Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 926078, 8 pages
http://dx.doi.org/10.1155/2013/926078
Research Article
Viscosity Approximation Methods and Strong
Convergence Theorems for the Fixed Point of Pseudocontractive
and Monotone Mappings in Banach Spaces
Yan Tang
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Correspondence should be addressed to Yan Tang; ttyy7999@163.com
Received 22 January 2013; Revised 26 May 2013; Accepted 27 May 2013
Academic Editor: Filomena Cianciaruso
Copyright © 2013 Yan Tang. 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.
Suppose that 𝐶 is a nonempty closed convex subset of a real reflexive Banach space 𝐸 which has a uniformly Gateaux differentiable
norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element
of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone
mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper
extend most of the results that have been proposed for this class of nonlinear mappings.
1. Introduction
Let 𝐸 be a real Banach space with dual 𝐸∗ . A normalized
∗
duality mapping 𝐽 : 𝐸 → 2𝐸 is defined by
󵄩 󵄩2
𝐽𝑥 = {𝑓∗ ∈ 𝐸∗ : ⟨𝑥, 𝑓∗ ⟩ = ‖𝑥‖2 = 󵄩󵄩󵄩𝑓∗ 󵄩󵄩󵄩 } ,
(1)
where ⟨⋅, ⋅⟩ denotes the generalized duality pairing between
𝐸 and 𝐸∗ . It is well known that 𝐸 is smooth if and only if
𝐽 is single valued, and if 𝐸 is uniformly smooth, then 𝐽 is
uniformly continuous on bounded subsets of 𝐸. Moreover, if
𝐸 is reflexive and strictly convex Banach space with a strictly
convex dual then 𝐽−1 is single valued, one-to-one surjective,
and it is the duality mapping from 𝐸∗ into 𝐸, and then 𝐽𝐽−1 =
𝐼𝐸∗ and 𝐽𝐽−1 = 𝐼𝐸 .
A mapping 𝐴 : 𝐷(𝐴) ⊂ 𝐸 → 𝐸∗ is said to be monotone
if, for each 𝑥, 𝑦 ∈ 𝐷(𝐴), the following inequality holds:
⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 0.
(2)
A mapping 𝐴 : 𝐷(𝐴) ⊂ 𝐸 → 𝐸∗ is said to be strongly
monotone, if there exists a positive real number 𝛼 > 0 such
that for all 𝑥, 𝑦 ∈ 𝐷(𝐴)
󵄩2
󵄩
⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 𝛼󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩 .
(3)
Obviously, the class of monotone mappings includes the
class of the 𝛼-inverse strongly monotone mappings.
Let 𝐶 be closed convex subset of Banach space 𝐸. A
mapping 𝑇 is said to be pseudocontractive if, for any 𝑥, 𝑦 ∈
𝐷(𝑇), there exists 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that
󵄩2
󵄩
(4)
⟨𝑇𝑥 − 𝑇𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 .
A mapping 𝑇 is said to be 𝜅-strictly pseudo-contractive,
if, for any 𝑥, 𝑦 ∈ 𝐷(𝑇), there exist 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) and a
constant 0 ≤ 𝜅 ≤ 1 such that
󵄩
󵄩2
⟨𝑥 − 𝑦 − (𝑇𝑥 − 𝑇𝑦) , 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝜅󵄩󵄩󵄩(𝐼 − 𝑇) 𝑥 − (𝐼 − 𝑇) 𝑦󵄩󵄩󵄩 .
(5)
A mapping 𝑇 : 𝐶 → 𝐶 is called nonexpansive if
󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩
(6)
󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩 ≤ 󵄩󵄩𝑥 − 𝑦󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐶.
Clearly, the class of pseudo-contractive mappings
includes the class of strict pseudo-contractive mappings and
non-expansive mappings. We denote by 𝐹(𝑇) the set of fixed
points of 𝑇; that is, 𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}.
A mapping 𝑓 : 𝐶 → 𝐶 is called contractive with a
contraction coefficient if there exists a constant 𝜌 ∈ (0, 1)
such that
󵄩
󵄩
󵄩
󵄩󵄩
(7)
󵄩󵄩𝑓 (𝑥) − 𝑓 (𝑦)󵄩󵄩󵄩 ≤ 𝜌 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐶.
2
Journal of Applied Mathematics
Let 𝐸 be a real Banach space with dual 𝐸∗ . The norm on 𝐸
is said to be uniformly Gateaux differentiable if for each 𝑦 ∈
𝑆1 (0) := {𝑥 ∈ 𝐸 : ‖𝑥‖ = 1} the limit lim𝑡 → 0 (‖𝑥 + 𝑡𝑦‖ − ‖𝑥‖)/𝑡
exists uniformly for 𝑥 ∈ 𝑆1 (0).
For finding an element of the set of fixed points of the
non-expansive mappings, Halpern [1] was the first to study
the convergence of the scheme in 1967:
𝑥𝑛+1 = 𝛼𝑛+1 𝑢 + (1 − 𝛼𝑛+1 ) 𝑇 (𝑥𝑛 ) .
(8)
Viscosity approximation methods are very important
because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential
equations. In Hilbert spaces, many authors have studied the
fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity
approximation methods, and obtained a series of good results
(see [2–17]).
In 2000, Moudifi [18] introduced the viscosity approximation methods and proved the strong convergence of the
following iterative algorithm in Hilbert spaces under some
suitable conditions:
𝑥𝑛+1 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝑇 (𝑥𝑛 ) .
(9)
Suppose that 𝐴 is monotone mapping from 𝐶 into 𝐸.
The classical variational inequality problem is formulated as
finding a point 𝑢 ∈ 𝐶 such that ⟨V − 𝑢, 𝐴𝑢⟩ ≥ 0, for all
V ∈ 𝐶. The set of solutions of variational inequality problems
is denoted by 𝑉𝐼(𝐶, 𝐴).
Takahashi and Toyoda [19, 20] introduced the following
scheme in Hilbert spaces and studied the weak and strong
convergence theorem of the elements of 𝐹(𝑇) ∩ 𝑉𝐼(𝐶, 𝐴),
respectively, under different conditions:
𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛+1 ) 𝑇𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝑥𝑛 ) ,
(10)
where 𝑇 is non-expansive mapping and 𝐴 is 𝛼-inverse strong
monotone operator.
Recently, Zegeye and Shahzad [21] introduced the following algorithm and obtained the strong convergence theorem
but still in Hilbert spaces:
𝑥𝑛+1 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + (1 − 𝛼𝑛 ) 𝑇𝑟𝑛 𝐹𝑟𝑛 𝑥𝑛 ,
(11)
where 𝑇𝑟𝑛 , 𝐹𝑟𝑛 are nonexpansive mappings.
Our concern now is the following: is it possible to construct a new sequence in Banach spaces which converges
strongly to a common element of fixed points of a finite family
of pseudocontractive mappings and the solution set of a variational inequality problems for finite family of monotone
mappings?
where {𝜃𝑛 } is a sequence in (0,1) and {𝜎𝑛 } is a real sequence such
that
(i) ∑∞
𝑛=0 𝜃𝑛 = ∞;
(ii) lim sup𝑛 → ∞ 𝜎𝑛 /𝜃𝑛 ≤ 0 or ∑∞
𝑛=0 𝜎𝑛 < ∞.
