughout this paper, we denote by φ the functional on E × E defined by
2
φ x, y x2 − 2 x, J y y ,
∀x, y ∈ E.
1.3
It is obvious from 1.3 that
2
2
x − y ≤ φ x, y ≤ x y ,
∀x, y ∈ E.
1.4
Definition 1.1. Let C be a nonempty subset of E, and let T be a mapping from C into E. A point
p ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {xn }∞
n0 which
converges weakly to p and limn → ∞ xn − Txn 0. The set of asymptotic fixed points of T is
denoted by FT.
We say that a mapping T is relatively nonexpansive see, e.g., 4–9 if the
following conditions are satisfied:
R1 FT /
∅,
R2 φp, Tx ≤ φp, x, for all x ∈ C, p ∈ FT,
R3 FT FT.
If T satisfies R1 and R2, then T is said to be relatively quasi-nonexpansive. It is easy to
see that the class of relatively quasi-nonexpansive mappings contains the class of relatively
nonexpansive mappings. Many authors have studied the methods of approximating the fixed
points of relatively quasi-nonexpansive mappings see, e.g., 10–12 and the references cited
therein. Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasinonexpansive mappings are the same, for φx, y x−y2 , for all x, y ∈ H, and this implies
that
φ p, Tx ≤ φ p, x ⇐⇒ Tx − p ≤ x − p,
∀x ∈ C, p ∈ FT.
1.5
The examples of relatively quasi-nonexpansive mappings are given in 11.
Let F be a bifunction of C × C into Ê. The equilibrium problem see, e.g., 13–25 is to
∗
find x ∈ C such that
F x∗ , y ≥ 0,
1.6
for all y ∈ C. We will denote the solutions set of 1.6 by EPF. Numerous problems in
physics, optimization, and economics reduce to find a solution of problem 1.6. The equilibrium problems include fixed point problems, optimization problems, and variational
inequality problems as special cases see, e.g., 26.
International Journal of Mathematics and Mathematical Sciences
3
In 7, Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach
space which is also uniformly smooth: x0 ∈ C,
yn J −1 αn Jxn 1 − αn JTxn ,
Hn w ∈ C : φ w, yn ≤ φw, xn ,
1.7
Wn {w ∈ C : xn − w, Jx0 − Jxn ≥ 0},
xn1 ΠHn ∩Wn x0 ,
n ≥ 0.
∅.
They proved that {xn }∞
n0 converges strongly to ΠFT x0 , where FT /
In 27, Plubtieng and Ungchittrakool introduced the following hybrid projection
algorithm for a pair of relatively nonexpansive mappings: x0 ∈ C,
1
2
3
zn J −1 βn Jxn βn JTxn βn JSxn ,
yn J −1 αn Jx0 1 − αn Jzn ,
Cn z ∈ C : φ z, yn ≤ φz, xn αn x0 2 2w, Jxn − Jx0 ,
1.8
Qn {z ∈ C : xn − z, Jxn − Jx0 ≤ 0},
xn1 PCn ∩Qn x0 ,
1
2
3
1
2
3
where {αn }, {βn }, {βn }, and {βn } are sequences in 0, 1 satisfying βn βn βn 1 and
T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on
E. They proved under the appropriate conditions on the parameters that the sequence {xn }
generated by 1.8 converges strongly to a common fixed point of T and S.
In 9, Takahashi and Zembayashi introduced the following hybrid iterative scheme
for approximation of fixed point of relatively nonexpansive mapping which is also a solution
to an equilibrium problem in a uniformly convex real Banach space which is also uniformly
smooth: x0 ∈ C, C1 C, x1 ΠC1 x0 ,
yn J −1 αn Jxn 1 − αn JTxn ,
1
y − un , Jun − Jyn ≥ 0, ∀y ∈ C,
F un , y rn
Cn1 w ∈ Cn : φw, un ≤ φw, xn ,
xn1 ΠCn1 x0 ,
1.9
n ≥ 1,
where J is the duality mapping on E. Then, they proved that {xn }∞
n0 converges strongly to
∅.
ΠF x0 , where F EPF ∩ FT /
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Furthermore, in 28, Qin et al. introduced the following hybrid iterative algorithm for
approximation of common fixed point of two countable families of closed relatively quasinonexpansive mappings in a uniformly convex and uniform smooth real Banach space:
1
2
3
zi,n J −1 βn,i Jxn βn,i JTi xn βn,i JSixn ,
yi,n J −1 αn,i Jx0 1 − αn,i Jzi,n ,
Cn,i z ∈ C : φ z, yi,n ≤ φz, xn αn,i x0 2 2z, Jxn − Jx0 ,
Cn 1.10
Cn,i ,
i∈I
Q0 C,
Qn {z ∈ Qn−1 : xn − z, Jx0 − Jxn ≥ 0},
xn1 ΠCn ∩Qn x0 ,
n ≥ 0.
They proved that the sequence {xn } converges strongly to a common fixed point of the countable families {Ti } and {Si } of closed relatively quasi-nonexpansive mappings in a uniformly
1
convex and uniformly smooth Banach space under some appropriate conditions on {βn,i },
2
3
{βn,i }, {βn,i }, and {αn,i }.
Recently, Li et al. 29 introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized
f-projection operator in a uniformly smooth real Banach space which is also uniformly
convex: x0 ∈ C, C0 C,
yn J −1 αn Jxn 1 − αn JTxn ,
Cn1 w ∈ Cn : G w, Jyn ≤ Gw, Jxn ,
f
xn1 ΠCn1 x0 ,
1.11
n ≥ 0.
They proved a strong convergence theorem for finding an element in the fixed points set of
T. We remark here that the results of Li et al. 29 extended and improved on the results of
Matsushita and Takahashi 7.
