[ DOI: 10.7508/ijmsi.2016.01.007 ]
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Iranian Journal of Mathematical Sciences and Informatics
Vol. 11, No. 1 (2016), pp 69-83
DOI: 10.7508/ijmsi.2016.01.007
An Explicit Viscosity Iterative Algorithm for Finding Fixed
Points of Two Noncommutative Nonexpansive Mappings
H. R. Sahebi, A. Razani∗
Department of Mathematics, Science and Research Branch, Islamic Azad
University, Tehran, Iran.
E-mail: [email protected]
E-mail: [email protected]
Abstract. We suggest an explicit viscosity iterative algorithm for finding a common element in the set of solutions of the general equilibrium
problem system (GEPS) and the set of all common fixed points of two
noncommuting nonexpansive self mappings in the real Hilbert space.
Keywords: General equilibrium problems, Strongly positive linear bounded
operator, α−Inverse strongly monotone mapping, Fixed point, Hilbert space.
2000 Mathematics subject classification: 47H09, 47H10, 47J20.
1. Introduction
Let H be a real Hilbert space with inner product h., .i and norm ||.||. Recall
that a mapping T with domain D(T ) and range R(T ) in H is called nonexpansive iff for all x, y ∈ D(T ),
kT x − T yk ≤ kx − yk.
F (T ) denotes the set of fixed points of T . Moreover, H satisfies the Opial’s
condition [6], if for any sequence {xn } with xn ⇀ x, the inequality
lim inf kxn − xk < lim inf kxn − yk,
n→∞
∗ Corresponding
n→∞
Author
Received 06 June 2014; Accepted 17 January 2015
c
2016
Academic Center for Education, Culture and Research TMU
69
[ DOI: 10.7508/ijmsi.2016.01.007 ]
70
H. R. Sahebi, A. Razani
holds for every y ∈ H with x 6= y.
Recall that f is said to be weakly contractive [2] iff for all x, y ∈ D(T ),
kf (x) − f (y)k ≤ kx − yk − φ(kx − yk),
for some φ : [0, ∞) → [0, ∞) is a continuous and strictly increasing function
such that φ is positive on (0, ∞) and φ(0) = 0. A mapping A is a strongly
positive linear bounded operator on H if there exists a constant γ̄ > 0 such
that
hAx, xi ≥ γ̄kxk2 , for all x ∈ H.
Moreover, B : C → H is called α−inverse strongly monotone if there exists a
positive real number α > 0 such that for all x, y ∈ C
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hBx − By, x − yi ≥ αkBx − Byk2 .
Let C be a nonempty closed convex subset of H. A : H → H be an inverse
strongly monotone mapping and F : C × C → R be a bifunction. The general
equilibrium problem is to find x̃ ∈ C such that for all y ∈ C,
F (x̃, y) + hAx, y − xi ≥ 0.
There are several other problems, for example, the complementarity problem,
fixed point problem and optimization problem, which can also be written in
the form of an EP. In other words, the general equilibrium problem system
(GEPS) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc [1, 4].
To study the generalized equilibrium problem, we assume that the bifunction
F satisfies the following conditions:
(A1)
(A2)
(A3)
(A4)
F (x, x) = 0, for all x ∈ C;
F is monotone, i.e., F (x, y) + F (y, x) ≤ 0 for all x, y ∈ C;
for each x, y, z ∈ C, lim supt→0− F (tz + (1 − t)x, y) ≤ F (x, y);
for each x ∈ C y 7→ F (x, y) is convex and weakly lower semi-continuous.
Recently, Yao and Chen [10] introduced a new iteration for two averaged self
mappings S and T on a closed convex subset C as follows
(
x0 = x ∈ C;
Pn Pn−i
2
j i−j
xn+1 = αn x + (1 − αn ) (n+1)(n+2)
∨ (ST )i T j−i )xn ,
i=0
j=0 ((ST ) S
where n ≥ 0 and
(ST )j S i−j ∨ (ST )i T j−i =
(ST )j S i−j
(ST )i T j−i
if i ≥ j
if i < j.
(1.1)
By improving this idea, Jankaew et al. [5] considered the following iteration:
n n−i
xn+1 = αn f (xn )+βn xn +γn
XX
2
((ST )j S i−j ∨(ST )i T j−i )xn ,
(n + 1)(n + 2) i=0 j=0
(1.2)
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[ DOI: 10.7508/ijmsi.2016.01.007 ]
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71
where {αn }, {βn }, {γn } ⊂ (0, 1), αn + βn + γn = 1, f is a contraction mapping on C. They proved that the iteration process (1.2) converges strongly
to common fixed point of the mapping S and T which solves some variational
inequality.