Then lim𝑛 → ∞ 𝑎𝑛 = 0.
Lemma 2 (see, e.g., [10]). Let 𝐶 be a nonempty, closed, and
convex subset of uniformly smooth strictly convex real Banach
space 𝐸 with dual 𝐸∗ . Let 𝐴 : 𝐶 → 𝐸∗ be a continuous monotone mapping; define mapping 𝐹𝑟 as follows: 𝑥 ∈ 𝐸, 𝑟 ∈ (0, ∞)
𝐹𝑟 (𝑥) = {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴𝑧⟩ +
∀𝑦 ∈ 𝐶} .
(13)
Then the following hold:
(i) 𝐹𝑟 is well defined and single valued;
(ii) 𝐹𝑟 is a firmly non-expansive mapping; that is,
⟨𝐹𝑟 𝑥 − 𝐹𝑟 𝑦, 𝐽𝐹𝑟 𝑥 − 𝐽𝐹𝑟 𝑦⟩ ≤ ⟨𝐹𝑟 𝑥 − 𝐹𝑟 𝑦, 𝐽𝑥 − 𝐽𝑦⟩;
(iii) 𝐹(𝐹𝑟 ) = 𝑉𝐼(𝐶, 𝐴);
(iv) 𝑉𝐼(𝐶, 𝐴) is closed and convex.
Lemma 3 (see, e.g., [22]). Let 𝐶 be a nonempty closed convex
subset of uniformly smooth strictly convex real Banach space
𝐸. Let 𝑇 : 𝐶 → 𝐸, be a continuous strictly pseudocontractive
mapping; define mapping 𝑇𝑟 as follows: 𝑥 ∈ 𝐸, 𝑟 ∈ (0, ∞)
𝑇𝑟 (𝑥)
= {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝑇𝑧⟩ −
1
⟨𝑦 − 𝑧, (1 + 𝑟) 𝐽𝑧 − 𝐽𝑥⟩ ≤ 0,
𝑟
∀𝑦 ∈ 𝐶} .
(14)
Then the following hold:
(i) 𝑇𝑟 is single valued;
(ii) 𝑇𝑟 is a firmly nonexpansive mapping; that is,
⟨𝑇𝑟 𝑥 − 𝑇𝑟 𝑦, 𝐽𝑇𝑟 𝑥 − 𝐽𝑇𝑟 𝑦⟩ ≤ ⟨𝑇𝑟 𝑥 − 𝑇𝑟 𝑦, 𝐽𝑥 − 𝐽𝑦⟩;
(iii) 𝐹(𝑇𝑟 ) = 𝐹(𝑇);
(iv) 𝐹(𝑇) is closed and convex.
Lemma 4 (see, e.g., [23]). Let {𝑥𝑛 } and {𝑧𝑛 } be bounded
sequence in a Banach space, and let {𝛽𝑛 } be a sequence in [0, 1]
which satisfies the following condition:
0 < lim inf 𝛽𝑛 < lim sup 𝛽𝑛 < 1.
2. Preliminaries
𝑛→∞
Lemma 1 (see, e.g., [5]). Let {𝑎𝑛 } be a sequence of nonnegative
real numbers satisfying the following relation:
𝑛 ≥ 0,
𝑛→∞
(15)
Suppose
In the sequel, we will use the following lemmas.
𝑎𝑛+1 ≤ (1 − 𝜃𝑛 ) 𝑎𝑛 + 𝜎𝑛 ,
1
⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0,
𝑟
(12)
𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑧𝑛 ,
𝑛 ≥ 0,
󵄩 󵄩
󵄩
󵄩
lim (󵄩󵄩𝑧 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩) ≤ 0.
𝑛 → ∞ 󵄩 𝑛+1
Then lim𝑛 → ∞ ‖𝑧𝑛 − 𝑥𝑛 ‖ = 0.
(16)
Journal of Applied Mathematics
3
Lemma 5 (see, e.g., [16]). Let 𝐶 be a nonempty closed and
convex subset of a real smooth Banach space 𝐸. A mapping
Π𝐶 : 𝐸 → 𝐶 is a generalized projection. Let 𝑥 ∈ 𝐸; then
𝑥0 = Π𝐶𝑥 if and only if
⟨𝑧 − 𝑥0 , 𝐽𝑥 − 𝐽𝑥0 ⟩ ≤ 0,
∀𝑧 ∈ 𝐶.
(17)
Lemma 6. Let 𝐸 be a real Banach space with dual 𝐸∗ . 𝐽 : 𝐸 →
∗
2𝐸 is the generalized duality pairing; then, for all 𝑥, 𝑦 ∈ 𝐸,
󵄩2
󵄩󵄩
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑗 (𝑥 + 𝑦)⟩ , ∀𝑗 (𝑥 + 𝑦) ∈ 𝐽 (𝑥 + 𝑦) .
(18)
3. Main Results
Let 𝐶 be a nonempty closed convex and bounded subset of
a smooth, strictly convex and reflexive real Banach space 𝐸
with dual 𝐸∗ . Let 𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, . . . , 𝑚 be a finite
family of continuous monotone mappings, and let 𝑇𝑖 : 𝐶 →
𝐶, 𝑖 = 1, 2, . . . , 𝑚, be a finite family of continuous strictly
pseudo-contractive mappings. For the rest of this paper, 𝑇𝑖𝑟𝑛 𝑥
and 𝐹𝑖𝑟𝑛 𝑥 are mappings defined as follows: for 𝑥 ∈ 𝐸, 𝑟𝑛 ∈
(0, ∞),
𝑇𝑖𝑟𝑛 (𝑥)
:= {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝑇𝑖 𝑧⟩ −
1
⟨𝑦 − 𝑧, (1 + 𝑟𝑛 ) 𝐽𝑧 − 𝐽𝑥⟩ ≤ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶} .
Clearly ∩𝑚
𝑖=1 𝐹(Γ𝑖 ) ⊂ 𝐹(Γ). Now, we prove that 𝐹(Γ) ⊂
Let 𝑥 ∈ 𝐹(Γ) and 𝑝 ∈ ∩𝑚
𝑖=1 𝐹(Γ𝑖 ). Then
∩𝑚
𝑖=1 𝐹(Γ𝑖 ).
𝑚
0 = ⟨Γ𝑥 − 𝑥, 𝑗 (𝑥 − 𝑝)⟩ = ⟨∑𝛼𝑖 Γ𝑖 𝑥 − 𝑥, 𝑗 (𝑥 − 𝑝)⟩
𝑖=1
(22)
𝑚
= ∑𝛼𝑖 ⟨Γ𝑖 𝑥 − 𝑥, 𝑗 (𝑥 − 𝑝)⟩ ,
𝑖=1
and non-expansivity of each Γ𝑖 implies that ⟨Γ𝑖 𝑥−𝑥, 𝑗(𝑥−𝑝)⟩ =
0 for each 𝑖 = 1, 2, . . . , 𝑚, which implies Γ𝑖 𝑥 = 𝑥 for each
𝑖 = 1, 2, . . . , 𝑚. Therefore, 𝑥 ∈ ∩𝑚
𝑖=1 𝐹(Γ𝑖 ), and hence 𝐹(Γ) =
𝐹(Γ
).