Quite recently, motivated by the results of Takahashi and Zembayashi 9, Cholamjiak
and Suantai 30 proved the following strong convergence theorem by hybrid iterative
scheme for approximation of common fixed point of a countable family of closed relatively
quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems
in uniformly convex and uniformly smooth Banach space.
Theorem 1.2. Let E be a uniformly convex real Banach space which is also uniformly smooth, and let
C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m, let Fk be a bifunction from
International Journal of Mathematics and Mathematical Sciences
5
countable family of closed and relatively
C × C satisfying (A1)–(A4). Suppose {Ti }∞
i1 is an infinitely ∞
∅. Suppose
quasi-nonexpansive mappings of C into itself such that Ω : m
k1 EPFk ∩ i1 FTi /
is
iteratively
generated
by
x
∈
C,
C
C,
{xn }∞
0
0
n0
yi,n J −1 αn Jxn 1 − αn JTi xn ,
F2
F1
m−1
ui,n TrFmm,n TrFm−1
,n · · · Tr2 ,n Tr1 ,n yi,n ,
Cn1 z ∈ Cn : sup φz, ui,n ≤ φz, xn i≥1
xn1 ΠCn1 x0 ,
,
1.12
n ≥ 0.
∞
Assume that {αn }∞
n1 and {rk,n }n1 k 1, 2, . . . , m are sequences which satisfy the following
conditions:
i lim supn → ∞ αn < 1,
ii lim infn → ∞ rk,n > 0 k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
Motivated by the above-mentioned results and the on-going research, it is our purpose
in this paper to prove a strong convergence theorem for two countable families of closed
relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium
problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee
property using the properties of generalized f-projection operator. Our results extend the
results of Matsushita and Takahashi 7, Takahashi and Zembayashi 9, Qin et al. 28,
Cholamjiak and Suantai 30, Li et al. 29, and many other recent known results in the
literature.
2. Preliminaries
Let E be a real Banach space. The modulus of smoothness of E is the function ρE : 0, ∞ →
0, ∞ defined by
ρE t : sup
1 x y x − y − 1 : x ≤ 1, y ≤ t .
2
2.1
E is uniformly smooth if and only if
ρE t
0.
t→0 t
lim
2.2
Let dim E ≥ 2. The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by
x y
δE : inf 1 − : x y 1; x − y .
2 2.3
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International Journal of Mathematics and Mathematical Sciences
E is uniformly convex if, for any ∈ 0, 2, there exists a δ δ > 0 such that if x, y ∈ E with
x ≤ 1, y ≤ 1, and x − y ≥ , then 1/2x y ≤ 1 − δ. Equivalently, E is uniformly
convex if and only if δE > 0 for all ∈ 0, 2. A normed space E is called strictly convex if
for all x, y ∈ E, x / y, x y 1, we have λx 1 − λy < 1, for all λ ∈ 0, 1.
Let E be a smooth, strictly convex, and reflexive real Banach space, and let C be
a nonempty, closed, and convex subset of E. Following Alber 31, the generalized projection
ΠC from E onto C is defined by
ΠC x : arg min φ y, x ,
y∈C
∀x ∈ E.
2.4
The existence and uniqueness of ΠC follows from the property of the functional φx, y and
strict monotonicity of the mapping J see, e.g., 3, 31–34. If E is a Hilbert space, then ΠC is
the metric projection of H onto C.
Next, we recall the concept of generalized f-projector operator, together with its
properties. Let G : C × E∗ → Ê ∪ {∞} be a functional defined as follows:
2
G ξ, ϕ ξ2 − 2 ξ, ϕ ϕ 2ρfξ,
2.5
where ξ ∈ C, ϕ ∈ E∗ , ρ is a positive number, and f : C → Ê ∪ {∞} is proper, convex,
and lower semicontinuous. From the definitions of G and f, it is easy to see the following
properties:
i Gξ, ϕ is convex and continuous with respect to ϕ when ξ is fixed,
ii Gξ, ϕ is convex and lower semicontinuous with respect to ξ when ϕ is fixed.
Definition 2.1 see Wu and Huang 35. Let E be a real Banach space with its dual E∗ . Let C
f
be a nonempty, closed, and convex subset of E. We say that ΠC : E∗ → 2C is a generalized
f-projection operator if
f
ΠC ϕ u ∈ C : G u, ϕ inf G ξ, ϕ ,
∀ϕ ∈ E∗ .
ξ∈C
2.6
For the generalized f-projection operator, Wu and Huang 35 proved the following
theorem basic properties.
Lemma 2.2 see Wu and Huang 35. Let E be a real reflexive Banach space with its dual E∗ . Let
C be a nonempty, closed, and convex subset of E. Then, the following statements hold:
f
i ΠC is a nonempty closed convex subset of C for all ϕ ∈ E∗ ,
f
ii if E is smooth, then, for all ϕ ∈ E∗ , x ∈ ΠC if and only if
x − y, ϕ − Jx ρf y − ρfx ≥ 0,
∀y ∈ C,
2.7
iii if E is strictly convex and f : C → Ê ∪ {∞} is positive homogeneous (i.e., ftx tfx
f
for all t > 0 such that tx ∈ C where x ∈ C), then ΠC is a single-valued mapping.
International Journal of Mathematics and Mathematical Sciences
7
Fan et al. 36 showed that the condition f is positive homogeneous which appeared
in Lemma 2.2 can be removed.
Lemma 2.3 see Fan et al. 36. Let E be a real reflexive Banach space with its dual E∗ and C
f
a nonempty, closed, and convex subset of E. Then, if E is strictly convex, then ΠC is a single-valued
mapping.
Recall that J is a single-valued mapping when E is a smooth Banach space. There exists
a unique element ϕ ∈ E∗ such that ϕ Jx for each x ∈ E. This substitution in 2.5 gives
Gξ, Jx ξ2 − 2ξ, Jx x2 2ρfξ.
2.8
Now, we consider the second generalized f-projection operator in a Banach space.
