Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 821737, 7 pages
http://dx.doi.org/10.1155/2013/821737
Research Article
Approximation Analysis for a Common Fixed Point of
Finite Family of Mappings Which Are Asymptotically 𝑘-Strict
Pseudocontractive in the Intermediate Sense
H. Zegeye1 and N. Shahzad2
1
2
Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to N. Shahzad; nshahzad@kau.edu.sa
Received 16 February 2013; Accepted 18 April 2013
Academic Editor: Luigi Muglia
Copyright © 2013 H. Zegeye and N. Shahzad. 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 introduce an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous
asymptotically 𝑘𝑖 -strict pseudocontractive mappings in the intermediate sense for 𝑖 = 1, 2, . . . , 𝑁. The projection of 𝑥0 onto the
intersection of closed convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1 is not required. Moreover, the restriction that the interior of common
fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this
important class of nonlinear mappings.
1. Introduction and Preliminaries
Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. A
mapping 𝑇 : 𝐶 → 𝐻 is called Lipschitzian if there exists
𝐿 > 0 such that ‖𝑇𝑥 − 𝑇𝑦‖ ≤ 𝐿‖𝑥 − 𝑦‖, for all 𝑥, 𝑦 ∈ 𝐶.
If 𝐿 = 1, then 𝑇 is called nonexpansive, and if 𝐿 ∈ [0, 1),
𝑇 is called contraction. 𝑇 is called uniformly 𝐿-Lipschitzian
if there exists 𝐿 > 0 such that ‖𝑇𝑛 𝑥 − 𝑇𝑛 𝑦‖ ≤ 𝐿‖𝑥 −
𝑦‖, for all 𝑥, 𝑦 ∈ 𝐶 and all 𝑛 ≥ 1. Clearly, every contraction mapping is nonexpansive and every nonexpansive
mapping is uniformly 𝐿-Lipschitzian with 𝐿 = 1 and hence
Lipschitzian.
A mapping 𝑇 : 𝐶 → 𝐻 is said to be asymptotically
nonexpansive if there exists a sequence {𝛾𝑛 } ⊂ [0, ∞) with
𝛾𝑛 → 0 such that ‖𝑇𝑛 𝑥 − 𝑇𝑛 𝑦‖ ≤ (1 + 𝛾𝑛 )‖𝑥 − 𝑦‖ for all
integers 𝑛 ≥ 1 and all 𝑥, 𝑦 ∈ 𝐶. 𝑇 is said to be asymptotically
nonexpansive in the intermediate sense if there exist sequences
{𝛾𝑛 }, {𝑐𝑛 } ⊂ [0, ∞) with 𝛾𝑛 → 0, 𝑐𝑛 → 0 such that
‖𝑇𝑛 𝑥 − 𝑇𝑛 𝑦‖ ≤ (1 + 𝛾𝑛 )‖𝑥 − 𝑦‖ + 𝑐𝑛 for all integers 𝑛 ≥ 1
and all 𝑥, 𝑦 ∈ 𝐶.
A mapping 𝑇 : 𝐶 → 𝐻 is said to be asymptotically 𝑘strict pseudocontractive if there exist a constant 𝑘 ∈ [0, 1) and
a sequence {𝛾𝑛 } ⊂ [0, ∞) with 𝛾𝑛 → 0, as 𝑛 → ∞, such that
󵄩󵄩 𝑛
𝑛 󵄩2
󵄩󵄩𝑇 𝑥 − 𝑇 𝑦󵄩󵄩󵄩
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩
󵄩2
+ 𝑘󵄩󵄩󵄩(𝐼 − 𝑇𝑛 )𝑥 − (𝐼 − 𝑇𝑛 )𝑦󵄩󵄩󵄩 ,
(1)
∀𝑥, 𝑦 ∈ 𝐶.
The class of asymptotically 𝑘-strict pseudocontractive mappings which includes the class of asymptotically nonexpansive, and hence the class of nonexpansive mappings was
introduced by Liu [1] in 1996 (see, also [2]). Kim and Xu
[3] proved that the fixed point set of asymptotically 𝑘-strict
pseudocontractions is closed and convex. Recall that a fixed
point of a map 𝑇 : 𝐶 → 𝐻 is a set {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥},
and it is denoted by 𝐹(𝑇). In addition, it is noted in [3]
that every asymptotically 𝑘-strict pseudocontractive mapping
with sequence {𝛾𝑛 } is a uniformly 𝐿-Lipschitzian mapping
with 𝐿 := sup{(𝑘 + √1 + (1 − 𝑘)𝛾𝑛 )/(1 − 𝑘) : 𝑛 ∈ N}.
2
Journal of Applied Mathematics
A mapping 𝑇 is said to be an asymptotically 𝑘-strict
pseudocontractive in the intermediate sense if
󵄩2
󵄩
󵄩2
󵄩
lim sup sup (󵄩󵄩󵄩𝑇𝑛 𝑥 − 𝑇𝑛 𝑦󵄩󵄩󵄩 − (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
𝑛→∞
𝑥,𝑦∈𝐶
󵄩
󵄩
−𝑘 󵄩󵄩󵄩(𝐼 − 𝑇𝑛 ) 𝑥 − (𝐼 − 𝑇𝑛 ) 𝑦󵄩󵄩󵄩) ≤ 0,
(2)
where 𝑘 ∈ [0, 1) and {𝛾𝑛 } ⊂ [0, ∞) such that 𝛾𝑛 → 0, as
𝑛 → ∞. If we put
𝑥,𝑦∈𝐶
(3)
󵄩
󵄩2
−𝑘󵄩󵄩󵄩(𝐼 − 𝑇𝑛 ) 𝑥 − (𝐼 − 𝑇𝑛 ) 𝑦󵄩󵄩󵄩 ) } .