The
proof
is
complete.
∩𝑚
𝑖
𝑖=1
Theorem 8. Let 𝐶 be a nonempty closed convex subset of a
real reflexive Banach space 𝐸 which has a uniformly Gateaux
differentiable norm. Let 𝑇𝑖 : 𝐶 → 𝐶, 𝑖 = 1, 2, . . . , 𝑚, be a
finite family of continuous strictly pseudocontractive mappings,
let 𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, . . . , 𝑚, be a finite family of continuous monotone mappings such that 𝐹 = 𝐹1 ∩ 𝐹2 ≠ 0, and
let 𝑓 : 𝐶 → 𝐶 be a contraction with a contraction coefficient
𝜌 ∈ (0, 1). 𝑇𝑖𝑟𝑛 and 𝐹𝑖𝑟𝑛 are defined as (19) and (20), respectively.
Let {𝑥𝑛 } be a sequence generated by 𝑥0 ∈ 𝐶
𝑚
𝑦𝑛 = 𝜆 𝑛 𝑥𝑛 + (1 − 𝜆 𝑛 ) ∑𝜇𝑖 𝐹𝑖𝑟𝑛 𝑥𝑛 ,
𝑖=1
(19)
(23)
𝑚
𝑥𝑛+1 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝜎𝑖 𝑇𝑖𝑟𝑛 𝑦𝑛 ,
𝑖=1
Consider
𝐹𝑖𝑟𝑛 (𝑥)
1
:= {𝑧 ∈ 𝐶 : ⟨𝑦 − 𝑧, 𝐴 𝑖 𝑧⟩ + ⟨𝑦 − 𝑧, 𝐽𝑧 − 𝐽𝑥⟩ ≥ 0,
𝑟𝑛
where 𝜆 𝑛 ∈ [0, 1] and {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences of nonnegative real numbers in [0, 1], and
(20)
∀𝑦 ∈ 𝐶} .
𝑚
Denote 𝐹1 = ∩𝑚
𝑖=1 𝐹(𝑇𝑖𝑟𝑛 ), 𝐹2 = ∩𝑖=1 𝐹(𝐹𝑖𝑟𝑛 ).
Lemma 7. Let 𝐶 be a nonempty closed convex and bounded
subset of a smooth Banach space 𝐸, and let Γ𝑖 : 𝐶 → 𝐶, 𝑖 =
1, 2, . . . , 𝑚, be a finite family of non-expansive mappings such
that ∩𝑚
𝑖=1 𝐹(Γ𝑖 ) ≠ 0. Suppose that 𝛼 = inf{𝛼𝑖 } > 0; then there
exists non-expansive mapping Γ : 𝐶 → 𝐶 such that 𝐹(Γ) =
∩𝑚
𝑖=1 𝐹(Γ𝑖 ).
Proof. Let {𝛼𝑖 } be any sequence of positive real numbers
𝑚
satisfying ∑𝑚
𝑖=1 𝛼𝑖 = 1 and set Γ = ∑𝑖=1 𝛼𝑖 Γ𝑖 . Since each Γ𝑖
is non-expansive for any 𝑖 ∈ {1, 2, . . . , 𝑚}, we have that Γ is
well-defined nonexpansive mapping (see, e.g., [20]), and
󵄩󵄩 𝑚
󵄩󵄩
𝑚
󵄩󵄩
󵄩 󵄩󵄩
󵄩󵄩
󵄩󵄩Γ𝑥 − Γ𝑦󵄩󵄩󵄩 = 󵄩󵄩󵄩∑𝛼𝑖 Γ𝑖 𝑥 − ∑𝛼𝑖 Γ𝑖 𝑦󵄩󵄩󵄩
󵄩󵄩󵄩𝑖=1
󵄩󵄩󵄩
𝑖=1
(21)
𝑚
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩
≤ ∑𝛼𝑖 󵄩󵄩Γ𝑖 𝑥 − Γ𝑖 𝑦󵄩󵄩 ≤ 󵄩󵄩𝑥 − 𝑦󵄩󵄩 .
𝑖=1
Next, we claim that 𝐹(Γ) = ∩𝑚
𝑖=1 𝐹(Γ𝑖 ).
𝑚
(i) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 𝑛 ≥ 0, ∑𝑚
𝑖=1 𝜇𝑖 = 1, ∑𝑖=1 𝜎𝑖 = 1, 𝜇𝑖 ≥ 0,
𝜎𝑖 ≥ 0;
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 = ∞;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 < lim sup𝑛 → ∞ 𝛽𝑛 < 1;
(iv) lim sup𝑛 → ∞ 𝑟𝑛 > 0, ∑∞
𝑛=1 |𝑟𝑛+1 − 𝑟𝑛 | < ∞.
Then the sequence {𝑥𝑛 } converges strongly to an element 𝑥 =
Π𝐹 𝑓(𝑥), and also 𝑥 is the unique solution of the variational
inequality
⟨(𝑓 − 𝐼) (𝑥) , 𝑗 (𝑦 − 𝑥)⟩ ≤ 0,
∀𝑦 ∈ 𝐹.
(24)
Proof. First we prove that {𝑥𝑛 } is bounded. Take 𝑝 ∈ 𝐹,
because 𝐹𝑖𝑟𝑛 is non-expansive; then we have that
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩 ≤ 𝜆 𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜆 𝑛 )
𝑚
󵄩
󵄩 󵄩
󵄩
× ∑𝜇𝑖 󵄩󵄩󵄩󵄩𝐹𝑖𝑟𝑛 𝑥𝑛 − 𝐹𝑖𝑟𝑛 𝑝󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 .
𝑖=1
(25)
4
Journal of Applied Mathematics
For 𝑛 ≥ 0, because 𝑇𝑖𝑟𝑛 and 𝐹𝑖𝑟𝑛 are nonexpansive and 𝑓 is
contractive, we have from (25) that
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑚
󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩𝛼𝑛 (𝑓 (𝑥𝑛 ) − 𝑝) + 𝛽𝑛 (𝑥𝑛 − 𝑝) + 𝛾𝑛 (∑𝜎𝑖 𝑇𝑖𝑟𝑛 𝑦𝑛 − 𝑝)󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
≤ 𝛼𝑛 󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑓 (𝑝)󵄩󵄩 + 𝛼𝑛 󵄩󵄩𝑓 (𝑝) − 𝑝󵄩󵄩
󵄩
󵄩
󵄩
󵄩
+ 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝜌𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑝) − 𝑝󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ [1 − (1 − 𝜌) 𝛼𝑛 ] 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑝) − 𝑝󵄩󵄩󵄩
󵄩
󵄩 𝑓 (𝑝) − 𝑝
≤ max {󵄩󵄩󵄩𝑥0 − 𝑝󵄩󵄩󵄩 ,
}.