Definition 2.4. Let E be a real Banach space and C a nonempty, closed, and convex subset of
f
E. We say that ΠC : E → 2C is a generalized f-projection operator if
f
ΠC x
u ∈ C : Gu, Jx infGξ, Jx ,
ξ∈C
∀x ∈ E.
2.9
Obviously, the definition of T : C → C is a relatively quasi-nonexpansive mapping
and is equivalent to
R 1 FT / ∅,
R 2 Gp, JTx ≤ Gp, Jx, for all x ∈ C, p ∈ FT.
Lemma 2.5 see Li et al. 29. Let E be a Banach space, and let f : E → Ê ∪ {∞} be a lower
semicontinuous convex functional. Then, there exists x∗ ∈ E∗ and α ∈ Ê such that
fx ≥ x, x∗ α,
∀x ∈ E.
2.10
f
We know that the following lemmas hold for operator ΠC .
Lemma 2.6 see Li et al. 29. Let C be a nonempty, closed, and convex subset of a smooth and
reflexive Banach space E. Then, the following statements hold:
f
i ΠC x is a nonempty closed and convex subset of C for all x ∈ E,
f
ii for all x ∈ E, x ∈ ΠC x if and only if
x − y, Jx − J x ρf y − ρfx ≥ 0,
∀y ∈ C,
2.11
f
iii if E is strictly convex, then ΠC x is a single-valued mapping.
Lemma 2.7 see Li et al. 29. Let C be a nonempty, closed, and convex subset of a smooth and
f
reflexive Banach space E. Let x ∈ E and x ∈ ΠC x. Then,
φy, x
Gx,
Jx ≤ G y, Jx ,
∀y ∈ C.
2.12
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International Journal of Mathematics and Mathematical Sciences
The fixed points set FT of a relatively quasi-nonexpansive mapping is closed and
convex as given in the following lemma.
Lemma 2.8 see Chang et al. 37. Let C be a nonempty, closed, and convex subset of a uniformly
smooth and strictly convex real Banach space E which also has Kadec-Klee property. Let T be a closed
relatively quasi-nonexpansive mapping of C into itself. Then, FT is closed and convex.
Also, this following lemma will be used in the sequel.
Lemma 2.9 see Cho et al. 38. Let E be a uniformly convex real Banach space. For arbitrary
r > 0, let Br 0 : {x ∈ E : x ≤ r} and λ, μ, γ ∈ 0, 1 such that λ μ γ 1. Then, there exists
a continuous strictly increasing convex function
g : 0, 2r −→ Ê, g0 0,
2.13
such that, for every x, y, z ∈ Br 0, the following inequality holds:
λx μy γ z2 ≤ λx2 μy2 − λμg x − y .
For solving the equilibrium problem for a bifunction F : C × C →
that F satisfies the following conditions:
2.14
Ê, let us assume
A1 Fx, x 0 for all x ∈ C,
A2 F is monotone, that is, Fx, y Fy, x ≤ 0 for all x, y ∈ C,
A3 for each x, y ∈ C, limt → 0 Ftz 1 − tx, y ≤ Fx, y,
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
Lemma 2.10 see Blum and Oettli 26. Let C be a nonempty closed convex subset of a smooth,
strictly convex, and reflexive Banach space E, and let F be a bifunction of C × C into Ê satisfying
(A1)–(A4). Let r > 0 and x ∈ E. Then, there exists z ∈ C such that
1
F z, y y − z, Jz − Jx ≥ 0, ∀y ∈ K.
r
2.15
Lemma 2.11 see Takahashi and Zembayashi 39. Let C be a nonempty closed convex subset
of a smooth, strictly convex, and reflexive Banach space E. Assume that F : C × C → Ê satisfies
(A1)–(A4). For r > 0 and x ∈ E, define a mapping TrF : E → C as follows:
TrF x 1
z ∈ C : F z, y y − z, Jz − Jx ≥ 0, ∀y ∈ C
r
2.16
for all z ∈ E. Then, the following hold:
1 TrF is singlevalued,
2 TrF is firmly nonexpansive-type mapping, that is, for any x, y ∈ E,
TrF x − TrF y, JTrF x − JTrF y ≤ TrF x − TrF y, Jx − Jy ,
2.17
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9
3 FTrF EPF,
4 EPF is closed and convex.
Lemma 2.12 see Takahashi and Zembayashi 39. Let C be a nonempty closed convex subset
of a smooth, strictly convex, and reflexive Banach space E. Assume that F : C × C → Ê satisfies
(A1)–(A4), and let r > 0. Then, for each x ∈ E and q ∈ FTrF ,
φ q, TrF x φ TrF x, x ≤ φ q, x .
2.18
For the rest of this paper, the sequence {xn }∞
n0 converges strongly to p and will be
converges
weakly
to p and will be denoted by xn p
denoted by xn → p as n → ∞, {xn }∞
n0
1 2 3
1
2
3
and we will assume that βn,i , βn,i , βn,i ∈ 0, 1, for all i 1, 2, 3, . . . such that βn,i βn,i βn,i 1, for all n ≥ 0.
We recall that a Banach space E has Kadec-Klee property if, for any sequence {xn }∞
n0 ⊂ E
and x ∈ E with xn x and xn → x, xn → x as n → ∞. We note that every uniformly
convex Banach space has the Kadec-Klee property. For more details on Kadec-Klee property,
the reader is referred to 2, 33.
Lemma 2.13 see Li et al. 29. Let E be a Banach space and y ∈ E. Let f : E → Ê ∪ {∞} be a
proper, convex, and lower semicontinuous mapping with convex domain Df. If {xn } is a sequence in
Df such that xn x ∈ intDf and limn → ∞ Gxn , Jy Gx, Jy, then limn → ∞ xn x.