A typical problem is to minimize a quadratic function over the set of the fixed
points of a nonexpansive mappings on a real Hilbert space H:
1
hAx, xi − h(x)
2
where A is strongly positive linear bounded operator and h is a potential function for γf , i. e., h′ (x) = γf, for all x ∈ H. In this paper, we consider and
analyze an iterative scheme for finding a common element of the set of solutions
of the general equilibrium problem system (GEPS) and the set of all common
fixed points of two noncommutative nonexpansive self mapping in the framework of a real Hilbert space. The results in this paper generalize and improve
some well known results in Jankaew et al.[5] and others.
In order to prove our main results, we need the following lemmas.
min
Lemma 1.1. [3] Let C be a nonempty closed convex subset of H and F :
C × C → R be a bifunction satisfying (A1) − (A4). Then for any r > 0 and
x ∈ H there exists z ∈ C such that
1
F (z, y) + hy − z, z − xi ≥ 0, ∀y ∈ C.
r
Further, define
1
Tr x = {z ∈ C : F (z, y) + hy − z, z − xi ≥ 0, ∀y ∈ C}
r
for all r > 0 and x ∈ H. Then
(a) Tr is single-valued;
(b) Tr is firmly nonexpansive, i.e., for any x, y ∈ H
kTr x − Tr yk2 ≤ hTr x − Tr y, x − yi;
(c) F (Tr ) = GEP (F );
(d) kTs x − Tr xk ≤ s−r
s kTs x − xk;
(e) GEP (F ) is closed and convex.
Remark 1.2. It is clear that for any x ∈ H and r > 0, by Lemma 1.1(a), there
exists z ∈ H such that
1
F (z, y) + hy − z, z − xi ≥ 0, ∀y ∈ H.
(1.3)
r
Replacing x with x − rψx in (1.3), we obtain
1
F (z, y) + hψx, y − zi + hy − z, z − xi ≥ 0, ∀y ∈ H.
r
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[ DOI: 10.7508/ijmsi.2016.01.007 ]
72
H. R. Sahebi, A. Razani
Lemma 1.3. [9] Assume {an } is a sequence of nonnegative numbers such that
an+1 ≤ (1 − αn )an + δn ,
where {αn } is a sequence in (0, 1) and {δn } is a sequence in real number such
that
∞
X
αn = ∞,
(i) lim αn = 0,
n→∞
n=1
∞
X
δn
(ii) lim sup
|δn | < ∞,
≤ 0 or
n→∞ αn
n=1
Then lim an = 0.
n→∞
Lemma 1.4. [5] Let C be a nonempty bounded closed convex subset of a Hilbert
space H, and let S, T be two nonexpansive mappings of C into itself such that
T
F (ST ) = F (S) F (T ) 6= ∅. Let {xn } be a sequence defined as follows:
xn+1
=
αn f (xn ) + βn xn
n n−i
+γn
XX
2
((ST )j S i−j ∨ (ST )i T j−i )xn ,
(n + 1)(n + 2) i=0 j=0
and put
n n−i
Λn =
XX
2
((ST )j S i−j ∨ (ST )i T j−i )xn .
(n + 1)(n + 2) i=0 j=0
Then,
lim sup
kΛn (x) − ST Λn (x)k = 0.
n→∞ x∈C
2. Explicit Viscosity Iterative Algorithm
In this section, we introduce an explicit viscosity iterative algorithm for
finding a common element of the set of solution for an equilibrium problem
system involving a bifunction defined on a closed convex subset and the set of
fixed points for two noncommutative nonexpansive mappings.
Theorem 2.1. Let x0 ∈ C, {un,i } ⊂ C and C be a nonempty closed convex
subset of a real Hilbert space H, F1 , F2 , . . . , Fk be bifunctions from C × C
to R satisfying (A1) − (A4), Ψ1 , Ψ2 , . . . , Ψk be µi −inverse strongly monotone
mapping on C, f be a weakly contractive mapping with a function φ on H, A
be a strongly positive linear bounded operator with coefficient γ̄ such that γ̄ ≤
kAk ≤ 1, B be strongly positive linear bounded operator on H with coefficient
β̄ ∈ (0, 1] such that kBk = β̄, S, T be nonexpansive mappings on C, such that
T
F (ST ) = F (T S) = F (T ) F (S) 6= ∅. Let {xn } be a sequence generated in the
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[ DOI: 10.7508/ijmsi.2016.01.007 ]
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73
following manner:

















F1 (un,1 , y) + hΨ1 xn , y − un,1 i + r1n hy − un,1 , un,1 − xn i ≥ 0, for all y ∈ C
F2 (un,2 , y) + hΨ2 xn , y − un,2 i + r1n hy − un,2 , un,2 − xn i ≥ 0, for all y ∈ C
..
.