󵄩󵄩 𝑛
𝑛 󵄩2
󵄩󵄩𝑇 𝑥 − 𝑇 𝑦󵄩󵄩󵄩
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
∀𝑥, 𝑦 ∈ 𝐶.
(4)
We note that the class of asymptotically 𝑘-strict pseudocontractive mappings is properly contained in a class of
asymptotically 𝑘-strict pseudocontractive mapping in the
intermediate sense (see examples in [4]). The class of
asymptotically 𝑘-strict pseudocontractive mappings in the
intermediate sense was introduced by Sahu et al. [4]. They
obtained a weak convergence theorem of modified Mann
iterative processes for these class of mappings. In [4], Sahu
et al. established the following classical result.
Theorem SXY 1 (see [4]). Let 𝐻 be a real Hilbert space,
and let 𝐶 ⊂ 𝐻 be a nonempty closed and convex set.
Let 𝑇 be a uniformly continuous and asymptotically 𝑘-strict
pseudocontractive mapping in the intermediate sense with
sequences {𝛾𝑛 } and {𝑐𝑛 } such that 𝐹(𝑇) is nonempty and
∑𝑛𝑛=1 𝛾𝑛 < ∞. Assume that {𝛼𝑛 } is a sequence in (0, 1) such
that 0 < 𝛿 ≤ 𝛼𝑛 ≤ 1 − 𝑘 − 𝛿 < 1 and ∑𝑛𝑛=1 𝛼𝑛 𝑐𝑛 < ∞. Let {𝑥𝑛 }
be a sequence defined by 𝑥1 = 𝑥 ∈ 𝐶 and
𝑛 ≥ 1.
𝑛 ≥ 1,
𝑄𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝑢 − 𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = 𝑃𝐶𝑛 ∩ 𝑄𝑛 (𝑢) ,
It follows that 𝑐𝑛 → 0, as 𝑛 → ∞, and (2) is reduced to the
following:
𝑥𝑛+1 = (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇𝑛 𝑥𝑛 ,
𝑦𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇𝑛 𝑥𝑛 ,
󵄩
󵄩2 󵄩
󵄩2
𝐶𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩𝑦𝑛 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + 𝜃𝑛 } ,
󵄩2
󵄩
󵄩2
󵄩
𝑐𝑛 := max {0, sup (󵄩󵄩󵄩𝑇𝑛 𝑥 − 𝑇𝑛 𝑦󵄩󵄩󵄩 − (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
󵄩
󵄩2
+ 𝑘󵄩󵄩󵄩(𝐼 − 𝑇𝑛 ) 𝑥 − (𝐼 − 𝑇𝑛 ) 𝑦󵄩󵄩󵄩 + 𝑐𝑛 ,
Theorem SXY2 (see [4]). Let 𝐻 be a real Hilbert space, and
let 𝐶 ⊂ 𝐻 be a nonempty, closed, and convex set. Let 𝑇 be
a uniformly continuous asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense with sequences {𝛾𝑛 } and
{𝑐𝑛 } such that 𝐹(𝑇) is nonempty and bounded. Assume that {𝛼𝑛 }
is a sequence in [0, 1] such that 0 < 𝛿 ≤ 𝛼𝑛 ≤ 1 − 𝑘, for all
𝑛 ∈ 𝑁. Let {𝑥𝑛 } be a sequence in 𝐶 defined by 𝑢 = 𝑥1 ∈ 𝐶 and
(5)
Then, {𝑥𝑛 } converges weakly to a fixed point of 𝑇.
But it is worth mentioning that the convergence obtained
is a weak convergence. Furthermore, we observe from the
proof of Theorem SXY1 that if, in addition, 𝐶 or 𝑇 has some
compactness assumption, we obtain that the sequence {𝑥𝑛 }
given by (5) converges strongly to a fixed point of 𝑇.
Attempts to modify the Mann iteration method (5) so
that strong convergence is guaranteed, without compactness
assumption on 𝐶 or 𝑇, have recently been made. Sahu et
al. [4] established the following hybrid Mann algorithm
for approximating fixed points of asymptotically 𝑘-strict
pseudocontractive mappings in the intermediate sense.
(6)
𝑛 ≥ 1,
where 𝜃𝑛 = 𝑐𝑛 + 𝛾𝑛 𝛿𝑛 and 𝛿𝑛 = sup{‖𝑥𝑛 − 𝑧‖ : 𝑧 ∈ 𝐹(𝑇)} < ∞.
Then, {𝑥𝑛 } converges strongly to 𝑃𝐹(𝑇) (𝑢).
Recently, Hu and Cai [5] studied the strong convergence
of the modified Mann iteration process (5) for a finite family
of asymptotically 𝑘-strict pseudocontractive mappings in
the intermediate sense. More precisely, they obtained the
following theorem.
Theorem HC (see [5]). Let 𝐻 be a real Hilbert space, let
𝐶 ⊂ 𝐻 be a nonempty, closed, and convex set. Let 𝑇𝑖 :
𝐶 → 𝐶 be uniformly continuous asymptotically 𝑘𝑖 -strict
pseudocontractive mappings in the intermediate sense for some
0 ≤ 𝑘𝑖 < 1 with sequences {𝛾𝑛,𝑖 } and {𝑐𝑛,𝑖 } such that
lim𝑛 → ∞ 𝛾𝑛,𝑖 = 0 and lim𝑛 → ∞ 𝑐𝑛,𝑖 = 0 for 𝑖 = 1, 2, . . ., 𝑁.