3. Main Results
Theorem 3.1. Let E be a uniformly smooth and strictly convex real Banach space which also has
Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m,
∞
two countable
let Fk be a bifunction from C × C satisfying (A1)–(A4). Suppose {Ti }∞
i1 and {Si }i1 are
families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω : m
k1 EPFk ∩
∞
∞
FT
∩
FS
∅.
Let
f
:
E
→
Ê
be
a
convex
and
lower
semicontinuous
mapping
/
i
i
i1
i1
is
iteratively
generated
by
x
∈
C,
C
C,
C1 with C ⊂ intDf, and suppose {xn }∞
0
1,i
n0
f
∞
∩i1 C1,i , x1 ΠC1 x0 ,
1
2
3
zn,i J −1 βn,i Jxn βn,i JTi xn βn,i JSixn ,
yn,i J −1 αn,i Jxn 1 − αn,i Jzn,i ,
F2
F1
m−1
un,i TrFmm,n TrFm−1
,n · · · Tr2 ,n Tr1 ,n yn,i ,
Cn1,i {z ∈ Cn,i : Gz, Jun,i ≤ Gz, Jxn },
Cn1 ∞
Cn1,i ,
i1
f
xn1 ΠCn1 x0 ,
n ≥ 1,
3.1
10
International Journal of Mathematics and Mathematical Sciences
with the conditions
1 2
i lim infn → ∞ βn,i βn,i > 0,
1 3
ii lim infn → ∞ βn,i βn,i > 0,
iii 0 ≤ αn,i ≤ α < 1 for some α ∈ 0, 1,
iv {rk,n }∞
n1 ⊂ 0, ∞ k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0 k 1, 2, . . . , m.
f
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
Proof. We first show that Cn , for all n ≥ 1 is closed and convex. It is obvious that C1,i C is
closed and convex. Suppose Ck,i is closed and convex for some k > 1. For each z ∈ Ck,i , we
see that Gz, Juk,i ≤ Gz, Jxk is equivalent to
2z, Jxk − z, Juk,i ≤ xk 2 − uk,i 2 .
3.2
By the construction of the set Ck1,i , we see that Ck1,i is closed and convex. Therefore, Ck1 ∞
i1 Ck1,i is also closed and convex. Hence, Cn , for all n ≥ 1 is closed and convex.
F2
F1
k−1
0
By taking θnk TrFkk,n TrFk−1
,n · · · Tr2 ,n Tr1 ,n , k 1, 2, . . . , m and θn I for all n ≥ 1, we obtain
m
un,i θn yn,i .
We next show that Ω ⊂ Cn , for all n ≥ 1. For n 1, we have Ω ⊂ C C1 . Then, for
each x∗ ∈ Ω, we obtain
Gx∗ , Jun,i G x∗ , Jθnm yn,i ≤ G x∗ , Jyn,i
Gx∗ , αn,i Jxn 1 − αn,i Jzn,i x∗ 2 −2αn,i x∗ , Jxn −21 − αn,i x∗ , Jzn,i αn,i Jxn 1 − αn,i Jzn,i 2 2ρfx∗ ≤ x∗ 2 −2αn,i x∗ , Jxn −21 − αn,i x∗ , Jzn,i αn,i xn 2 1−αn,i zn,i 2 2ρfx∗ αn,i Gx∗ , Jxn 1 − αn,i Gx∗ , Jzn,i
1
2
3
αn,i Gx∗ , Jxn 1 − αn,i G x∗ , βn,i Jxn βn,i JTi xn βn,i JSi xn
1
≤ αn,i Gx∗ , Jxn 1 − αn,i x∗ 2 − 2βn,i x∗ , Jxn 2
3
1
− 2βn,i x∗ , JTi xn − 2βn,i x∗ , JSi xn βn,i xn 2
2
3
βn,i Ti xn 2 βn,i Si xn 2 2ρfx∗ 1
2
3
αn,i Gx∗ , Jxn 1 − αn,i βn,i Gx∗ , Jxn βn,i Gx∗ , JTi xn βn,i Gx∗ , JSixn ≤ Gx∗ , Jxn .
3.3
So, x∗ ∈ Cn . This implies that Ω ⊂ Cn , for all n ≥ 1. Therefore, {xn } is well defined.
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11
We now show that limn → ∞ Gxn , Jx0 exists. Since f : E → Ê is convex and lower
semicontinuous, applying Lemma 2.5, we see that there exists u∗ ∈ E∗ and α ∈ Ê such that
f y ≥ y, u∗ α,
∀y ∈ E.
3.4
It follows that
Gxn , Jx0 xn 2 − 2xn , Jx0 x0 2 2ρfxn ≥ xn 2 − 2xn , Jx0 x0 2 2ρxn , u∗ 2ρα
xn 2 − 2 xn , Jx0 − ρu∗ x0 2 2ρα
3.5
≥ xn 2 − 2xn Jx0 − ρu∗ x0 2 2ρα
2
2
xn − Jx0 − ρu∗ x0 2 − Jx0 − ρu∗ 2ρα.
f
Since xn ΠCn x0 , it follows from 3.5 that
2
2
Gx∗ , Jx0 ≥ Gxn , Jx0 ≥ xn − Jx0 − ρu∗ x0 2 − Jx0 − ρu∗ 2ρα
3.6
∞
for each x∗ ∈ F. This implies that {xn }∞
n0 is bounded and so is {Gxn , Jx0 }n0 . By the
f
construction of Cn , we have that Cn1 ⊂ Cn and xn1 ΠCn1 x0 ∈ Cn . It then follows from
Lemma 2.7 that
φxn1 , xn Gxn , Jx0 ≤ Gxn1 , Jx0 .
3.7
φxn1 , xn ≥ xn1 − xn 2 ≥ 0,
3.8
It is obvious that
∞
and so {Gxn , Jx0 }∞
n0 is nondecreasing. It follows that the limit of {Gxn , Jx0 }n0 exists.