1−𝜌
(26)
Therefore, {𝑥𝑛 } is bounded. Consequently, we get that
{𝐹𝑖𝑟𝑛 𝑥𝑛 }, {𝑇𝑖𝑟𝑛 𝑦𝑛 } and {𝑦𝑛 }, {𝑓(𝑥𝑛 )} are bounded.
Next, we show that ‖𝑥𝑛+1 − 𝑥𝑛 ‖ → 0.
Consider
󵄩
󵄩󵄩
󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩
Since 𝐴 𝑖 , 𝑖 ∈ {1, 2, . . . , 𝑚} are monotone mappings, which
implies that
⟨V𝑖,𝑛+1 − V𝑖𝑛 ,
𝐽V𝑖𝑛 − 𝐽𝑥𝑛 𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1
−
⟩ ≥ 0,
𝑟𝑛
𝑟𝑛+1
therefore we have that
⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝐽V𝑖𝑛 − 𝐽𝑥𝑛 −
Let V𝑖𝑛 = 𝐹𝑖𝑟𝑛 𝑥𝑛 , V𝑖,𝑛+1 = 𝐹𝑖𝑟𝑛+1 𝑥𝑛+1 ; by the definition of
mapping 𝐹𝑖𝑟𝑛 , we have that
⟨𝑦 − V𝑖𝑛 , 𝐴 𝑖 V𝑖𝑛 ⟩ +
1
⟨𝑦 − V𝑖𝑛 , 𝐽V𝑖𝑛 − 𝐽𝑥𝑛 ⟩ ≥ 0,
𝑟𝑛
⟨𝑦 − V𝑖,𝑛+1 , 𝐴 𝑖 V𝑖𝑛+1 ⟩ +
1
𝑟𝑛+1
∀𝑦 ∈ 𝐶,
(28)
⟨𝑦 − V𝑖,𝑛+1 , 𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1 ⟩ ≥ 0,
∀𝑦 ∈ 𝐶.
(29)
Putting 𝑦 := V𝑖,𝑛+1 in (28) and leting 𝑦 := V𝑖𝑛 in (29), we
have that
1
⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝐴 𝑖 V𝑖𝑛 ⟩ + ⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝐽V𝑖𝑛 − 𝐽𝑥𝑛 ⟩ ≥ 0,
𝑟𝑛
⟨V𝑖𝑛 − V𝑖,𝑛+1 , 𝐴 𝑖 V𝑖,𝑛+1 ⟩
+
≤ ⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝐽𝑥𝑛+1 − 𝐽𝑥𝑛 + (1 −
(31)
𝐽V − 𝐽𝑥𝑛 𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1
+ ⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝑖𝑛
−
⟩ ≥ 0.
𝑟𝑛
𝑟𝑛+1
(34)
󵄩
󵄩 󵄩
󵄩
≤ 󵄩󵄩󵄩V𝑖,𝑛+1 − V𝑖𝑛 󵄩󵄩󵄩 { 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
󵄨󵄨
𝑟 󵄨󵄨󵄨 󵄩
󵄨
󵄩
+ 󵄨󵄨󵄨1 − 𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩V𝑖,𝑛+1 − 𝑥𝑛+1 󵄩󵄩󵄩} .
󵄨󵄨
𝑟𝑛+1 󵄨󵄨
Without loss of generality, let 𝑏 be a real number such that
𝑟𝑛 > 𝑏 > 0; then we have that
󵄨
𝑟 󵄨󵄨󵄨 󵄩
󵄩 󵄩
󵄩 󵄨󵄨
󵄩
󵄩󵄩
󵄩󵄩V𝑖,𝑛+1 − V𝑖𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄨󵄨󵄨1 − 𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩V𝑖,𝑛+1 − 𝑥𝑛+1 󵄩󵄩󵄩
󵄨󵄨
󵄨
𝑟𝑛+1 󵄨
(35)
󵄩󵄩
󵄩󵄩 1 󵄨󵄨
󵄨󵄨
≤ 󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩 + 󵄨󵄨𝑟𝑛+1 − 𝑟𝑛 󵄨󵄨 𝐾,
𝑏
where 𝐾 = sup{‖V𝑖,𝑛+1 − 𝑥𝑛+1 ‖}.
Then we have from (35) and (27) that
󵄨
󵄨
󵄩 󵄩
󵄩 (1 − 𝜆 𝑛+1 ) 󵄨󵄨󵄨𝑟𝑛+1 − 𝑟𝑛 󵄨󵄨󵄨
󵄩󵄩
𝐾
󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 +
𝑏
󵄩󵄩
󵄩󵄩
𝑚
󵄩󵄩
󵄨󵄨 󵄩󵄩󵄩
󵄨󵄨
+ 󵄨󵄨𝜆 𝑛+1 − 𝜆 𝑛 󵄨󵄨 󵄩󵄩𝑥𝑛 − ∑𝜇𝑖 𝐹𝑖𝑟𝑛 𝑥𝑛 󵄩󵄩󵄩 .
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
(36)
On the other hand, let 𝑢𝑖𝑛 = 𝑇𝑖𝑟𝑛 𝑦𝑛 , 𝑢𝑖,𝑛+1 = 𝑇𝑖𝑟𝑛+1 𝑦𝑛+1 ; we
have that
⟨𝑦 − 𝑢𝑖𝑛 , 𝑇𝑖 𝑢𝑖𝑛 ⟩ −
1
⟨𝑦 − 𝑢𝑖𝑛 , (1 + 𝑟𝑛 ) 𝐽𝑢𝑖𝑛 − 𝐽𝑦𝑛 ⟩ ≤ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶,
1
⟨V − V𝑖,𝑛+1 , 𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1 ⟩ ≥ 0.
𝑟𝑛+1 𝑖𝑛
⟨V𝑖,𝑛+1 − V𝑖𝑛 , 𝐴 𝑖 V𝑖𝑛 − 𝐴 𝑖 V𝑖,𝑛+1 ⟩
𝑟𝑛
)
𝑟𝑛+1
× (𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1 ) ⟩
(30)
Adding (30), we have that
(33)
That is,
󵄩󵄩
󵄩2
󵄩󵄩V𝑖,𝑛+1 − V𝑖𝑛 󵄩󵄩󵄩
𝑖=1
(27)
𝑟𝑛 (𝐽V𝑖,𝑛+1 − 𝐽𝑥𝑛+1 )
𝑟𝑛+1
+𝐽V𝑖,𝑛+1 − 𝐽V𝑖,𝑛+1 ⟩ ≥ 0.
𝑚
󵄩
󵄩
󵄩
󵄩
≤ 𝜆 𝑛+1 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝜆 𝑛+1 ) ∑𝜇𝑖 󵄩󵄩󵄩󵄩𝐹𝑖𝑟𝑛+1 𝑥𝑛+1 − 𝐹𝑖𝑟𝑛 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑚
󵄩󵄩
󵄨
󵄨 󵄩󵄩
+ 󵄨󵄨󵄨𝜆 𝑛+1 − 𝜆 𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩𝑥𝑛 − ∑𝜇𝑖 𝐹𝑖𝑟𝑛 𝑥𝑛 󵄩󵄩󵄩 .