Fk (un,k , y) + hΨk xn , y − un,k i + r1n hy − un,k , un,k − xn i ≥ 0, for all y ∈ C

Pk



ωn = k1 i=1 un,i ,






Pn Pn−i

2

j i−j

∨ (ST )i T j−i )ωn ,
Λn = (n+1)(n+2)

i=0
j=0 ((ST ) S


xn+1 = αn γf (xn ) + βn Bxn + ((1 − εn )I − βn B − αn A)Λn .
where {αn } ⊂ (0, 1), {βn } and {εn } are the sequences in [0, 1) such that εn ≤
αn and {rn } ⊂ (0, ∞) is a real sequence satisfying the following conditions:
(C1) lim αn = 0,
n→∞
∞
X
αn
n=1
P∞
n=1
= ∞,
(C2) lim βn = 0,
|βn+1 − βn | < ∞,
n→∞
P∞
(C3)
inf rn > 0 and 0 < b < rn < a < 2µi for
n=1 |rn+1 − rn | < ∞ and lim
n→∞
1 ≤ i ≤ k,
P∞
(C4)
n=1 |εn+1 − εn | < ∞,
εn
(C5) lim
= 0.
n→∞ αn
Then
(i) the sequence {xn } is bounded.
(ii) lim kxn+1 − xn k = 0.
n→∞
(iii) lim kΨi xn − Ψi x∗ k = 0. for i ∈ {1, 2, . . . , k}.
n→∞
(iv) lim kxn − Λn k = 0.
n→∞
Proof. (i) Without loss of generality, we assume that αn < (1−εn −βn kBk)kAk−1 .
Since A, B are two strongly positive bounded linear operator on H, we have
kAk = sup{|hAx, xi| : x ∈ H, kxk = 1},
kBk = sup{|hBx, xi| : x ∈ H, kxk = 1}.
Also, (1 − εn )I − βn B − αn A is positive. Indeed,
h((1 − εn )I − βn B − αn A)x, xi
=
(1 − εn )hx, xi − βn hBx, xi − αn hAx, xi
≥
1 − εn − βn kBk − αn kAk > 0.
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74
H. R. Sahebi, A. Razani
Notice that
k(1 − εn )I
−
βn B − αn Ak
=
sup{h((1 − εn )I − βn B − αn A)x, xi : x ∈ H, kxk = 1}
=
sup{(1 − εn )hx, xi − βn hBx, xi − αn hAx, xi : x ∈ H, kxk = 1}
≤
1 − εn − βn β̄ − αn γ̄
≤
1 − βn β̄ − αn γ̄.
Let Q = PF (ST ) T GEP (Fi ,Ψi ) . It is clear that Q(I − A + γf ) is a contraction.
Hence, there exists a unique element z ∈ H such that z = Q(I − A + γf )z.
Tk
T
Let x∗ ∈ i=1 F (ST ) GEP (Fi , Ψi ). For any i = 1, 2, . . . , k, I − rn Ψi is a
nonexpansive mapping and kun,i −x∗ k ≤ kxn −x∗ k. Also kωn −x∗ k ≤ kxn −x∗ k.
Thus
kxn+1 − x∗ k
=
kαn γf (xn ) + βn Bxn + ((1 − εn )I − βn B − αn A)Λn − x∗ k
≤
αn kγf (xn ) − Ax∗ k + βn kBkkxn − x∗ k
+k((1 − εn )I − βn B − αn A)kkΛn − x∗ k + εn kx∗ k
≤
αn γkf (xn ) − f (x∗ )k + αn kγf (x∗ ) − Ax∗ k + βn β̄kxn − x∗ k
+(1 − βn β̄ − αn γ̄)kxn − x∗ k + αn kx∗ k
≤
αn γkxn − x∗ k − φ(kxn − x∗ ) + αn kγf (x∗ ) − Ax∗ k
+βn β̄kxn − x∗ k + (1 − βn β̄ − αn γ̄)kxn − x∗ k + αn kx∗ k
≤
≤
(1 − (γ̄ − γ)αn )kxn − x∗ k + αn (kγf (x∗ ) − Ax∗ k + kx∗ k)
kγf (x∗ ) − Ax∗ k
}.
max{kxn − x∗ k,
γ̄ − γ
By induction
kxn − x∗ k ≤ max{kx1 − x∗ k,
kγf (x∗ ) − Ax∗ k
}.
γ̄ − γ
and the sequence {xn } is bounded and also {f (xn )}, {ωn } and {Λn } are bounded.
(ii) Note that un,i can be written as un,i = Trn ,i (xn − rn ψi xn ). It follows
from Lemma 1.1 that
kun+1,i − un,i k ≤ kxn+1 − xn k + 2Mi |rn+1 − rn |,
(2.1)
where
Mi = max{sup{
kTrn+1,i (I − rn Ψi )xn − Trn,i (I − rn Ψi )xn k
}, sup{kΨi xn k}}.
rn+1
[ DOI: 10.7508/ijmsi.2016.01.007 ]
An Explicit Viscosity Iterative Algorithm for ...