∞
Now since {xn }n0 is bounded in C and E is reflexive, we may assume that xn p,
and since Cn is closed and convex for each n ≥ 1, it is easy to see that p ∈ Cn for each n ≥ 1.
f
f
Again since xn ΠCn x0 , from the definition of ΠCn , we obtain
Gxn , Jx0 ≤ G p, Jx0 ,
∀n ≥ 1.
3.9
Since
lim inf Gxn , Jx0 lim inf xn 2 − 2xn , Jx0 x0 2 2ρfxn n→∞
n→∞
2
≥ p − 2 p, Jx0 x0 2 2ρf p G p, Jx0 ,
3.10
12
International Journal of Mathematics and Mathematical Sciences
then we obtain
G p, Jx0 ≤ lim inf Gxn , Jx0 ≤ lim supGxn , Jx0 ≤ G p, Jx0 .
n→∞
n→∞
3.11
This implies that limn → ∞ Gxn , Jx0 Gp, Jx0 . By Lemma 2.13, we obtain limn → ∞ xn p. In view of Kadec-Klee property of E, we have that limn → ∞ xn p.
∞
∞
We next show that p ∈ m
k1 EPFk ∩ i1 FTi ∩ i1 FSi . We first show that
∞
∞
f
p ∈ i1 FTi ∩ i1 FSi . By the fact that Cn1 ⊂ Cn and xn1 ΠCn1 x0 ∈ Cn , we obtain
φxn1 , un,i ≤ φxn1 , xn .
3.12
φxn1 , un,i ≤ φxn1 , xn ≤ Gxn1 , Jx0 − Gxn , Jx0 .
3.13
Now, 3.7 implies that
Taking the limit as n → ∞ in 3.13, we obtain
lim φxn1 , xn 0.
n→∞
3.14
Therefore,
lim φxn1 , un,i 0,
n→∞
∀i ≥ 1.
3.15
It then yields that limn → ∞ xn1 − un,i 0, for all i ≥ 1. Since limn → ∞ xn1 p, we
have
lim un,i p,
n→∞
∀i ≥ 1.
3.16
Hence,
lim Jun,i Jp,
n→∞
∀i ≥ 1.
3.17
∗
∗
This implies that {Jun,i}∞
n0 , i ≥ 1 is bounded in E . Since E is reflexive, and so E is
∗
reflexive, we can then assume that Jun,i f0 ∈ E , for all i ≥ 1. In view of reflexivity of
E, we see that JE E∗ . Hence, there exists x ∈ E such that Jx f0 . Since
φxn1 , un,i xn1 2 − 2xn1 , Jun,i un,i 2
3.18
xn1 2 − 2xn1 , Jun,i Jun,i 2 ,
International Journal of Mathematics and Mathematical Sciences
13
taking the limit inferior of both sides of 3.18 and in view of weak lower semicontinuity of
· , we have
2
2 2
0 ≥ p − 2 p, f0 f0 p − 2 p, Jx Jx2
3.19
2
p − 2 p, Jx x2 φ p, x ,
that is, p x. This implies that f0 Jp and so Jun,i Jp, for all i ≥ 1. It follows from
limn → ∞ Jun,i Jp, for all i ≥ 1 and Kadec-Klee property of E∗ that Jun,i → Jp, for all i ≥
1. Note that J −1 : E∗ → E is hemicontinuous; it yields that un,i p, for all i ≥ 1. It
then follows from limn → ∞ un,i p, for all i ≥ 1 and Kadec-Klee property of E that
limn → ∞ un,i p, for all i ≥ 1. Hence,
lim xn − un,i 0,
n→∞
∀i ≥ 1.
3.20
Since J is uniformly norm-to-norm continuous on bounded sets and limn → ∞ xn − un,i 0, for all i ≥ 1, we obtain
lim Jxn − Jun,i 0,
n→∞
∀i ≥ 1.
3.21
Since {xn } is bounded, so are {zn,i }, {JTi xn }, and {JSi xn }. Also, since E is uniformly smooth,
E∗ is uniformly convex. Then, from Lemma 2.9, we have
Gx∗ , Jun,i G x∗ , Jθnm yn,i ≤ G x∗ , Jyn,i
Gx∗ , αn,i Jxn 1 − αn,i Jzn,i x∗ 2 −2αn,i x∗ , Jxn −21−αn,i x∗ , Jzn,i αn,i Jxn1 − αn,i Jzn,i 2 2ρfx∗ ≤ x∗ 2 −2αn,i x∗ , Jxn −21−αn,i x∗ , Jzn,i αn,i xn 2 1 − αn,i zn,i 2 2ρfx∗ αn,i Gx∗ , Jxn 1 − αn,i Gx∗ , Jzn,i
1
2
3
αn,i Gx∗ , Jxn 1 − αn,i G x∗ , βn,i Jxn βn,i JTi xn βn,i JSi xn
1
2
≤ αn,i Gx∗ , Jxn 1−αn,i x∗ 2 −2βn,i x∗ , Jxn − 2βn,i x∗ , JTi xn 3
1
2
3
−2βn,i x∗ , JSi xn βn,i xn 2 βn,i Ti xn 2 βn,i Si xn 2
1 2
−βn,i βn,i gJxn − JTi xn 2ρfx∗ 14
International Journal of Mathematics and Mathematical Sciences
1
2
αn,i Gx∗ , Jxn 1 − αn,i βn,i Gx∗ , Jxn βn,i Gx∗ , JTi xn 3
1 2
βn,i Gx∗ , JSixn − βn,i βn,i gJxn − JTi xn 1
2
≤ αn,i Gx∗ , Jxn 1 − αn,i βn,i Gx∗ , Jxn βn,i Gx∗ , Jxn 3
1 2
βn,i Gx∗ , Jxn − βn,i βn,i gJxn − JTi xn 1 2
αn,i Gx∗ , xn 1 − αn,i Gx∗ , Jxn − βn,i βn,i gJxn − JTi xn 1 2
≤ Gx∗ , Jxn − 1 − αn,i βn,i βn,i gJxn − JTi xn .