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
(32)
(37)
⟨𝑦 − 𝑢𝑖,𝑛+1 , 𝑇𝑖 𝑢𝑖,𝑛+1 ⟩ −
1
⟨𝑦 − 𝑢𝑖,𝑛+1 , (1 + 𝑟𝑛+1 )
𝑟𝑛+1
× 𝐽𝑢𝑖,𝑛+1 − 𝐽𝑦𝑛+1 ⟩ ≤ 0,
∀𝑦 ∈ 𝐶.
(38)
Journal of Applied Mathematics
5
Let 𝑦 := 𝑢𝑖,𝑛+1 in (37), and let 𝑦 := 𝑢𝑖𝑛 in (38); we have
that
1
⟨𝑢
− 𝑢𝑖𝑛 , (1 + 𝑟𝑛 ) 𝐽𝑢𝑖𝑛 − 𝐽𝑦𝑛 ⟩ ≤ 0,
𝑟𝑛 𝑖,𝑛+1
⟨𝑢𝑖,𝑛+1 − 𝑢𝑖𝑛 , 𝑇𝑖 𝑢𝑖𝑛 ⟩ −
⟨𝑢𝑖𝑛 − 𝑢𝑖,𝑛+1 , 𝑇𝑖 𝑢𝑖,𝑛+1 ⟩
−
1
⟨𝑢 − 𝑢𝑖,𝑛+1 , (1 + 𝑟𝑛+1 ) 𝐽𝑢𝑖,𝑛+1 − 𝐽𝑦𝑛+1 ⟩ ≤ 0.
𝑟𝑛+1 𝑖𝑛
(39)
Adding (39) and because 𝑇𝑖 , 𝑖 ∈ {1, 2, . . . , 𝑚} is pseudocontractive, we have that
⟨𝑢𝑖,𝑛+1 − 𝑢𝑖𝑛 ,
𝐽𝑢𝑖𝑛 − 𝐽𝑦𝑛 𝐽𝑢𝑖,𝑛+1 − 𝐽𝑦𝑛+1
⟩ ≥ 0.
−
𝑟𝑛
𝑟𝑛+1
(40)
Therefore we have
𝑟𝑛 (𝐽𝑢𝑖,𝑛+1 − 𝐽𝑦𝑛+1 )
⟨𝑢𝑖,𝑛+1 − 𝑢𝑖𝑛 , 𝐽𝑢𝑖𝑛 − 𝐽𝑦𝑛 −
𝑟𝑛+1
(41)
+𝐽𝑢𝑖,𝑛+1 − 𝐽𝑢𝑖,𝑛+1 ⟩ ≥ 0.
Hence we have that
󵄩󵄩
󵄩 󵄩
󵄩 1󵄨
󵄨
󵄩󵄩𝑢𝑖,𝑛+1 − 𝑢𝑖𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩 + 󵄨󵄨󵄨𝑟𝑛+1 − 𝑟𝑛 󵄨󵄨󵄨 𝑀,
𝑏
(42)
𝛾𝑛+1 󵄨󵄨
󵄨
× ((1 − 𝜆 𝑛+1 ) 𝐾 + 𝑀) +
󵄨𝜆 − 𝜆 𝑛 󵄨󵄨󵄨
1 − 𝛽𝑛+1 󵄨 𝑛+1
󵄩󵄩
󵄩󵄩
𝑚
󵄩󵄩
󵄩󵄩
󵄩
× 󵄩󵄩𝑥𝑛 − ∑𝜇𝑖 𝐹𝑖𝑟𝑛 𝑥𝑛 󵄩󵄩󵄩 .
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
(47)
Hence we have from (35), (36), and (42) that
󵄩󵄩
󵄩
󵄩󵄩𝑦𝑛+1 − 𝑦𝑛 󵄩󵄩󵄩 󳨀→ 0,
󵄩󵄩
󵄩
󵄩󵄩𝑢𝑖,𝑛+1 − 𝑢𝑖𝑛 󵄩󵄩󵄩 󳨀→ 0,
󵄩󵄩
󵄩󵄩
󵄩󵄩V𝑖,𝑛+1 − V𝑖𝑛 󵄩󵄩 󳨀→ 0.
(48)
In addition, since 𝑥𝑛+1 = 𝛼𝑛 𝑓(𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝑚
𝑖=1 𝜎𝑖 𝑢𝑖𝑛 ,
𝑦𝑛 = 𝜆 𝑛 𝑥𝑛 + (1 − 𝜆 𝑛 ) ∑𝑚
𝜇
V
,
for
all
𝑝
∈
𝐹,
we
have
from
𝑖=1 𝑖 𝑖𝑛
the monotonicity of 𝐴 𝑖 , the non-expansivity of 𝑇𝑖𝑟𝑛 , and the
convexity of ‖ ⋅ ‖2 that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩2
𝑚
󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩𝛼𝑛 𝑓 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝜎𝑖 𝑢𝑖𝑛 − 𝑝󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩
󵄩
󵄩2
󵄩2
+ 𝛾𝑛 𝜆 𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + (1 − 𝜆 𝑛 ) 𝛾𝑛 󵄩󵄩󵄩V𝑛 − 𝑝󵄩󵄩󵄩
(43)
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + (𝛽𝑛 + 𝛾𝑛 𝜆 𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩
󵄩2
󵄩
󵄩2 󵄩
+ (1 − 𝜆 𝑛 ) 𝛾𝑛 (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛 − V𝑖𝑛 󵄩󵄩󵄩 )
󵄩
󵄩
󵄩2
󵄩2 󵄩
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − (1 − 𝜆 𝑛 ) 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − V𝑖𝑛 󵄩󵄩󵄩 .
(49)
𝛾𝑛
𝛾𝑛+1
−
)𝑢 .
1 − 𝛽𝑛+1 1 − 𝛽𝑛 𝑖𝑛
Hence we have from (43), (42), and (36) that
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
󵄨
(𝜌 − 1) 𝛼𝑛+1 󵄩󵄩
𝛼𝑛 󵄨󵄨󵄨󵄨
󵄩 󵄨󵄨 𝛼
≤
󵄨󵄨
󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄨󵄨󵄨 𝑛+1 −
1 − 𝛽𝑛+1
󵄨󵄨 1 − 𝛽𝑛+1 1 − 𝛽𝑛 󵄨󵄨
󵄨
󵄨
𝛾𝑛+1 󵄨󵄨󵄨𝑟𝑛+1 − 𝑟𝑛 󵄨󵄨󵄨
󵄩󵄩
󵄩󵄩 󵄩󵄩 󵄩󵄩
× {󵄩󵄩𝑓 (𝑥𝑛 )󵄩󵄩 + 󵄩󵄩𝑢𝑖𝑛 󵄩󵄩} +
1 − 𝛽𝑛+1
𝑏
Therefore we have that
󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 = 󵄨󵄨󵄨1 − 𝛽𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩𝑧𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0.
󵄩
󵄩 󵄨
󵄨󵄩
󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩2
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + 𝛽𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 𝛾𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑝󵄩󵄩󵄩
𝛾𝑛+1
(𝑢
− 𝑢𝑖𝑛 )
1 − 𝛽𝑛+1 𝑖,𝑛+1
+(
(46)
𝑖=1
𝛼𝑛+1
(𝑓 (𝑥𝑛+1 ) − 𝑓 (𝑥𝑛 ))
1 − 𝛽𝑛+1
+
Hence we have from Lemma 4 that
󵄩
󵄩
lim sup 󵄩󵄩󵄩𝑧𝑛 − 𝑥𝑛 󵄩󵄩󵄩 = 0.