Let M =
1
k
k
X
2Mi < ∞. Next, we estimate kωn+1 − ωn k,
i=1
k
kωn+1 − ωn k ≤
1X
kun+1,i − un,i k ≤ kxn+1 − xn k + M |rn+1 − rn |.
k i=1
(2.2)
Now, we prove that lim kxn+1 − xn k = 0. We observe that
n→∞
kxn+2 − xn+1 k
=
kαn+1 γf (xn+1 ) + βn+1 Bxn+1
+((1 − εn+1 )I − βn+1 B − αn+1 A)Λn+1 − αn+1 γf (xn )
−βn Bxn − ((1 − εn )I − βn B − αn A)Λn k
=
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k((1 − εn+1 )I − βn+1 B − αn+1 A)(Λn+1 − Λn )
+{(εn − εn+1 )Λn + (βn − βn+1 )BΛn + (αn − αn+1 )AΛn }
+αn+1 γ(f (xn+1 ) − f (xn )) + (αn+1 − αn )γf (xn )
+βn+1 B(xn+1 − xn ) + (βn+1 − βn )Bxn k
≤
k(1 − εn+1 )I − βn+1 B − αn+1 AkkΛn+1 − Λn k
+kεn − εn+1 kkΛn k + |βn − βn+1 |kBkΛn k
+|αn − αn+1 |kAΛn k + αn+1 γkf (xn+1 ) − f (xn )k
+|αn+1 − αn |γkf (xn )k + βn+1 kBkkxn+1 − xn k
+|βn+1 − βn |kBkkxn k
≤
(1 − βn+1 β̄ − αn+1 γ̄)kΛn+1 − Λn k + |εn − εn+1 |kΛn k
+|βn − βn+1 |β̄kΛn k + |αn − αn+1 |kAΛn k
+αn+1 γkxn+1 − xn k − αn+1 γφ(kxn+1 − xn k)
+|αn+1 − αn |γkf (xn )k + βn+1 β̄kxn+1 − xn k
+|βn+1 − βn |β̄kxn k
≤
(1 − βn+1 β̄ − αn+1 γ̄)kΛn+1 − Λn k + |εn − εn+1 |kΛn k
+|βn − βn+1 |β̄K + K|αn − αn+1 |
+(αn+1 γ + βn+1 β̄)kxn+1 − xn k − αn+1 γφ(kxn+1 − xn k)
where K = sup{max{kΛn k + kxn k, γkf (xn )k + kAΛn k}, ∀n ≥ 0} < ∞.
Let ∆n = kβn − βn+1 kβ̄K + K|αn − αn+1 | + |εn − εn+1 |kΛn k, then
kxn+2 − xn+1 k ≤
(1 − βn+1 β̄ − αn+1 γ̄)kΛn+1 − Λn k
+(αn+1 γ + βn+1 β̄)kxn+1 − xn k
−αn+1 γφ(kxn+1 − xn k) + ∆n ,
(2.3)
From [5], we conclude
kΛn+1 − Λn k ≤ kωn+1 − ωn k +
4
4
kωn+1 − x∗ k +
kx∗ k.
n+3
n+3
(2.4)
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H. R. Sahebi, A. Razani
Substituting (2.2) and (2.4) into (2.3), thus
kxn+2 − xn+1 k
≤
(1 − βn+1 β̄ − αn+1 γ̄){kxn+1 − xn k + M |rn+1 − rn |
4
4
kωn+1 − x∗ k +
kx∗ k}
+
n+3
n+3
+(αn+1 γ + βn+1 β̄)kxn+1 − xn k − αn+1 γφ(kxn+1 − xn k)
+∆n ,
for some positive constant M . It follows that
kxn+2 − xn+1 k
≤
(1 − αn+1 (γ̄ − γρ))kxn+1 − xn k
+M (1 − βn+1 β̄ − αn+1 γ̄)|rn+1 − rn |
4
+(1 − βn+1 β̄ − αn+1 γ̄)
kωn+1 − x∗ k
n+3
4
kx∗ k + ∆n .
+(1 − βn+1 β̄ − αn+1 γ̄)
n+3
By Lemma 1.3,
lim kxn+1 − xn k = 0.
n→∞
(2.5)
(iii) For any i ∈ {1, 2, . . . , k},
kun,i − x∗ k2
≤
k(xn − x∗ ) − rn (Ψi xn − Ψi x∗ )k2
=
kxn − x∗ k2 − 2rn hxn − x∗ , Ψi xn − Ψi x∗ i + rn2 kΨi xn − Ψi x∗ k2
≤
kxn − x∗ k2 − rn (2µi − rn )kΨi xn − Ψi x∗ k2 ,
thus
kωn − x∗ k2 =
≤
k
X
1
(un,i − x∗ )k2
k
k
i=1
k
X
1
kun,i − x∗ k2
k
i=1
≤
∗ 2
kxn − x k −
1
k
k
X
i=1
rn (2µi − rn )kΨi xn − Ψi x∗ )k2 .