3.22
It then follows that
1 2
1 2
1 − αβn,i βn,i gJxn − JTi xn ≤ 1 − αn,i βn,i βn,i gJxn − JTi xn ≤ Gx∗ , Jxn − Gx∗ , Jun,i.
3.23
But
Gx∗ , Jxn − Gx∗ , Jun,i xn 2 − un,i 2 − 2x∗ , Jxn − Jun,i
≤ xn 2 − un,i 2 2x∗ , Jxn − Jun,i 3.24
∗
≤ |xn − un,i |xn un,i 2x Jxn − Jun,i ≤ xn − un,i xn un,i 2x∗ Jxn − Jun,i .
From limn → ∞ xn − un,i 0 and limn → ∞ Jxn − Jun,i 0, we obtain
Gx∗ , Jxn − Gx∗ , Jun,i −→ 0,
n −→ ∞.
3.25
1 2
Using the condition lim infn → ∞ βn,i βn,i > 0, we have
lim gJxn − JTi xn 0,
n→∞
∀i ≥ 1.
3.26
By property of g, we have limn → ∞ Jxn − JTi xn 0, for all i ≥ 1. Since J −1 is also uniformly
norm-to-norm continuous on bounded sets, we have
lim xn − Ti xn 0,
n→∞
∀i ≥ 1.
3.27
International Journal of Mathematics and Mathematical Sciences
15
Similarly, we can show that
lim xn − Si xn 0,
n→∞
∀i ≥ 1.
3.28
∞
Since xn → p and Ti , Si are closed, we have p ∈ ∞
i1 FTi ∩ i1 FSi .
m
Next, we show that p ∈ k1 EPFk . Now, by Lemma 2.12, we obtain
φ un,i , yn,i φ θnm yn,i , yn,i
≤ φ x∗ , yn,i − φ x∗ , θnm yn,i
≤ φx∗ , xn − φx∗ , un,i −→ 0,
3.29
n −→ ∞.
It then yields that limn → ∞ un,i − yn,i 0. Since limn → ∞ un,i p, i ≥ 1, we have
lim yn,i p,
n→∞
i ≥ 1.
3.30
Hence,
lim Jyn,i Jp,
n→∞
i ≥ 1.
3.31
∗
∗
This implies that {Jyn,i }∞
n0 is bounded in E . Since E is reflexive, and so E is reflexive, we
∗
can then assume that Jyn,i f1 ∈ E . In view of reflexivity of E, we see that JE E∗ . Hence,
there exists z ∈ E such that Jz f1 . Since
2
φ un,i , yn,i un,i 2 − 2 un,i , Jyn,i yn,i 2
un,i 2 − 2 un,i , Jyn,i Jyn,i ,
3.32
taking the limit inferior of both sides of 3.32 and in view of weak lower semicontinuity of
· , we have
2
2 2
0 ≥ p − 2 p, f1 f1 p − 2 p, Jz Jz2
2
p − 2 p, Jz z2 φ p, z ,
3.33
that is, p z. This implies that f1 Jp and so Jyn,i Jp. It follows from limn → ∞ Jyn,i Jp and Kadec-Klee property of E∗ that Jyn,i → Jp. Note that J −1 : E∗ → E is hemicontinuous; it yields that yn,i p. It then follows from limn → ∞ yn,i p and KadecKlee property of E that limn → ∞ yn,i p, i ≥ 1. By the fact that θnk , k 1, 2, . . . , m is relatively
nonexpansive and using Lemma 2.12 again, we have that
φ θnk yn,i , yn,i ≤ φ x∗ , yn,i − φ x∗ , θnk yn,i
≤ φx∗ , xn − φ x∗ , θnk yn,i .
3.34
16
International Journal of Mathematics and Mathematical Sciences
Observe that
φx∗ , un,i φ x∗ , θnm yn,i
Fk
Fk−1
F2
F1
m−1
φ x∗ , TrFmm,n TrFm−1
,n · · · Trk ,n Trk−1 ,n · · · Tr2 ,n Tr1 ,n yn,i
3.35
k
m−1
φ x∗ , TrFmm,n TrFm−1
,n · · · θn yn,i
≤ φ x∗ , θnk yn,i .
Using 3.35 in 3.34, we obtain
φ θnk yn,i , yn,i ≤ φx∗ , xn − φx∗ , un,i −→ 0,
n −→ ∞.
3.36
It then yields that limn → ∞ θnk yn,i − yn,i 0. Since limn → ∞ yn,i p, we have
lim θnk yn,i p,
n→∞
k 1, 2, . . . , m.
3.37
This implies that {θnk yn,i }∞
n0 is bounded in E. Since E is reflexive, we can then assume that
θnk yn,i w ∈ E. Since
2
2
φ θnk yn,i , yn,i θnk yn,i − 2 θnk yn,i , Jyn,i yn,i 2
2
θnk yn,i − 2 θnk yn,i , Jyn,i Jyn,i ,
3.38
taking the limit inferior of both sides of 3.38 and in view of weak lower semicontinuity of
· , we have
2
2
0 ≥ w2 − 2 w, Jp p w2 − 2 w, Jp Jp
φ w, p ,
3.39
that is, p w. This implies that θnk yn,i p. It follows from limn → ∞ θnk yn,i p and KadecKlee property of E that
θnk yn,i −→ p,
n −→ ∞, k 1, 2, . . . , m.
3.40
Similarly, limn → ∞ p − θnk−1 yn,i 0, k 1, 2, . . . , m. This further implies that
lim θnk yn,i − θnk−1 yn,i 0,
n→∞
i ≥ 1.