Let 𝑘 = max{𝑘𝑖 : 1 ≤ 𝑖 ≤ 𝑁}, 𝛾𝑛 = max{𝛾𝑛,𝑖 : 1 ≤ 𝑖 ≤ 𝑁},
and 𝑐𝑛 = max{𝑐𝑛,𝑖 : 1 ≤ 𝑖 ≤ 𝑁}. Assume that 𝐹 := ⋂𝑁
𝑛=1 𝐹(𝑇𝑖 )
is nonempty and bounded. Let {𝛽𝑛 } be a sequence in [0, 1] such
that 0 < 𝛿 ≤ 𝛽𝑛 ≤ 1 − 𝑘 for all 𝑛 ∈ 𝑁. Let {𝑥𝑛 } be a sequence
defined by 𝑥0 ∈ 𝐶 and
𝑘(𝑛)
𝑦𝑛 = (1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇𝑖(𝑛)
𝑥𝑛 ,
𝑛 ≥ 1,
󵄩
󵄩2 󵄩
󵄩2
𝐶𝑛 = {𝑧 ∈ 𝐶 : 󵄩󵄩󵄩𝑦𝑛 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 + 𝜃𝑛 } ,
(7)
𝑄𝑛 = {𝑧 ∈ 𝐶 : ⟨𝑥𝑛 − 𝑧, 𝑥0 − 𝑥𝑛 ⟩ ≥ 0} ,
𝑥𝑛+1 = 𝑃𝐶𝑛 ∩ 𝑄𝑛 (𝑥0 ) ,
𝑛 ≥ 1,
where 𝜃𝑛 = 𝛾𝑘(𝑛) 𝜌𝑛2 + 𝑐𝑘(𝑛) → 0, as 𝑛 → ∞, for 𝜌𝑛 =
sup{‖𝑥𝑛 − 𝑧‖ : 𝑧 ∈ 𝐹} < ∞, 𝑛 = (ℎ − 1)𝑁 + 𝑖, where
𝑖 = 𝑖(𝑛) ∈ {1, 2, . . . , 𝑁}, ℎ = ℎ(𝑛) ≥ 1 a positive integer such
that ℎ(𝑛) → ∞, as 𝑛 → ∞. Then, {𝑥𝑛 } converges strongly to
𝑃𝐹 (𝑥0 ).
But we observe that the iterative algorithms (6) and
(7) generate a sequence {𝑥𝑛 } by projecting 𝑥0 onto the
intersection of closed convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1
which is not easy to compute.
Attempts to remove projection mapping onto the intersection of closed convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1 have
recently been made. In [6], Zegeye et al. studied the strong
convergence of the modified Mann iteration process for the
class of asymptotically 𝑘-strict pseudocontractive mappings
Journal of Applied Mathematics
3
in the intermediate sense. More precisely, they proved the
following theorem.
Theorem ZRS (see [6]). Let 𝐶 be a nonempty, closed, and
convex subset of a real Hilbert space 𝐻, and let 𝑇 : 𝐶 →
𝐶 be a uniformly 𝐿-Lipschitzian and asymptotically 𝑘-strict
pseudocontractive mapping in the intermediate sense with
sequences {𝛾𝑛 } ⊂ [0, ∞) and {𝑐𝑛 } ⊂ [0, ∞). Assume that the
interior of 𝐹(𝑇) is nonempty. Let {𝑥𝑛 } be a sequence defined by
𝑥1 = 𝑥 ∈ 𝐶 and
𝑦𝑛 = 𝛽𝑛 𝑇𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑥𝑛 ,
𝑥𝑛+1 = 𝛼𝑛 𝑇𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑥𝑛 ,
𝑛 ≥ 1,
(8)
where {𝛼𝑛 } and {𝛽𝑛 } satisfy certain conditions. Then, {𝑥𝑛 }
converges strongly to a fixed point of 𝑇.
But it is worth to mention that the assumption interior of
𝐹(𝑇) is nonempty is severe restriction.
It is our purpose, in this paper, to construct an iteration
scheme which converges strongly to a common fixed point
of a finite family of uniformly continuous asymptotically 𝑘𝑖 strict pseudocontractive mappings in the intermediate sense
for 𝑖 = 1, 2, . . ., 𝑁. The projection of 𝑥0 onto the intersection
of closed convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1 is not
required. Furthermore, the restriction that the interior of
𝐹(𝑇) is nonempty is not required. Our theorems improve
and unify most of the results that have been proved for this
important class of nonlinear mappings.
In order to prove our results, we need the following
lemmas.
Lemma 1. Let 𝐻 be a real Hilbert space. Then, for any given 𝑥,
𝑦 ∈ 𝐻, the following inequality holds:
󵄩2
󵄩󵄩
2
󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑥 + 𝑦⟩ .
(9)
Lemma 2 (see [7]). Let H be a real Hilbert space. Then, for all
𝑥, 𝑦 ∈ 𝐻 and 𝛼𝑖 , ∈ [0, 1] for 𝑖 = 1, 2, . . ., 𝑛 such that 𝛼0 +
𝛼1 + ⋅ ⋅ ⋅ + 𝛼𝑛 = 1, the following equality holds:
󵄩2
󵄩󵄩
󵄩󵄩𝛼0 𝑥0 + 𝛼1 𝑥1 + ⋅ ⋅ ⋅ + 𝛼𝑛 𝑥𝑛 󵄩󵄩󵄩
𝑛
󵄩
󵄩2
󵄩 󵄩2
= ∑𝛼𝑖 󵄩󵄩󵄩𝑥𝑖 󵄩󵄩󵄩 − ∑ 𝛼𝑖 𝛼𝑗 󵄩󵄩󵄩󵄩𝑥𝑖 − 𝑥𝑗 󵄩󵄩󵄩󵄩 .