(2.6)
[ DOI: 10.7508/ijmsi.2016.01.007 ]
An Explicit Viscosity Iterative Algorithm for ...
From (2.6),
kxn+1 − x∗ k2
=
kαn (γf (xn ) − Ax∗ ) + βn B(xn − x∗ )
+((1 − εn )I − βn B − αn A)(Λn − x∗ ) − εn x∗ k2
≤
kαn (γf (xn ) − Ax∗ ) + βn B(xn − x∗ )
+((1 − εn )I − βn B − αn A)(ωn − x∗ ) + εn x∗ k2
≤
αn kγf (xn ) − Ax∗ k2 + βn kBk2 kxn − x∗ k2
+(1 − βn β̄ − αn γ̄)kΛn − x∗ k2 + ε2n kx∗ k2
≤
αn kγf (xn ) − Ax∗ k2 + βn β̄ 2 kxn − x∗ k2
+(1 − βn β̄ − αn γ̄)kωn − x∗ k2 + ε2n kx∗ k2
≤
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αn kγf (xn ) − Ax∗ k2 + βn β̄kxn − x∗ k2
+(1 − βn β̄ − αn γ̄){kxn − x∗ k2
k
−
≤
1X
rn (2µi − rn )kΨi xn − Ψi x∗ )k2 } + ε2n kx∗ k2
k i=1
αn kγf (xn ) − Ax∗ k2 + kxn − x∗ k2
k
−(1 − βn β̄ − αn γ̄)
+ε2n kx∗ k2
1X
rn (2µi − rn )kΨi xn − Ψi x∗ )k2
k i=1
and hence
(1
−βn β̄ − αn γ̄) k1
k
X
b(2µi − a)kΨi xn − Ψi x∗ k2
i=1
≤ αn kγf (xn ) − Ax∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2 + ε2n kx∗ k2
≤ αn kγf (xn ) − Ax∗ k2 + kxn+1 − xn k(kxn+1 − x∗ k − kxn − x∗ k)
+ε2n kx∗ k2 .
Since αn → 0 and εn ≤ αn then εn → 0 as n → ∞. The inequality (2.5)
implies that
lim kΨi xn − Ψi x∗ k = 0, ∀i = 1, 2, . . . , k.
n→∞
(2.7)
(iv) By Lemma 1.1
kun,i − x∗ k2 ≤ kxn − x∗ k2 − kxn − un,i k2 + 2rn kxn − un,i kkΨi xn − Ψi x∗ k(2.8)
and hence
kωn − x∗ k2 =
≤
≤
Pk 1
∗ 2
i=1 k (un,i − x )k
P
k
1
∗ 2
k
i=1 kun,i − x
k
Pk
1
∗ 2
kxn − x k − k i=1 kun,i − xn k2
Pk
+ k1 i=1 2rn kxn − un,i kkΨi xn − Ψi x∗ k.
k
(2.9)
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78
H. R. Sahebi, A. Razani
From (2.9),
kxn+1 − x∗ k2
≤
αn kγf (xn ) − Ax∗ k2 + βn β̄kxn − x∗ k2
+(1 − βn β̄ − αn γ̄)kωn − x∗ k2 + ε2n kx∗ k2
≤
αn kγf (xn ) − Ax∗ k2 + βn β̄kxn − x∗ k2
+(1 − βn β̄ − αn γ̄){kxn − x∗ k2 −
+
thus
k
1X
2rn kxn − un,i kkΨi xn − Ψi x∗ k} + ε2n kx∗ k2
k i=1
(1 − βn β̄ − αn γ̄)
≤
k
1X
kun,i − xn k2
k i=1
k
1X
kun,i − xn k2
k i=1
αn kγf (xn ) − Ax∗ k2 + kxn − x∗ k2 − kxn+1 − x∗ k2
k
≤
1X
2rn kxn − un,i kkΨi xn − Ψi x∗ k + ε2n kx∗ k2
+(1 − βn β̄ − αn γ̄)
k i=1
αn kγf (xn ) − Ax∗ k2 + kxn+1 − xn k(kxn+1 − x∗ k − kxn − x∗ k)
+(1 − βn β̄ − αn γ̄)
k
1X
2rn kxn − un,i kkΨi xn − Ψi x∗ k + ε2n kx∗ k2
k i=1
From the condition (C1), (2.5) and (2.7), we get
lim kun,i − xn k = 0.
(2.10)
lim kωn − xn k = 0.
(2.11)
n→∞
It is easy to prove
n→∞
By definition of the sequence {xn }, we obtain
kxn − Λn k
≤
kxn+1 − xn k + kxn+1 − Λn k
≤
kxn+1 − xn k + kαn γf (xn ) + βn Bxn
+((1 − εn )I − βn B − αn A)Λn − Λn k
≤
kxn+1 − xn k + αn kγf (xn ) − AΛn k + βn β̄kxn − Λn k + εn kΛn k.