3.41
International Journal of Mathematics and Mathematical Sciences
17
Also, since J is uniformly norm-to-norm continuous on bounded sets and using 3.41, we
obtain
lim Jθnk yn,i − Jθnk−1 yn,i 0,
i ≥ 1.
n→∞
3.42
Since lim infn → ∞ rk,n > 0 k 1, 2, . . . , m,
lim
k
Jθn yn,i − Jθnk−1 yn,i n→∞
rk,n
0.
3.43
By Lemma 2.11, we have that for each k 1, 2, . . . , m
1 Fk θnk yn,i , y y − θnk yn,i , Jθnk yn,i − Jθnk−1 yn,i ≥ 0,
rk,n
∀y ∈ C.
3.44
Furthermore, using A2, we obtain
1 y − θnk yn,i , Jθnk yn,i − Jθnk−1 yn,i ≥ Fk y, θnk yn,i .
rk,n
3.45
By A4, 3.43, and θnk yn,i → p, we have for each k 1, 2, . . . , m
Fk y, p ≤ 0,
∀y ∈ C.
3.46
For fixed y ∈ C, let zt,y : ty 1 − tp for all t ∈ 0, 1. This implies that zt,y ∈ C. This yields
that Fk zt,y , p ≤ 0. It follows from A1 and A4 that
0 Fk zt,y , zt,y ≤ tFk zt,y , y 1 − tFk zt,y , p
≤ tFk zt,y , y ,
3.47
and hence
0 ≤ Fk zt,y , y .
3.48
From condition A3, we obtain
Fk p, y ≥ 0,
∀y ∈ C.
This implies that p ∈ EPFk , k 1, 2, . . . , m. Thus, p ∈
m
∞
∞
k1 EPFk ∩ n0 FTi ∩ i1 FSi .
m
k1
3.49
EPFk . Hence, we have p ∈ Ω 18
International Journal of Mathematics and Mathematical Sciences
f
Finally, we show that p ΠΩ x0 . Since Ω m
k1
∞
EPFk ∩ ∞
n0 FTi ∩ i1 FSi f
is a closed and convex set, from Lemma 2.6, we know that ΠF x0 is single valued and denote
w
f
ΠΩ x0 .
Since xn f
ΠCn x0
and w ∈ Ω ⊂ Cn , we have
Gxn , Jx0 ≤ Gw, Jx0 ,
∀n ≥ 1.
3.50
We know that Gξ, Jϕ is convex and lower semicontinuous with respect to ξ when ϕ is fixed.
This implies that
G p, Jx0 ≤ lim inf Gxn , Jx0 ≤ lim sup Gxn , Jx0 ≤ Gw, Jx0 .
n→∞
3.51
n→∞
f
From the definition of ΠΩ x0 and p ∈ Ω, we see that p w. This completes the proof.
f
Take fx 0 for all x ∈ E in Theorem 3.1, then Gξ, Jx φξ, x and ΠC x0 ΠC x0 .
Then we obtain the following corollary.
Corollary 3.2. Let E be a uniformly smooth and strictly convex real Banach space which also has
Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m,
∞
let Fk be a bifunction from C × C satisfying (A1)–(A4). Suppose {Ti }∞
two countable
i1 and {Si }i1 are
families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω : m
k1 EPFk ∩
∞
∞
∅. Suppose {xn }∞
n0 is iteratively generated by x0 ∈ C, C1,i C, C1 n1 FTi ∩ n1 FSi /
∩∞
i1 C1,i , x1 ΠC1 x0 ,
1
2
3
zn,i J −1 βn,i Jxn βn,i JTi xn βn,i JSixn ,
yn,i J −1 αn,i Jxn 1 − αn,i Jzn,i ,
F2
F1
m−1
un,i TrFmm,n TrFm−1
,n · · · Tr2 ,n Tr1 ,n yn,i ,
Cn1,i z ∈ Cn,i : φz, un,i ≤ φz, xn ,
Cn1 ∞
3.52
Cn1,i ,
i1
xn1 ΠCn1 x0 ,
n ≥ 1,
with the conditions
1 2
i lim infn → ∞ βn,i βn,i > 0,
1 3
ii lim infn → ∞ βn,i βn,i > 0,
iii 0 ≤ αn,i ≤ α < 1 for some α ∈ 0, 1,
iv {rk,n }∞
n1 ⊂ 0, ∞ k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0 k 1, 2, . . . , m.
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
International Journal of Mathematics and Mathematical Sciences
19
Corollary 3.3 see Li et al. 29. Let E be a uniformly convex real Banach space which is also
uniformly smooth. Let C be a nonempty, closed, and convex subset of E. Suppose T is a relatively
nonexpansive mapping of C into itself such that Ω : FT /
∅. Let f : E → Ê be a convex and
lower semicontinuous mapping with C ⊂ intDf, and suppose {xn }∞
n0 is iteratively generated by
x0 ∈ C, C0 C,
Cn1
yn J −1 αn Jxn 1 − αn JTxn ,
w ∈ Cn : G w, Jyn ≤ Gw, Jxn ,
f
xn1 ΠCn1 x0 ,
3.53
n ≥ 0.
∞
Suppose {αn }∞
n1 is a sequence in 0, 1 such that lim supn → ∞ αn < 1. Then, {xn }n0 converges
strongly to ΠΩ x0 .
Corollary 3.4 see Takahashi and Zembayashi 9. Let E be a uniformly convex real Banach
space which is also uniformly smooth. Let C be a nonempty, closed, and convex subset of E. Let F be
a bifunction from C × C satisfying (A1)–(A4). Suppose T is a relatively nonexpansive mapping of C
into itself such that Ω : EPF ∩ FT /
∅. Let {xn }∞
n0 be iteratively generated by x0 ∈ C, C1 C,
x1 ΠC1 x0 ,
yn J −1 αn,i Jxn 1 − αn,i JTxn ,
1
y − un , Jun − Jyn ≥ 0, ∀y ∈ C,
F un , y rn
Cn1 w ∈ Cn : φw, un ≤ φw, xn ,
xn1 ΠCn1 x0 ,
3.54
n ≥ 1,
where J is the duality mapping on E. Suppose {αn,i }∞
n1 is a sequence in 0, 1 such that
⊂
0,
∞
satisfying
lim infn → ∞ rn > 0. Then, {xn }∞
lim infn → ∞ αn,i 1 − αn,i > 0 and {rn }∞
n1
n0
converges strongly to ΠΩ x0 .