𝑖=0
(10)
𝑎𝑘 ≤ 𝑎𝑚𝑘 +1 .
(11)
In fact, 𝑚𝑘 = max{𝑗 ≤ 𝑘 : 𝑎𝑗 < 𝑎𝑗+1 }.
Lemma 4 (see [9]). Let {𝑎𝑛 } be a sequence of nonnegative real
numbers satisfying the following relation:
𝑎𝑛+1 ≤ (1 − 𝛼𝑛 ) 𝑎𝑛 + 𝛼𝑛 𝛿𝑛 ,
𝑛 ≥ 𝑛0 ,
Lemma 5 (see [4]). Let 𝐶 be a nonempty closed convex
subset of a Hilbert space 𝐻, and let 𝑇 : 𝐶 → 𝐶 be a
continuous asymptotically 𝑘-strict pseudocontractive mapping
in the intermediate sense. Then, 𝐹(𝑇) is closed and convex.
Lemma 6 (see [4]). Let 𝐶 be a nonempty closed convex subset
of a Hilbert space 𝐻, and let 𝑇 : 𝐶 → 𝐶 be a uniformly
continuous asymptotically 𝑘-strict pseudocontractive mapping
in the intermediate sense. Let {𝑥𝑛 } be a sequence in 𝐶 such that
‖𝑥𝑛 − 𝑇𝑛 𝑥𝑛 ‖ → 0 and ‖𝑥𝑛 − 𝑥𝑛+1 ‖ → 0, as 𝑛 → ∞. Then,
‖𝑥𝑛 − 𝑇𝑥𝑛 ‖ → 0, as 𝑛 → ∞.
Lemma 7 (see [4]). Let 𝐶 be a nonempty closed convex subset
of a Hilbert space 𝐻, and 𝑇 : 𝐶 → 𝐶 be a continuous
asymptotically 𝑘-strict pseudocontractive mapping in the intermediate sense. Then if {𝑥𝑛 } is a sequence in 𝐶 such that 𝑥𝑛
converges weakly 𝑥 ∈ 𝐶 and lim sup𝑚 → ∞ lim sup𝑛 → ∞ ‖𝑥𝑛 −
𝑇𝑚 𝑥𝑛 ‖ = 0, then (𝐼 − 𝑇)𝑥 = 0.
Lemma 8 (see [10]). Let 𝐶 be a nonempty closed, convex
subset of a Hilbert space 𝐻 and let 𝑃𝐶 be the metric projection
mapping from 𝐻 onto 𝐶. Given 𝑧 = 𝑃𝐶𝑥 if and only if ⟨𝑦 −
𝑧, 𝑥 − 𝑧⟩ ≤ 0, for all 𝑦 ∈ 𝐶.
2. Main Result
Theorem 9. Let 𝐶 be a nonempty, closed and convex subset of a
real Hilbert space 𝐻. Let 𝑇𝑖 : 𝐶 → 𝐶 be uniformly continuous
asymptotically 𝑘𝑖 -strict pseudocontractive mappings in the
intermediate sense for some 0 ≤ 𝑘𝑖 < 1 with sequences {𝛾𝑛,𝑖 }
and {𝑐𝑛,𝑖 }, for 𝑖 = 1, 2, . . ., 𝑁. Assume that 𝐹 := ⋂𝑁
𝑖=1 𝐹(𝑇𝑖 ) is
nonempty. Let {𝑥𝑛 } be a sequence generated by
𝑥0 = 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,
𝑁
0≤𝑖,𝑗≤𝑛
Lemma 3 (see [8]). Let {𝑎𝑛 } be sequences of real numbers such
that there exists a subsequence {𝑛𝑖 } of {𝑛} such that 𝑎𝑛𝑖 < 𝑎𝑛𝑖 +1
for all 𝑖 ∈ 𝑁. Then, there exists a nondecreasing sequence
{𝑚𝑘 } ⊂ 𝑁 such that 𝑚𝑘 → ∞, and the following properties
are satisfied by all (sufficiently large) numbers 𝑘 ∈ 𝑁:
𝑎𝑚𝑘 ≤ 𝑎𝑚𝑘 +1 ,
where {𝛼𝑛 } ⊂ (0, 1) and {𝛿𝑛 } ⊂ 𝑅 satisfying the following conditions: lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 = ∞ and lim sup𝑛 → ∞ 𝛿𝑛 ≤ 0.
Then, lim𝑛 → ∞ 𝑎𝑛 = 0.
(12)
𝑦𝑛 = 𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖𝑛 𝑥𝑛 ,
(13)
𝑖=1
𝑥𝑛+1 = 𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 ,
where 0 < 𝛼𝑛 ≤ 𝑎0 < 1 such that lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 =
∞, lim𝑛 → ∞ [𝛾𝑛,𝑖 /𝛼𝑛 ] = 0 = lim𝑛 → ∞ [𝑐𝑛,𝑖 /𝛼𝑛 ], and 0 < 𝛿 ≤
𝛽𝑛,𝑖 ≤ 1 − 𝑘 − 𝛿 < 1 satisfying 𝛽𝑛,0 + 𝛽𝑛,1 + ⋅ ⋅ ⋅ + 𝛽𝑛,𝑁 = 1 for each
𝑛 ≥ 1 and 𝑘 := max{𝑘𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then, {𝑥𝑛 } converges
strongly to an element of 𝐹.