𝑛→∞
𝑚
𝑧𝑛+1 − 𝑧𝑛
𝛼
𝛼𝑛
) 𝑓 (𝑥𝑛 )
+ ( 𝑛+1 −
1 − 𝛽𝑛+1 1 − 𝛽𝑛
(45)
𝑛→∞
󵄩
󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩𝛼𝑛 (𝑓 (𝑥𝑛 ) − 𝑝) + 𝛽𝑛 (𝑥𝑛 − 𝑝)󵄩󵄩󵄩 + 𝛾𝑛 ∑𝜎𝑖 󵄩󵄩󵄩𝑢𝑖𝑛 − 𝑝󵄩󵄩󵄩
where 𝑀 = sup{‖𝑢𝑖𝑛 − 𝑦𝑛 ‖}.
Let 𝑥𝑛+1 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 )𝑧𝑛 . Hence we have that
=
Noticing the conditions (ii) and (iv), we have that
󵄩 󵄩
󵄩
󵄩
lim sup (󵄩󵄩󵄩𝑧𝑛+1 − 𝑧𝑛 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩) = 0.
So we have that
󵄩
󵄩2
(1 − 𝜆 𝑛 ) 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − V𝑖𝑛 󵄩󵄩󵄩
󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩2 󵄩
󵄩 󵄩
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝑓 (𝑥𝑛 ) − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑥𝑛+1 󵄩󵄩󵄩 × (󵄩󵄩󵄩𝑥𝑛 − 𝑝󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛+1 − 𝑝󵄩󵄩󵄩) .
(50)
(44)
Since 𝛼𝑛 → 0, so we have from (47) that
󵄩󵄩󵄩𝑥𝑛 − V𝑖𝑛 󵄩󵄩󵄩 󳨀→ 0.
󵄩
󵄩
(51)
In a similar way, we have that
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑛 − 𝑢𝑖𝑛 󵄩󵄩󵄩 󳨀→ 0.
(52)
6
Journal of Applied Mathematics
Let V𝑡 = 𝑡V + (1 − 𝑡)𝑤, 𝑡 ∈ [0, 1], for all V ∈ 𝐶. Because
{𝑇𝑖 }, 𝑖 ∈ {1, 2, . . . , 𝑚} are pseudocontractive mappings, we
have that
Consequently, we have that
󵄩󵄩
󵄩 󵄨
󵄨󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 = 󵄨󵄨󵄨1 − 𝜆 𝑛 󵄨󵄨󵄨 󵄩󵄩󵄩𝑥𝑛 − V𝑖𝑛 󵄩󵄩󵄩 󳨀→ 0,
󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑦𝑛 − 𝑢𝑖𝑛 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑥𝑛 − 𝑢𝑖𝑛 󵄩󵄩󵄩 󳨀→ 0.
(53)
Since 𝐸 is a reflexive real Banach space and the sequence
{𝑥𝑛 } is bounded, so there exists the subsequence {𝑥𝑛𝑘 } of {𝑥𝑛 }
and 𝑤 ∈ 𝐶 such that 𝑥𝑛𝑘 ⇀ 𝑤. And because 𝑥𝑛 → V𝑖𝑛 ,
𝑛 → ∞, therefore V𝑖𝑛𝑘 ⇀ 𝑤. Next we show that 𝑤 ∈ 𝐹.
Because V𝑖𝑛 = 𝐹𝑖𝑟𝑛 𝑥𝑛 , by the definition of mapping 𝐹𝑖𝑟𝑛 , we
have that
1
⟨𝑦 − V𝑖𝑛 , 𝐴 𝑖 V𝑖𝑛 ⟩+ ⟨𝑦 − V𝑖𝑛 , 𝐽V𝑖𝑛 − 𝐽𝑥𝑛 ⟩ ≥ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶,
𝐽V − 𝐽𝑥𝑛𝑘
⟨𝑦 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑖𝑛𝑘 ⟩ + ⟨𝑦 − V𝑖𝑛 , 𝑖𝑛𝑘
⟩ ≥ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶.
(54)
Let V𝑡 = 𝑡V + (1 − 𝑡)𝑤, 𝑡 ∈ [0, 1], for all V ∈ 𝐶; we have that
⟨V𝑡 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑡 ⟩ ≥ ⟨V𝑡 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑡 ⟩ − ⟨V𝑡 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑖𝑛𝑘 ⟩
− ⟨V𝑡 − V𝑖𝑛𝑘 ,
𝐽V𝑖𝑛𝑘 − 𝐽𝑥𝑛𝑘
⟩
𝑟𝑛
(55)
0 ≤ lim ⟨V𝑡 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑡 ⟩ = ⟨V𝑡 − 𝑤, 𝐴 𝑖 V𝑡 ⟩ .
(56)
Consequently we have that
(57)
If 𝑡 → 0, by the continuity of 𝐴 𝑖 , we have that ⟨V −
𝑤, 𝐴 𝑖 𝑤⟩ ≥ 0; that is, 𝑤 ∈ 𝑉𝐼(𝐶, 𝐴 𝑖 ), and then 𝑤 ∈ 𝐹2 .
Similarly, because 𝑢𝑖𝑛 = 𝑇𝑖𝑟𝑛 𝑦𝑛 , by the definition of mapping 𝑇𝑖𝑟𝑛 , we have that
⟨𝑦 − 𝑢𝑖𝑛 , 𝑇𝑖 𝑢𝑖𝑛 ⟩ −
1
⟨𝑦 − 𝑢𝑖𝑛 , (1 + 𝑟𝑛 ) 𝐽𝑢𝑖𝑛 − 𝐽𝑦𝑛 ⟩ ≤ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶,
⟨𝑦 − 𝑢𝑖𝑛𝑘 , 𝑇𝑖 𝑢𝑖𝑛𝑘 ⟩ −
1
⟨V − 𝑢𝑖𝑛𝑘 , (1 + 𝑟𝑛 ) 𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 ⟩
𝑟𝑛 𝑡
= ⟨V𝑡 − 𝑢𝑖𝑛𝑘 , 𝑇𝑖 𝑢𝑖𝑛𝑘 − 𝑇𝑖 V𝑡 ⟩
− ⟨V𝑡 − 𝑢𝑖𝑛𝑘 ,
1 + 𝑟𝑛
1
𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 ⟩
𝑟𝑛
𝑟𝑛
󵄩
󵄩2 1
≥ −󵄩󵄩󵄩V𝑡 − 𝑢𝑖𝑛𝑘 󵄩󵄩󵄩 − ⟨V𝑡 − 𝑢𝑖𝑛𝑘 , 𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 ⟩
𝑟𝑛
− ⟨V𝑡 − 𝑢𝑖𝑛𝑘 , 𝐽𝑢𝑖𝑛𝑘 ⟩
= − ⟨V𝑡 − 𝑢𝑖𝑛𝑘 , 𝐽V𝑡 ⟩ −
1
⟨V − 𝑢𝑖𝑛𝑘 , 𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 ⟩ .