Then
kxn − Λn k
≤
1
αn
kxn+1 − xn k +
kγf (xn ) − AΛn k
1 − βn β̄
1 − βn β¯n
εn
+
kΛn k.
1 − βn β̄
Thanks to the conditions (C1) − (C2) and (2.5), we conclude that
lim kxn − Λn k = 0.
n→∞
(2.12)
[ DOI: 10.7508/ijmsi.2016.01.007 ]
An Explicit Viscosity Iterative Algorithm for ...
Also
kωn − Λn k ≤ kωn − xn k + kxn − Λn k,
hence
lim kωn − Λn k = 0.
n→∞
(2.13)
Theorem 2.2. Suppose all assumptions of Theorem 2.1 are holds. Then the
sequence {xn } is strongly convergent to a point x̄, where
Tk
T
x̄ ∈ i=1 F (ST ) GEP (Fi , Ψi ) solves the variational inequality
h(A − γf )x̄, x̄ − xi ≤ 0.
Equivalently, x̄ = PTk
F (ST )
i=1
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79
T
GEP (Fi ,Ψi ) (I
− A + γf )(x̄).
Proof. Observe that PTk F (ST ) T GEP (Fi ,Ψi ) (I − A + γf ) is a contraction of H
i=1
into itself. Since H is complete, there exists a unique element x̄ ∈ H such that
x̄ = PTk
i=1
F (ST )
T
GEP (Fi ,Ψi ) (I
− A + γf )(x̄).
Next, we prove
lim suph(A − γf )x̄, x̄ − Λn i ≤ 0
n→∞
Let x̃ = PTk
i=1
F (ST )
T
GEP (Fi ,Ψi ) x1 ,
set
E = {ȳ ∈ H : kȳ − x̃k ≤ kx1 − x̃k +
kγf (x̃) − Ax̃k \
} C.
γ̄ − γρ
It is clear, E is nonempty closed bounded convex subset of C and S(E) ⊂ E,
T (E) ⊂ E. Without loss of generality, we may assume S and T are mappings
of E into itself. Since {Λn } ⊂ E is bounded, there is a subsequence {Λnj } of
{Λn } such that
lim suph(A − γf )x̄, x̄ − Λn i = lim h(A − γf )x̄, x̄ − Λnj i.
j→∞
n→∞
(2.14)
As {Λnj } is also bounded, there exists a subsequence {Λnjl } of {Λnj } such that
Λnjl ⇀ ξ. Without loss of generality, let Λnj ⇀ ξ. Now, we prove the following
items:
T
(i): ξ ∈ F (ST ) = F (T ) F (S).
Assume ξ ∈
/ F (ST ). By Lemma 1.4 and Opial’s condition,
lim inf kΛnj − ξk
j→∞
<
lim inf kΛnj − ST (ξ)k
≤
lim inf (kΛnj − ST (Λnj )k + kST (Λnj ) − ST (ξ)k)
≤
lim inf kΛnj − ξk.
j→∞
j→∞
j→∞
That is a contradiction. Hence ξ = F (ST )ξ.
(ii): By the same argument as in the proof of [7, Theorem 3.2], we conclude
Tk
that ξ ∈ GEP (Fi , Ψi ), for all i = 1, 2, . . . , k. Then ξ ∈ i=1 GEP (Fi , Ψi ).
Now, in view of (2.14), we see
[ DOI: 10.7508/ijmsi.2016.01.007 ]
80
H. R. Sahebi, A. Razani
lim suph(A − γf )x̄, x̄ − Λn i = h(A − γf )x̄, x̄ − ξi ≤ 0,
n→∞
Finally, we show that {xn } is strongly convergent to x̄. As a matter of fact,
k
xn+1 − x̄k2 = kαn γf (xn ) + βn Bxn + ((1 − εn )I − βn B − αn A)Λn − x̄k2
=
kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄) − εn x̄k2
≤
kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄) + εn x̄k2
=
kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
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+((1 − εn )I − βn B − αn A)(Λn − x̄)k2
+2εn kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k + ε2n kx̄k2
=
kβn B(xn − x̄) + ((1 − εn )I − βn B − αn A)(Λn − x̄)k2
+2αn h((1 − εn )I − βn B − αn A)(Λn − x̄), γf (xn ) − Ax̄i
+2εn kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k
+ε2n kx̄k2 + αn2 kγf (xn ) − Ax̄k2 + 2αn βn hB(xn − x̄), γf (xn ) − Ax̄i
≤
(βn kBkkxn − x̄k + k(1 − εn )I − βn B − αn AkkΛn − x̄k)2
+2αn γhΛn − x̄, f (xn ) − f (x̄)i + 2εn kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k + ε2n kx̄k2 + αn2 kγf (xn ) − Ax̄k2