Proof. Let 𝑥∗ = 𝑃𝐹 𝑤. Let 𝛾𝑛 := max{𝛾𝑛,𝑖 : 𝑖 = 1, 2, . . . , 𝑁} and
𝑐𝑛 := max{𝑐𝑛,𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then, from (13), Lemma 2,
4
Journal of Applied Mathematics
and asymptotically 𝑘𝑖 -strict pseudocontractiveness of 𝑇𝑖 , for
each 𝑖 ∈ {1, 2, . . . , 𝑁}, we get that
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑦𝑛 − 𝑥 󵄩󵄩󵄩
󵄩󵄩2
󵄩󵄩
𝑁
󵄩󵄩
󵄩
𝑛
∗󵄩
󵄩
= 󵄩󵄩𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖 𝑥𝑛 − 𝑥 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑖=1
󵄩
󵄩
since there exists 𝑁0 > 0 such that 𝛾𝑛 /𝛼𝑛 ≤ 𝜖 and 𝑐𝑛 /𝛼𝑛 ≤ 1
for all 𝑛 ≥ 𝑁0 and for some 𝜖 > 0 satisfying (1 − 𝜖)𝛼𝑛 ≤ 1.
Thus, by induction,
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ max {(1 − 𝜖)−1 [󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + 1] , 󵄩󵄩󵄩𝑥0 − 𝑥∗ 󵄩󵄩󵄩 } , (16)
∀𝑛 ≥ 𝑁0 ,
𝑁
󵄩
󵄩2
󵄩
󵄩2
≤ 𝛽𝑛,0 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + ∑𝛽𝑛,𝑖 󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
which implies that {𝑥𝑛 }, and hence {𝑇𝑖𝑛 𝑥𝑛 } and {𝑦𝑛 } are
bounded. Moreover, from (13), (14) and Lemma 1, we obtain
that
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩
󵄩2
= 󵄩󵄩󵄩𝛼𝑛 (𝑤 − 𝑥∗ ) + (1 − 𝛼𝑛 ) (𝑦𝑛 − 𝑥∗ )󵄩󵄩󵄩
󵄩2
󵄩
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝑤 − 𝑥∗ , 𝑥𝑛+1 − 𝑥∗ ⟩
𝑖=1
𝑁
󵄩
󵄩2
− ∑𝛽𝑛,0 𝛽𝑛,𝑖 󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩
𝑖=1
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛽𝑛,0 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛽𝑛,0 ) (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑁
󵄩2
󵄩
+ ∑𝛽𝑛,𝑖 𝑘󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛽𝑛,0 ) 𝑐𝑛
𝑖=1
𝑁
󵄩2
󵄩
󵄩2
󵄩
≤ (1 − 𝛼𝑛 ) [(1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 𝛿2 ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩
𝑁
󵄩
󵄩2
− ∑𝛽𝑛,0 𝛽𝑛,𝑖 󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩
𝑖=1
𝑖=1
󵄩2
󵄩
≤ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
+(1−𝛽𝑛,0 ) 𝑐𝑛 ]+2𝛼𝑛 ⟨𝑤−𝑥∗ , 𝑥𝑛+1 −𝑥∗ ⟩
𝑁
󵄩
󵄩2
− ∑𝛽𝑛,𝑖 (1 − 𝑘 − 𝛿) 󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛽𝑛,0 ) 𝑐𝑛
󵄩2
󵄩
󵄩2
󵄩
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 𝛾𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑖=1
𝑁
󵄩2
󵄩
󵄩2
󵄩
≤ (1+𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 −𝑥∗ 󵄩󵄩󵄩 −𝛿2 ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 −𝑥𝑛 󵄩󵄩󵄩 +(1−𝛽𝑛,0 ) 𝑐𝑛 ,
𝑖=1
(14)
󵄩󵄩
∗ 󵄩2
󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩
󵄩2
󵄩
= 󵄩󵄩󵄩𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑖=1
∗
+ 2𝛼𝑛 ⟨𝑤 − 𝑥 , 𝑥𝑛+1 − 𝑥∗ ⟩
󵄩2
󵄩
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝑤 − 𝑥∗ , 𝑥𝑛+1 − 𝑥∗ ⟩
𝑁
󵄩
󵄩2
+ (𝛾𝑛 + 𝑐𝑛 ) 𝑀 − 𝛿2 (1 − 𝑎0 ) ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩
𝑖=1
(17)
󵄩
󵄩2
≤ 𝛼𝑛 󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 )
󵄩2
󵄩
≤ (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 2𝛼𝑛 ⟨𝑤 − 𝑥∗ , 𝑥𝑛+1 − 𝑥∗ ⟩
󵄩2
󵄩
× [ (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 − 𝛿2
+ (𝛾𝑛 + 𝑐𝑛 ) 𝑀
(18)
for some 𝑀 > 0 and for all 𝑛 ∈ 𝑁.
Now, following the method of proof of Lemma 3.2 of
Maingé [8], we consider two cases.