𝑟𝑛 𝑡
(59)
Because 𝑦𝑛𝑘 − 𝑢𝑖𝑛𝑘 → 0, so 𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 → 0; we have
that
𝑘→∞
(60)
Consequently we have that
Because 𝑥𝑛𝑘 − V𝑖𝑛𝑘 → 0, so (𝐽V𝑖𝑛𝑘 − 𝐽𝑥𝑛𝑘 )/𝑟𝑛 → 0, and
because 𝐴 𝑖 are monotone, we have that
⟨V − 𝑤, 𝐴 𝑖 V𝑡 ⟩ ≥ 0.
−
𝑘→∞
𝐽V𝑖𝑛𝑘 − 𝐽𝑥𝑛𝑘
⟩.
𝑟𝑛
𝑘→∞
≥ ⟨𝑢𝑖𝑛𝑘 − V𝑡 , 𝑇𝑖 V𝑡 ⟩ + ⟨V𝑡 − 𝑢𝑖𝑛𝑘 , 𝑇𝑖 𝑢𝑖𝑛𝑘 ⟩
lim ⟨𝑢𝑖𝑛𝑘 − V𝑡 , 𝑇𝑖 V𝑡 ⟩ ≥ lim ⟨𝑢𝑖𝑛𝑘 − V𝑡 , 𝐽V𝑡 ⟩ .
= ⟨V𝑡 − V𝑖𝑛𝑘 , 𝐴 𝑖 V𝑡 − 𝐴 𝑖 V𝑖𝑛𝑘 ⟩
− ⟨V𝑡 − V𝑖𝑛𝑘 ,
⟨𝑢𝑖𝑛𝑘 − V𝑡 , 𝑇𝑖 V𝑡 ⟩
1
⟨𝑦 − 𝑢𝑖𝑛𝑘 , (1 + 𝑟𝑛 ) 𝐽𝑢𝑖𝑛𝑘 − 𝐽𝑦𝑛𝑘 ⟩ ≤ 0,
𝑟𝑛
∀𝑦 ∈ 𝐶.
(58)
⟨𝑤 − V𝑡 , 𝑇𝑖 V𝑡 ⟩ ≥ ⟨𝑤 − V𝑡 , 𝐽V𝑡 ⟩ ,
⟨V − 𝑤, 𝑇𝑖 V𝑡 ⟩ ≤ ⟨V − 𝑤, 𝐽V𝑡 ⟩ .
(61)
If 𝑡 → 0, by the continuity of 𝑇𝑖 , we have that ⟨V−𝑤, 𝑇𝑖 𝑤−
𝐽𝑤⟩ ≥ 0, for all V ∈ 𝐶; we conclude that 𝑤 = 𝑇𝑖 𝑤; that is,
𝑤 ∈ 𝐹(𝑇𝑖 ), and then 𝑤 ∈ 𝐹1 . Consequently 𝑤 ∈ 𝐹 = 𝐹1 ∩ 𝐹2 .
Because 𝑥 = Π𝐹 𝑓(𝑥), we have from Lemma 5 that
lim sup⟨𝑓 (𝑥) − 𝑥, 𝑗 (𝑥𝑛+1 − 𝑥)⟩
𝑛→∞
(62)
= ⟨𝑓 (𝑥) − 𝑥, 𝑗 (𝑤 − 𝑥)⟩ ≤ 0.
Next we show that 𝑥𝑛 → 𝑥. Since 𝑢𝑖𝑛 = 𝑇𝑖𝑟𝑛 𝑦𝑛 , from
formula (23) and Lemma 6 we have that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑥󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩2
𝑚
󵄩󵄩
󵄩󵄩
󵄩
= 󵄩󵄩𝛼𝑛 (𝑓 (𝑥𝑛 ) − 𝑥) + 𝛽𝑛 (𝑥𝑛 − 𝑥) + 𝛾𝑛 (∑𝜎𝑖 𝑇𝑖𝑟𝑛 𝑦𝑛 − 𝑥)󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
󵄩󵄩
󵄩󵄩2
𝑚
󵄩󵄩
󵄩󵄩
≤ 󵄩󵄩󵄩𝛽𝑛 (𝑥𝑛 − 𝑥) + 𝛾𝑛 (∑𝜎𝑖 𝑢𝑖𝑛 − 𝑥)󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
+ 2𝛼𝑛 ⟨𝑓 (𝑥𝑛 ) − 𝑥, 𝑗 (𝑥𝑛+1 − 𝑥)⟩
Journal of Applied Mathematics
7
2󵄩
󵄩2
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝑓 (𝑥𝑛 ) − 𝑓 (𝑥) , 𝑗 (𝑥𝑛+1 − 𝑥)⟩
+ 2𝛼𝑛 ⟨𝑓 (𝑥) − 𝑥, 𝑗 (𝑥𝑛+1 − 𝑥)⟩
2󵄩
󵄩
󵄩2
󵄩󵄩
󵄩
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 + 2𝜌𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥󵄩󵄩󵄩
⟨(𝑓 − 𝐼) (𝑥) , 𝑗 (𝑦 − 𝑥)⟩ ≤ 0,
󵄩
󵄩2
≤ (1 − 2𝛼𝑛 (1 − 𝜌)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 + 𝜎𝑛 ,
(63)
where 𝑀1 = sup{‖𝑥𝑛 −𝑥‖}, 𝜎𝑛 = 𝛼𝑛2 𝑀12 +2𝜌𝛼𝑛 ‖𝑥𝑛+1 −𝑥𝑛 ‖𝑀1 +
2𝛼𝑛 ⟨𝑓(𝑥) − 𝑥, 𝑗(𝑥𝑛+1 − 𝑥)⟩. Let 𝜃𝑛 = 2𝛼𝑛 (1 − 𝜌); according to
Lemma 1 and formula (62), we have that lim𝑛 → ∞ ‖𝑥𝑛 −𝑥‖ = 0;
that is, the sequence {𝑥𝑛 } converges strongly to 𝑥 ∈ 𝐹.
According to formula (62) we conclude that 𝑥 is the
solution of the variational inequality (24). Now we show that
𝑥 is the unique solution of the variational inequality (24).
Suppose that 𝑦 ∈ 𝐹 is another solution of the variational
inequality (24). Because 𝑥 is the solution of the variational
inequality (24), that is, ⟨(𝑓 − 𝐼)𝑥, 𝑗(𝑦 − 𝑥)⟩ ≤ 0, for all 𝑦 ∈ 𝐹.
And because that 𝑦 ∈ 𝐹, then we have
⟨(𝑓 − 𝐼) 𝑥, 𝑗 (𝑦 − 𝑥)⟩ ≤ 0.
(64)
On the other hand, to the solution 𝑦 ∈ 𝐹, since 𝑥 ∈ 𝐹, so
⟨(𝑓 − 𝐼) 𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 0.
(65)
Adding (65) and (64), we have that
⟨𝑥 − 𝑦 − (𝑓 (𝑥) − 𝑓 (𝑦)) , 𝑗 (𝑥 − 𝑦)⟩ ≤ 0.