+2αn βn hB(xn − x̄), γf (xn ) − Ax̄i + 2αn hΛn − x̄, γf (xn ) − Ax̄i
−2αn h(εn I + βn B + αn A)(Λn − x̄), γf (xn ) − Ax̄i
≤
(βn β̄kxn − x̄k + (1 − βn β̄ − αn γ̄)kxn − x̄k)2 + 2αn γkxn − x̄k2
+2εn kαn (γf (xn ) − Ax̄) + βn B(xn − x̄)
+((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k + ε2n kx̄k2
+αn2 kγf (xn ) − Ax̄k2 + 2αn βn hB(xn − x̄), γf (xn ) − Ax̄i
+2αn hΛn − x̄, γf (xn ) − Ax̄i
−2αn h(εn I + βn B + αn A)(Λn − x̄), γf (xn ) − Ax̄i
=
(1 − 2(γ̄ − γ)αn )kxn − x̄k2 + αn2 γ̄ 2 kxn − x̄k2 + 2εn kαn (γf (xn ) − Ax̄)
+βn B(xn − x̄) + ((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k + ε2n kx̄k2
+αn2 kγf (xn ) − Ax̄k2 + 2αn βn hB(xn − x̄), γf (xn ) − Ax̄i
+2αn hΛn − x̄, γf (xn ) − Ax̄i
−2αn h(εn I + βn B + αn A)(Λn − x̄), γf (xn ) − Ax̄i
≤
(1 − 2(γ̄ − γ)αn )kxn − x̄k2 + αn ϑn ,
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[ DOI: 10.7508/ijmsi.2016.01.007 ]
An Explicit Viscosity Iterative Algorithm for ...
81
where
ζn
=
(γ̄ − γρ)αn ,
ϑn
=
αn γ̄ 2 kxn − x̄k2 + 2kαn (γf (xn ) − Ax̄)
+βn B(xn − x̄) + ((1 − εn )I − βn B − αn A)(Λn − x̄)kkx̄k + εn kx̄k2
+αn kγf (xn ) − Ax̄k2 + 2βn hB(xn − x̄), γf (xn ) − Ax̄i
+2αn hΛn − x̄, γf (xn ) − Ax̄i
−2αn h(εn I + βn B + αn A)(Λn − x̄), γf (xn ) − Ax̄i.
From conditions (C1) − (C2) and (C5) and Lemma 1.3, we obtain the sequence
{xn } strongly convergence to x̄.
Using Theorems 2.1 and 2.2, we obtain the following corollaries.
Corollary 2.3. [5, Theorem 3.1] Let C be a nonempty closed convex subset
of H. Suppose S and T are nonexpansive mappings of C into itself, such that
T
F (ST ) = F (T S) = F (S) F (T ) 6= ∅. Let f be a contraction mapping from C
to C and {xn } be a sequence generated by x0 = x ∈ C and
xn+1
=
αn f (xn ) + βn
n n−i
+(1 − αn − βn )
for all n ∈ N
XX
2
((ST )j S i−j ∨ (ST )i T j−i )xn
(n + 1)(n + 2) i=0 j=0
S
{0}, where {αn }, {βn } ⊂ [0, 1] satisfy
lim αn = 0,
n→∞
∞
X
αn = ∞.
n=0
If 0 < lim inf βn ≤ lim supβn < 1, then {xn } converges strongly to z ∈
n→∞
T n→∞
F (S) F (T ), where z = PF (S) T F (T ) f (z) is the unique solution of the variational inequality
\
h(I − f )z, z − xi ≥ 0, ∀x ∈ F (S) F (T ).
Proof. Setting Fi = Ψi ≡ 0, ∀i ∈ {1, 2, . . . , k}, A = B ≡ I, γ = γ̄ = 1,
εn = 0, and ωn = xn , φ(t) = (1 − ρ)t in Theorems 2.1 and 2.2. Thus the proof
is straightforward.
Corollary 2.4. [5, Corollary 3.4] Let C be a nonempty closed convex subset
of H. Suppose S and T are averaged mappings of C into itself, such that
T
F (S) F (T ) 6= ∅. Let f be a contraction mapping from C to C and {xn } be a
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[ DOI: 10.7508/ijmsi.2016.01.007 ]
82
H. R. Sahebi, A. Razani
sequence generated by x0 = x ∈ C and
xn+1
=
αn f (xn ) + βn
n n−i
+(1 − αn − βn )
for all n ∈ N
XX
2
((ST )j S i−j ∨ (ST )i T j−i )xn
(n + 1)(n + 2) i=0 j=0
S
{0}, where {αn }, {βn } ⊂ [0, 1] satisfy
lim αn = 0, lim βn = 0,
n→∞
n→∞
∞
X
αn = ∞.
n=0
If 0 < lim inf βn ≤ lim supβn < 1, then {xn } is strongly convergent to z ∈
n→∞
T n→∞
F (S) F (T ), where z = PF (S) T F (T ) f (z) is the unique solution of the variational inequality
\
h(I − f )z, z − xi ≥ 0, ∀x ∈ F (S) F (T ).