𝑁
󵄩
󵄩2
×∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛽𝑛,0 ) 𝑐𝑛 ]
𝑖=1
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛼𝑛 󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) (1 + 𝛾𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑁
󵄩
󵄩2
− (1 − 𝛼𝑛 ) 𝛿2 ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) (1 − 𝛽𝑛,0 ) 𝑐𝑛
𝑖=1
󵄩
󵄩2
󵄩2
󵄩
≤ 𝛼𝑛 [󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + 1] + (1 − 𝛼𝑛 (1 − 𝜖)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
Case 1. Suppose that there exists 𝑛0 ∈ 𝑁 such that {‖𝑥𝑛 −𝑥∗ ‖}
is nonincreasing for all 𝑛 ≥ 𝑛0 . In this situation, {‖𝑥𝑛 − 𝑥∗ ‖)}
is convergent. Then, from (17), we have that
𝑥𝑛 − 𝑇𝑖𝑛 𝑥𝑛 󳨀→ 0,
as 𝑛 󳨀→ ∞,
(19)
for 𝑖 = 1, 2, . . ., 𝑁. Moreover, from (13) and (19) and the fact
that 𝛼𝑛 → 0, we get that
𝑁
󵄩
󵄩2
− (1 − 𝛼𝑛 ) 𝛿2 ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩
𝑁
󵄩
󵄩 𝑛
󵄩
󵄩󵄩
󵄩󵄩𝑦𝑛 − 𝑥𝑛 󵄩󵄩󵄩 ≤ ∑𝛽𝑛,𝑖 󵄩󵄩󵄩𝑇𝑖 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0,
𝑖=1
󵄩2
󵄩2
󵄩
󵄩
≤ 𝛼𝑛 [󵄩󵄩󵄩𝑤 − 𝑥∗ 󵄩󵄩󵄩 + 1] + (1 − 𝛼𝑛 (1 − 𝜖)) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩
𝑁
󵄩
󵄩2
− 𝛿2 (1 − 𝛼𝑛 ) ∑󵄩󵄩󵄩𝑇𝑖𝑛 𝑥𝑛 − 𝑥𝑛 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) (1 − 𝛽𝑛,0 ) 𝑐𝑛
𝑖=1
(15)
𝑥𝑛+1 − 𝑦𝑛 = 𝛼𝑛 (𝑤 − 𝑦𝑛 ) 󳨀→ 0,
(20)
Journal of Applied Mathematics
as 𝑛 → ∞, and hence
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0,
5
This implies that
as 𝑛 󳨀→ ∞.
(21)
Furthermore, from (19), (21), and Lemma 6, we obtain that
𝑥𝑛 − 𝑇𝑖 𝑥𝑛 󳨀→ 0,
as 𝑛 󳨀→ ∞.
(22)
Let {𝑥𝑛𝑘 +1 } be a subsequence of {𝑥𝑛+1 } such that
lim sup𝑛 → ∞ ⟨𝑤−𝑥∗ , 𝑥𝑛+1 −𝑥∗ ⟩ = lim𝑘 → ∞ ⟨𝑤−𝑥∗ , 𝑥𝑛𝑘 +1 −𝑥∗ ⟩
and {𝑥𝑛𝑘 +1 } converges weakly to V. Then, from (21), we also
get that {𝑥𝑛𝑘 } converges weakly to V. Moreover, since 𝑇𝑖
is uniformly continuous and ‖𝑥𝑛 − 𝑇𝑖 𝑥𝑛 ‖ → 0, for all
𝑖 ∈ {1, 2, . . . , 𝑁}, we get that ‖𝑥𝑛 − 𝑇𝑖𝑚 𝑥𝑛 ‖ → 0, as 𝑛 → ∞,
for all 𝑚 ∈ 𝑁. Therefore, by Lemma 7, we obtain that
V ∈ ⋂𝑁
𝑖=1 𝐹(𝑇𝑖 ). Now, from Lemma 8, we have that
lim sup ⟨𝑤 − 𝑥∗ , 𝑥𝑛+1 − 𝑥∗ ⟩ = ⟨𝑤 − 𝑥∗ , V − 𝑥∗ ⟩ ≤ 0. (23)
𝑛→∞
Then, from (18), (23), and Lemma 4, we obtain that ‖𝑥𝑛 −
𝑥∗ ‖ → 0, as 𝑛 → ∞. Consequently, 𝑥𝑛 → 𝑥∗ .
Case 2. Suppose that there exists a subsequence {𝑛𝑖 } of {𝑛}
such that
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑥𝑛𝑖 − 𝑥∗ 󵄩󵄩󵄩 < 󵄩󵄩󵄩𝑥𝑛𝑖 +1 − 𝑥∗ 󵄩󵄩󵄩
(24)
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩2
𝛼𝑚𝑘 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 +1 − 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑚𝑘 ⟨𝑤 − 𝑥∗ , 𝑥𝑚𝑘 +1 − 𝑥∗ ⟩ + (𝛾𝑚𝑘 + 𝑐𝑚𝑘 ) 𝑀.
(29)
Then, using (25), inequality (29) implies that
󵄩
󵄩2
𝛼𝑚𝑘 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩
≤ 2𝛼𝑚𝑘 ⟨𝑤 − 𝑥∗ , 𝑥𝑚𝑘 +1 − 𝑥∗ ⟩ + (𝛾𝑚𝑘 + 𝑐𝑚𝑘 ) 𝑀.
(30)
In particular, since 𝛼𝑚𝑘 > 0, we get
𝛾 +𝑐
󵄩󵄩
󵄩2
󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩 ≤ 2 ⟨𝑤 − 𝑥∗ , 𝑥𝑚𝑘 +1 − 𝑥∗ ⟩ + 𝑚𝑘 𝑚𝑘 .
󵄩
󵄩
𝛼𝑚𝑘
(31)
(25)
Furthermore, using (27) and the fact that (𝛾𝑚𝑘 + 𝑐𝑚𝑘 )/𝛼𝑚𝑘 →
0, we obtain that ‖𝑥𝑚𝑘 − 𝑥∗ ‖ → 0, as 𝑘 → ∞. This together
with (28) give ‖𝑥𝑚𝑘 +1 −𝑥∗ ‖ → 0, as 𝑘 → ∞. But ‖𝑥𝑘 −𝑥∗ ‖ ≤
‖𝑥𝑚𝑘 +1 − 𝑥∗ ‖ for all 𝑘 ∈ 𝑁. Thus we obtain that 𝑥𝑘 → 𝑥∗ .