(66)
Hence
󵄩2
󵄩
(1 − 𝜌) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≤ 0.
(69)
(67)
Theorem 9. Let 𝐶 be a nonempty closed convex subset of a
real reflexive Banach space 𝐸 which has a uniformly Gateaux
differentiable norm. Let 𝑇𝑖 : 𝐶 → 𝐶, 𝑖 = 1, 2, . . . , 𝑚, be a
finite family of continuous strictly pseudocontractive mappings,
let 𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, . . . , 𝑚, be a finite family of continuous monotone mappings such that 𝐹 = 𝐹1 ∩ 𝐹2 ≠ 0, and
let 𝑓 : 𝐶 → 𝐶 be a contraction with a contraction coefficient
𝜌 ∈ (0, 1). 𝑇𝑖𝑟𝑛 and 𝐹𝑖𝑟𝑛 are defined as (19) and (20), respectively.
Let 𝑥𝑛 be a sequence generated by 𝑥0 ∈ 𝐶
𝑚
𝑥𝑛+1 = 𝛼𝑛 𝑓 (𝑥𝑛 ) + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝜀𝑖 𝑇𝑖𝑟𝑛 𝐹𝑖𝑟𝑛 𝑥𝑛 ,
(68)
𝑖=1
where {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences of nonnegative real
numbers in [0, 1], and
(i) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 𝑛 ≥ 0, ∑𝑚
𝑖=1 𝜀𝑖 = 1, 𝜀𝑖 ≥ 0;
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 = ∞;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 < lim sup𝑛 → ∞ 𝛽𝑛 < 1;
|𝑟𝑛+1 − 𝑟𝑛 | < ∞.
If, in Theorems 8 and 9, we let 𝑓 :≡ 𝑢 ∈ 𝐶 be a constant
mapping, we have the following corollaries.
Corollary 10. Let 𝐶 be a nonempty closed convex subset of a
real reflexive Banach space 𝐸 which has a uniformly Gateaux
differentiable norm. Let 𝑇𝑖 : 𝐶 → 𝐶, 𝑖 = 1, 2, . . . , 𝑚, be a finite
family of continuous strictly pseudo-contractive mappings, and
𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, . . . , 𝑚, be a finite family of continuous
monotone mappings such that 𝐹 = 𝐹1 ∩ 𝐹2 ≠ 0. 𝑇𝑖𝑟𝑛 and 𝐹𝑖𝑟𝑛
are defined as (19) and (20), respectively. Let 𝑥𝑛 be a sequence
generated by 𝑥0 ∈ 𝐶
𝑚
𝑦𝑛 = 𝜆 𝑛 𝑥𝑛 + (1 − 𝜆 𝑛 ) ∑𝜇𝑖 𝐹𝑖𝑟𝑛 𝑥𝑛 ,
𝑖=1
(70)
𝑚
𝑥𝑛+1 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝜎𝑖 𝑇𝑖𝑟𝑛 𝑦𝑛 ,
𝑖=1
where 𝜆 𝑛 ∈ [0, 1] and {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences of nonnegative real numbers in [0, 1], and
𝑚
(i) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 𝑛 ≥ 0, ∑𝑚
𝑖=1 𝜇𝑖 = 1, ∑𝑖=1 𝜎𝑖 = 1, 𝜇𝑖 ≥ 0,
𝜎𝑖 ≥ 0;
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 = ∞;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 < lim sup𝑛 → ∞ 𝛽𝑛 < 1;
Because 𝜌 ∈ (0, 1), hence we conclude that 𝑥 = 𝑦, and the
uniqueness of the solution is obtained.
(iv) lim sup𝑛 → ∞ 𝑟𝑛 > 0,
∀𝑦 ∈ 𝐹.
Proof. Putting 𝜆 𝑛 = 0 in Theorem 8, we can obtain the result.
+ 2𝛼𝑛 ⟨𝑓 (𝑥) − 𝑥, 𝑗 (𝑥𝑛+1 − 𝑥)⟩
∑∞
𝑛=1
Then the sequence 𝑥𝑛 converges strongly to an element 𝑥 =
Π𝐹 𝑓(𝑥), and also 𝑥 is the unique solution of the variational
inequality
(iv) lim sup𝑛 → ∞ 𝑟𝑛 > 0, ∑∞
𝑛=1 |𝑟𝑛+1 − 𝑟𝑛 | < ∞.
Then the sequence 𝑥𝑛 converges strongly to 𝑥 = Π𝐹 𝑢, and
also 𝑥 is the unique solution of the variational inequality
⟨𝑢 − 𝑥, 𝑗 (𝑦 − 𝑥)⟩ ≤ 0,
∀𝑦 ∈ 𝐹.
(71)
Corollary 11. Let 𝐶 be a nonempty closed convex subset of a
real reflexive Banach space 𝐸 which has a uniformly Gateaux
differentiable norm. Let 𝑇𝑖 : 𝐶 → 𝐶, 𝑖 = 1, 2, . . . , 𝑚, be a
finite family of continuous strictly pseudocontractive mappings,
and 𝐴 𝑖 : 𝐶 → 𝐸∗ , 𝑖 = 1, 2, . . . , 𝑚, be a finite family of
continuous monotone mappings such that 𝐹 = 𝐹1 ∩ 𝐹2 ≠ 0. 𝑇𝑖𝑟𝑛
and 𝐹𝑖𝑟𝑛 are defined as (19) and (20), respectively. Let 𝑥𝑛 be a
sequence generated by 𝑥0 ∈ 𝐶
𝑚
𝑥𝑛+1 = 𝛼𝑛 𝑢 + 𝛽𝑛 𝑥𝑛 + 𝛾𝑛 ∑𝜀𝑖 𝑇𝑖𝑟𝑛 𝐹𝑖𝑟𝑛 𝑥𝑛 ,
(72)
𝑖=1
where {𝛼𝑛 }, {𝛽𝑛 }, and {𝛾𝑛 } are sequences of nonnegative real
numbers in [0, 1], and
(i) 𝛼𝑛 + 𝛽𝑛 + 𝛾𝑛 = 1, 𝑛 ≥ 0, ∑𝑚
𝑖=1 𝜀𝑖 = 1, 𝜀𝑖 > 0;
(ii) lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 = ∞;
(iii) 0 < lim inf 𝑛 → ∞ 𝛽𝑛 < lim sup𝑛 → ∞ 𝛽𝑛 < 1;
(iv) lim sup𝑛 → ∞ 𝑟𝑛 > 0, ∑∞
𝑛=1 |𝑟𝑛+1 − 𝑟𝑛 | < ∞.
8
Journal of Applied Mathematics
Then the sequence 𝑥𝑛 converges strongly to 𝑥 = Π𝐹 𝑢, and
also 𝑥 is the unique solution of the variational inequality
⟨𝑢 − 𝑥, 𝑗 (𝑦 − 𝑥)⟩ ≤ 0,
∀𝑦 ∈ 𝐹.
(73)
Acknowledgments
This work is supported by the National Science Foundation
of China (11001287), Natural Science Foundation Project of
Chongqing (CSTC, 2012jjA00039).
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