Proof. Set S and T averaged mappings of C into itself, Fi = Ψi ≡ 0, for all
i ∈ {1, 2, . . . , k}, A = B ≡ I, γ = γ̄ = 1, εn = 0 and ωn = xn , φ(t) = (1 − ρ)t
in Theorems 2.1 and 2.2. Thus the proof is straightforward.
Corollary 2.5. [8, Theorem 1] Let C be a nonempty closed convex subset
of H. Suppose S and T are nonexpansive mappings of C into itself, such
T
that ST = T S and F (S) F (T ) 6= ∅. Let {xn } be a sequence generated by
x0 = x ∈ C and
n n−i
xn+1 = αn x + (1 − αn )
XX
2
SiT j xn .
(n + 1)(n + 2) i=0 j=0
for all n ∈ N , where {αn } ⊂ [0, 1] satisfy
lim αn = 0,
n→∞
∞
X
αn = ∞.
n=0
Then {xn } is strongly convergent to z ∈ F (S)
T
metric projection of H onto F (S) F (T ).
T
F (T ), where PF (S) T F (T ) is a
Proof. Setting ST = T S, Fi = Ψi ≡ 0, for all i ∈ {1, 2, . . . , k}, A = B ≡ I, γ =
γ̄ = 1, f (y) = x, for all y ∈ C, εn = βn = 0 and ωn = xn , φ(t) = (1 − ρ)t in
Theorems 2.1 and 2.2. Thus the proof is straightforward.
Corollary 2.6. [10, Theorem 4] Let H be a Hilbert space and C a nonempty
closed convex subset of H. Suppose S and T are averaged mappings of C into
T
itself such that F (S) F (T ) is nonempty. Suppose that {αn } satisfies
lim αn = 0,
n→∞
∞
X
n=0
αn = ∞.
[ DOI: 10.7508/ijmsi.2016.01.007 ]
An Explicit Viscosity Iterative Algorithm for ...
83
For an arbitrary x ∈ C, the sequence {xn } generated by
(
x0 = x,
Pn Pn−i
2
j i−j
xn+1 = αn x + (1 − αn ) (n+1)(n+2)
∨ (ST )i T j−i )xn .
i=0
j=0 ((ST ) S
is strongly convergent to a common fixed point P x of S and T , where P is the
T
metric projection of H onto F (S) F (T ).
Proof. Setting S and T be averaged mappings of C into itself and Fi = Ψi ≡ 0,
for all i ∈ {1, 2, . . . , k}, A = B ≡ I, γ = γ̄ = 1, f (y) = x, ∀y ∈ C, εn = βn = 0
and ωn = xn , φ(t) = (1 − ρ)t in Theorems 2.1 and 2.2. Thus the proof is
straightforward.
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Acknowledgments
We would like to thank the referees for their constructive comments and
suggestions.
References
1. S. Chandok, T. D. Narang, Common fixed points and invariant approximations for Cq commuting generalized nonexpansive mappings, Iranian Journal of Mathematical Sciences and Informatics, 7(1), (2012), 21-34.
2. Ya. I. Alber, S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert
spaces,in: I. Gohberg, Yu. Lyubich (Eds), Advances and Appl., 98, Birkhauser, Basel
(1977), 7-22.
3. P. Combettes, A. Histoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear
Convex Anal., 6, (2005), 117-136.
4. M. Eslamina, A. Abkar, Generalized weakly contractive multivalued mappings and common fixed points, Iranian Journal of Mathematical Sciences and Informatics, 8(2),
(2013), 75-84.
5. E. Jankaew, S. Somyot, A. Tepphun, Viscosity iterative methods for common fixed points
of two nonexpansive mappings without commutativity assumption in Hilbert spaces, Acta
Math. Sci . Ser. B Engl. Ed., 31B(2), (2011), 716-726.
6. Z. Opial, Weak convergence of successive approximations for nonexpansive mappings,
Bull. Amer. Math. Soc., 73, (1967), 591-597.
7. H. R. Sahebi, A. Razani, A solution of a general equilibrium problem, Acta Math. Sci.
Ser. B Engl. Ed., 33B(6), (2013), 1598-1614.
8. T. Shimizu, W. Takahashi, Strong convergence to common fixed points of nonexpansive
mappings, J. Math. Anal. Appl., 211, (1997), 71-83.
9. H. K. Xu. Viscosity, Approximation methods for nonexpansive mappings, J. Math. Anal.
Appl., 298, (2004), 279-291.
10. Y. Yao, R. Chen, Convergence to common fixed points of averaged mappings without
commutativity assumption in Hilbert spaces, Nonlinear Anal., 67, (2007), 1758-1763.
11. M. M. Zahedi, A review on hyper K-algebras, Iranian Journal of Mathematical Sciences
and Informatics, 1(1), (2006), 55-112.