Therefore, from the above two cases, we can conclude that
{𝑥𝑛 } converges strongly to an element of 𝐹, and the proof is
complete.
and ‖𝑥𝑘 − 𝑥∗ ‖ ≤ ‖𝑥𝑚𝑘 +1 − 𝑥∗ ‖, for all 𝑘 ∈ 𝑁. Then, from (17)
and the fact that 𝛼𝑛 → 0, we have
If in Theorem 9, we assume that 𝑁 = 1, then we get the
following corollary.
for all 𝑖 ∈ 𝑁. Then, by Lemma 3, there exist a nondecreasing
sequence {𝑚𝑘 } ⊂ 𝑁 such that 𝑚𝑘 → ∞,
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑚𝑘 +1 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩 󵄩
󵄩
𝑁
󵄩
󵄩2
𝑚
𝛿2 (1 − 𝑎0 ) ∑󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑇𝑖 𝑘 𝑥𝑚𝑘 󵄩󵄩󵄩󵄩
𝑖=1
󵄩2 󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 +1 − 𝑥∗ 󵄩󵄩󵄩󵄩
󵄩
󵄩2
+ 𝛼𝑚𝑘 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩 + 2𝛼𝑚𝑘 ⟨𝑤 − 𝑥∗ , 𝑥𝑚𝑘 +1 − 𝑥∗ ⟩
+ (𝛾𝑚𝑘 + 𝑐𝑚𝑘 ) 𝑀 󳨀→ 0,
as 𝑘 󳨀→ ∞.
𝑥0 = 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,
(26)
𝑚
Then, we get that 𝑥𝑚𝑘 − 𝑇𝑖 𝑘 𝑥𝑚𝑘 → 0, as 𝑛 → ∞, for each
𝑖 = 1, 2, . . ., 𝑁. Thus, as in Case 1, we obtain that 𝑥𝑚𝑘 − 𝑦𝑚𝑘 →
0 and 𝑥𝑚𝑘 +1 − 𝑥𝑚𝑘 → 0, as 𝑘 → ∞ and
∗
∗
lim sup ⟨𝑤 − 𝑥 , 𝑥𝑚𝑘 +1 − 𝑥 ⟩ ≤ 0.
𝑘→∞
Corollary 10. Let 𝐶 be a nonempty, closed, and convex subset
of a real Hilbert space 𝐻. Let 𝑇 : 𝐶 → 𝐶 be a uniformly
continuous asymptotically 𝑘-strict pseudocontractive mapping
in the intermediate sense for some 0 ≤ 𝑘 < 1 with sequences
{𝛾𝑛 } and {𝑐𝑛 }. Assume that 𝐹 := 𝐹(𝑇) is nonempty. Let {𝑥𝑛 } be
a sequence generated by
(27)
Now, from (18), we have that
󵄩2
󵄩󵄩
󵄩󵄩𝑥𝑚𝑘 +1 − 𝑥∗ 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
≤ (1 − 𝛼𝑚𝑘 ) 󵄩󵄩󵄩󵄩𝑥𝑚𝑘 − 𝑥∗ 󵄩󵄩󵄩󵄩
+ 2𝛼𝑚𝑘 ⟨𝑤 − 𝑥∗ , 𝑥𝑚𝑘 +1 − 𝑥∗ ⟩ + (𝛾𝑚𝑘 + 𝑐𝑚𝑘 ) 𝑀.
(28)
𝑦𝑛 = 𝛽𝑛 𝑥𝑛 + (1 − 𝛽𝑛 ) 𝑇𝑛 𝑥𝑛 ,
(32)
𝑥𝑛+1 = 𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 ,
where 0 < 𝛼𝑛 ≤ 𝑎0 < 1 such that lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 =
∞, lim𝑛 → ∞ [𝛾𝑛 /𝛼𝑛 ] = 0 = lim𝑛 → ∞ [𝑐𝑛 /𝛼𝑛 ], and 0 < 𝛿 ≤ 𝛽𝑛 ≤
1 − 𝑘 − 𝛿 < 1 for each 𝑛 ≥ 1. Then, {𝑥𝑛 } converges strongly to
an element of 𝐹.
If in Theorem 9, we assume that each 𝑇𝑖 is asymptotically
𝑘𝑖 -strict pseudocontractive mapping, then we get the following corollary.
Corollary 11. Let 𝐶 be a nonempty, closed, and convex subset
of a real Hilbert space 𝐻. Let 𝑇𝑖 : 𝐶 → 𝐶 be asymptotically
𝑘𝑖 -strict pseudocontractive mappings for some 0 ≤ 𝑘𝑖 < 1
6
Journal of Applied Mathematics
with sequences {𝛾𝑛,𝑖 }, for 𝑖 = 1, 2, . . ., 𝑁. Assume that 𝐹 :=
⋂𝑁
𝑖=1 𝐹(𝑇𝑖 ) is nonempty. Let {𝑥𝑛 } be a sequence generated by
for 𝑖 = 1, 2, . . ., 𝑁. Assume that 𝐹 := ⋂𝑁
𝑖=1 𝐹(𝑇𝑖 ) is nonempty.