4. Applications
Let A be a monotone operator from C into E∗ , the classical variational inequality is to find
x∗ ∈ C such that
y − x, Ax∗ ≥ 0,
∀y ∈ C.
4.1
The set of solutions of 4.1 is denoted by VIC, A.
Let ϕ : C → Ê be a real-valued function. The convex minimization problem is to find
x∗ ∈ C such that
ϕx∗ ≤ ϕ y ,
∀y ∈ C.
The set of solutions of 4.2 is denoted by CMPϕ.
4.2
20
International Journal of Mathematics and Mathematical Sciences
The following lemmas are special cases of Lemmas 2.8 and Lemma 2.9 of 39.
Lemma 4.1. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive
Banach space E. Assume that A : C → E∗ is a continuous and monotone operator. For r > 0 and
x ∈ E, define a mapping TrA : E → C as follows:
TrA x
1
z ∈ C : Az, y − z y − z, Jz − Jx ≥ 0, ∀y ∈ C .
r
4.3
Then, the following hold:
1 TrA is singlevalued,
2 FTrA VIC, A,
3 VIC, A is closed and convex,
4 φq, TrA x φTrA x, x ≤ φq, x, for all q ∈ FTrA .
Lemma 4.2. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive
Banach space E. Assume that ϕ : C → Ê is lower semicontinuous and convex. For r > 0 and x ∈ E,
ϕ
define a mapping Tr : E → C as follows:
ϕ
Tr x 1
z ∈ C : ϕ y y − z, Jz − Jx ≥ ϕz, ∀y ∈ C .
r
4.4
Then, the following hold:
ϕ
1 Tr is single valued,
ϕ
2 FTr CMPϕ,
3 CMPϕ is closed and convex,
ϕ
ϕ
ϕ
4 φq, Tr x φTr x, x ≤ φq, x, for all q ∈ FTr .
Then we obtain the following theorems from Theorem 3.1.
Theorem 4.3. Let E be a uniformly smooth and strictly convex real Banach space which also has
Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m,
∞
let Ak be a continuous and monotone operator from C into E∗ . Suppose {Ti }∞
i1 and {Si }i1 are
two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that
∞
∞
∅. Let f : E → Ê be a convex and lower
Ω : m
k1 VIC, Ak ∩ i1 FTi ∩ i1 FSi /
International Journal of Mathematics and Mathematical Sciences
21
semicontinuous mapping with C ⊂ intDf, and suppose {xn }∞
n0 is iteratively generated by
f
∞
x0 ∈ C, C1,i C, C1 ∩i1 C1,i , x1 ΠC1 x0 ,
1
2
3
zn,i J −1 βn,i Jxn βn,i JTi xn βn,i JSixn ,
yn,i J −1 αn,i Jxn 1 − αn,i Jzn,i ,
A2 A1
m−1
un,i TrAmm,n TrAm−1
,n · · · Tr2 ,n Tr1 ,n yn,i ,
Cn1,i {z ∈ Cn,i : Gz, Jun,i ≤ Gz, Jxn },
Cn1 ∞
4.5
Cn1,i ,
i1
f
xn1 ΠCn1 x0 ,
n ≥ 1,
with the conditions
1 2
i lim infn → ∞ βn,i βn,i > 0,
1 3
ii lim infn → ∞ βn,i βn,i > 0,
iii 0 ≤ αn,i ≤ α < 1 for some α ∈ 0, 1,
iv {rk,n }∞
n1 ⊂ 0, ∞ k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0 k 1, 2, . . . , m.
f
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
Theorem 4.4. Let E be a uniformly smooth and strictly convex real Banach space which also has
Kadec-Klee property. Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . . , m,
∞
let ϕk : C → Ê be lower semicontinuous and convex. Suppose {Ti }∞
i1 and {Si }i1 are two
countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω :
m
∞
∞
∅. Let f : E → Ê be a convex and lower semik1 CMPϕk ∩ i1 FTi ∩ i1 FSi /
continuous mapping with C ⊂ intDf, and suppose {xn }∞
n0 is iteratively generated by x0 ∈ C,
f
C1,i C, C1 ∞
C
,
x
Π
x
,
1
i1 1,i
C1 0
1
2
3
zn,i J −1 βn,i Jxn βn,i JTi xn βn,i JSixn ,
yn,i J −1 αn,i Jxn 1 − αn,i Jzn,i ,
ϕ
ϕ
ϕ
ϕ
m−1
2
1
un,i Trmm,n Trm−1
,n · · · Tr2 ,n Tr1 ,n yn,i ,
Cn1,i {z ∈ Cn,i : Gz, Jun,i ≤ Gz, Jxn },
Cn1 ∞
Cn1,i ,
i1
f
xn1 ΠCn1 x0 ,
with the conditions
n ≥ 1,
4.6
22
International Journal of Mathematics and Mathematical Sciences
1 2
i lim infn → ∞ βn,i βn,i > 0,
1 3
ii lim infn → ∞ βn,i βn,i > 0,
iii 0 ≤ αn,i ≤ α < 1 for some α ∈ 0, 1,
iv {rk,n }∞
n1 ⊂ 0, ∞ k 1, 2, . . . , m satisfying lim infn → ∞ rk,n > 0 k 1, 2, . . . , m.
f
Then, {xn }∞
n0 converges strongly to ΠΩ x0 .
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