Let {𝑥𝑛 } be a sequence generated by
𝑥0 = 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,
𝑥0 = 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,
𝑁
𝑁
𝑦𝑛 = 𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖𝑛 𝑥𝑛 ,
𝑦𝑛 = 𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖𝑛 𝑥𝑛 ,
(33)
(35)
𝑖=1
𝑖=1
𝑥𝑛+1 = 𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 ,
𝑥𝑛+1 = 𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 ,
where 0 < 𝛼𝑛 ≤ 𝑎0 < 1 such that lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 =
∞, lim𝑛 → ∞ [𝛾𝑛,𝑖 /𝛼𝑛 ] = 0, and 0 < 𝛿 ≤ 𝛽𝑛,𝑖 ≤ 1 − 𝑘 − 𝛿 < 1
satisfying 𝛽𝑛,0 + 𝛽𝑛,1 + ⋅ ⋅ ⋅ + 𝛽𝑛,𝑁 = 1 for each 𝑛 ≥ 1 and 𝑘 :=
max{𝑘𝑖 : 𝑖 = 1, 2, . . . , 𝑁}. Then, {𝑥𝑛 } converges strongly to an
element of 𝐹.
where 0 < 𝛼𝑛 ≤ 𝑎0 < 1 such that lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 =
∞, lim𝑛 → ∞ [𝛾𝑛,𝑖 /𝛼𝑛 ] = 0, and 0 < 𝛿 ≤ 𝛽𝑛,𝑖 ≤ 1 − 𝛿 < 1
satisfying 𝛽𝑛,0 + 𝛽𝑛,1 + ⋅ ⋅ ⋅ + 𝛽𝑛,𝑁 = 1 for each 𝑛 ≥ 1. Then, {𝑥𝑛 }
converges strongly to an element of 𝐹.
Proof. Since every asymptotically 𝑘𝑖 -strict pseudocontractive
mapping is uniformly continuous and asymptotically 𝑘𝑖 -strict
pseudocontractive mapping in the intermediate sense with
𝑐𝑛,𝑖 ≡ 0, for all 𝑛 ≥ 1 and each 𝑖 = 1, 2, . . . , 𝑁, the conclusion
follows from Theorem 9.
If in Theorem 9, we assume that each 𝑇𝑖 is asymptotically
nonexpansive in the intermediate sense we obtain the following corollary.
Corollary 12. Let 𝐶 be a nonempty, closed, and convex
subset of a real Hilbert space 𝐻. Let 𝑇𝑖 : 𝐶 → 𝐶 be
uniformly continuous asymptotically nonexpansive mappings
in the intermediate sense with sequences {𝛾𝑛,𝑖 } and {𝑐𝑛,𝑖 }, for
𝑖 = 1, 2, . . ., 𝑁. Assume that 𝐹 := ⋂𝑁
𝑖=1 𝐹(𝑇𝑖 ) is nonempty.
Let {𝑥𝑛 } be a sequence generated by
𝑥0 = 𝑤 ∈ 𝐶 𝑐ℎ𝑜𝑠𝑒𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,
Proof. Since every asymptotically nonexpansive mapping is
uniformly continuous and asymptotically 𝑘𝑖 -strict pseudocontractive mapping with 𝑘𝑖 = 0 and 𝑐𝑛,𝑖 = 0, for all 𝑛 ≥ 1 and
𝑖 = 1, 2, . . ., 𝑁, the conclusion follows from Theorem 9.
Remark 14. Our results extend and unify most of the results
that have been proved for this important class of nonlinear
mappings. In particular, Theorem 9 extends Theorem SXY1,
SXY2, HC, and Theorem ZRS in the sense that our convergence is either strong, does not require computation of closed
convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1, or does not require the
assumption that interior of set of fixed points is nonempty.
Remark 15. We also remark that Corollary 11 is more general
than Theorem 3.1 of Kim and Xu [3] and Corollary 13 is
more general than Theorem 2.2 of Kim and Xu [11] in the
sense that our convergence is either strong, does not require
computation of closed convex sets 𝐶𝑛 and 𝑄𝑛 for each 𝑛 ≥ 1,
or does not require the assumption that interior of set of fixed
points is nonempty.
𝑁
𝑦𝑛 = 𝛽𝑛,0 𝑥𝑛 + ∑𝛽𝑛,𝑖 𝑇𝑖𝑛 𝑥𝑛 ,
(34)
𝑖=1
𝑥𝑛+1 = 𝛼𝑛 𝑤 + (1 − 𝛼𝑛 ) 𝑦𝑛 ,
where 0 < 𝛼𝑛 ≤ 𝑎0 < 1 such that lim𝑛 → ∞ 𝛼𝑛 = 0, ∑∞
𝑛=1 𝛼𝑛 =
∞, lim𝑛 → ∞ [𝛾𝑛,𝑖 /𝛼𝑛 ] = 0 = lim𝑛 → ∞ [𝑐𝑛,𝑖 /𝛼𝑛 ], and 0 < 𝛿 ≤
𝛽𝑛,𝑖 ≤ 1 − 𝛿 < 1 satisfying 𝛽𝑛,0 + 𝛽𝑛,1 + ⋅ ⋅ ⋅ + 𝛽𝑛,𝑁 = 1 for each
𝑛 ≥ 1. Then, {𝑥𝑛 } converges strongly to an element of 𝐹.
Proof. Since every asymptotically nonexpansive mapping in
the intermediate sense is asymptotically 𝑘𝑖 -strict pseudocontractive mapping in the intermediate sense with 𝑘𝑖 = 0,
for each 𝑖 = 1, 2, . . ., 𝑁, the conclusion follows from
Theorem 9.
If in Theorem 9, we assume that each 𝑇𝑖 is asymptotically
nonexpansive, we obtain the following corollary.
Corollary 13. Let 𝐶 be a nonempty, closed, and convex
subset of a real Hilbert space 𝐻. Let 𝑇𝑖 : 𝐶 → 𝐶 be
asymptotically nonexpansive mappings with sequences {𝛾𝑛,𝑖 },
Acknowledgment
The research of N. Shahzad was partially supported by
the Deanship of Scientific Research (DSR), King Abdulaziz
University, Jeddah, Saudi Arabia.
References
[1] Q. H. Liu, “Convergence theorems of the sequence of iterates
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[3] T.-H. Kim and H.-K. Xu, “Convergence of the modified Mann’s
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