Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 817392, 6 pages
http://dx.doi.org/10.1155/2013/817392
Research Article
The Problem of Image Recovery by the Metric Projections in
Banach Spaces
Yasunori Kimura1 and Kazuhide Nakajo2
1
2
Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan
Sundai Preparatory School, Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8313, Japan
Correspondence should be addressed to Yasunori Kimura; yasunori@is.sci.toho-u.ac.jp
Received 24 September 2012; Accepted 28 December 2012
Academic Editor: Jaan Janno
Copyright © 2013 Y. Kimura and K. Nakajo. 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 consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty
closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our
convergence theorems extend the results proved by Bregman and Crombez.
1. Introduction
Let 𝐶1 , 𝐶2 , . . . , 𝐶𝑟 be nonempty closed convex subsets of
a real Hilbert space 𝐻 such that ⋂𝑟𝑖=1 𝐶𝑖 ≠ 0. Then, the
problem of image recovery may be stated as follows: the
original unknown image 𝑧 is known a priori to belong to the
intersection of {𝐶𝑖 }𝑟𝑖=1 ; given only the metric projections 𝑃𝐶𝑖 of
𝐻 onto 𝐶𝑖 for 𝑖 = 1, 2, . . . , 𝑟, recover 𝑧 by an iterative scheme.
Such a problem is connected with the convex feasibility
problem and has been investigated by a large number of
researchers.
Bregman [1] considered a sequence {𝑥𝑛 } generated by
cyclic projections, that is, 𝑥0 = 𝑥 ∈ 𝐻, 𝑥1 = 𝑃𝐶1 𝑥, 𝑥2 =
𝑃𝐶2 𝑥1 , 𝑥3 = 𝑃𝐶3 𝑥2 , . . . , 𝑥𝑟 = 𝑃𝐶𝑟 𝑥𝑟−1 , 𝑥𝑟+1 = 𝑃𝐶1 𝑥𝑟 , 𝑥𝑟+2 =
𝑃𝐶2 𝑥𝑟+1 , . . .. It was proved that {𝑥𝑛 } converges weakly to an
element of ⋂𝑟𝑖=1 𝐶𝑖 for an arbitrary initial point 𝑥 ∈ 𝐻.
Crombez [2] proposed a sequence {𝑦𝑛 } generated by 𝑦0 =
𝑦 ∈ 𝐻, 𝑦𝑛+1 = 𝛼0 𝑦𝑛 + ∑𝑟𝑖=1 𝛼𝑖 (𝑦𝑛 + 𝜆 𝑖 (𝑃𝐶𝑖 𝑦𝑛 − 𝑦𝑛 )) for
𝑛 = 0, 1, 2, . . ., where 0 < 𝛼𝑖 < 1 for all 𝑖 = 0, 1, 2, . . . , 𝑟 with
∑𝑟𝑖=0 𝛼𝑖 = 1 and 0 < 𝜆 𝑖 < 2 for every 𝑖 = 1, 2, . . . , 𝑟. It was
proved that {𝑦𝑛 } converges weakly to an element of ⋂𝑟𝑖=1 𝐶𝑖
for an arbitrary initial point 𝑦 ∈ 𝐻.
Later, Kitahara and Takahashi [3] and Takahashi and
Tamura [4] dealt with the problem of image recovery by convex combinations of nonexpansive retractions in a uniformly
convex Banach space 𝐸. Alber [5] took up the problem of
image recovery by the products of generalized projections
in a uniformly convex and uniformly smooth Banach space
𝐸 whose duality mapping is weakly sequentially continuous
(see also [6, 7]).
On the other hand, using the hybrid projection method
proposed by Haugazeau [8] (see also [9–11] and references
therein) and the shrinking projection method proposed by
Takahashi et al. [12] (see also [13]), Nakajo et al. [14] and
Kimura et al. [15] considered this problem by the metric
projections and proved convergence of the iterative sequence
to a common point of countable nonempty closed convex
subsets in a uniformly convex and smooth Banach space 𝐸
and in a strictly convex, smooth, and reflexive Banach space
𝐸 having the Kadec-Klee property, respectively. Kohsaka
and Takahashi [16] took up this problem by the generalized
projections and obtained the strong convergence to a common point of a countable family of nonempty closed convex
subsets in a uniformly convex Banach space whose norm is
uniformly Gâteaux differentiable (see also [17, 18]). Although
these results guarantee the strong convergence, they need to
use metric or generalized projections onto different subsets
for each step, which are not given in advance.
In this paper, we consider this problem by the metric
projections, which are one of the most familiar projections
to deal with. The advantage of our results is that we use
projections onto the given family of subsets only, to generate
2
Abstract and Applied Analysis
the iterative scheme. Our convergence theorems extend the
results of [1, 2] to a Banach space 𝐸, and they deduce the
weak convergence to a common point of a countable family
of nonempty closed convex subsets of 𝐸.
There are a number of results dealing with the image
recovery problem from the aspects of engineering using
nonlinear functional analysis (see, e.g., [19]). Comparing
with these researches, we may say that our approach is
more abstract and theoretical; we adopt a general Banach
space with several conditions for an underlying space, and
therefore, the technique of the proofs can be applied to
various mathematical results related to nonlinear problems
defined on Banach spaces.
for every 𝑦∗ ∈ 𝐸∗ , where 𝑞 ∈ R satisfies 1/𝑝 + 1/𝑞 = 1. For
𝑝 > 1, the duality mapping 𝐽𝑝 of a smooth Banach space 𝐸 is
said to be weakly sequentially continuous if 𝑥𝑛 ⇀ 𝑥 implies
∗
that 𝐽𝑝 𝑥𝑛 ⇀ 𝐽𝑝 𝑥 (see [20, 21] for details). The following is
proved by Xu [22] (see also [23]).
Theorem 1 (Xu [22]). Let 𝐸 be a smooth Banach space and
𝑝 > 1. Then, 𝐸 is 𝑝-uniformly convex if and only if there exists
a constant 𝑐 > 0 such that ‖𝑥 + 𝑦‖𝑝 ≥ ‖𝑥‖𝑝 +𝑝⟨𝑦, 𝐽𝑝 𝑥⟩+𝑐‖𝑦‖𝑝
holds for every 𝑥, 𝑦 ∈ 𝐸.
Remark 2. For a 𝑝-uniformly convex and smooth Banach
space 𝐸, we have that the constant 𝑐 in the theorem above
satisfies 𝑐 ≤ 1. Let
2. Preliminaries
Throughout this paper, let N be the set of all positive integers,
R the set of all real numbers, 𝐸 a real Banach space with norm
‖⋅‖, and 𝐸∗ the dual of 𝐸. For 𝑥 ∈ 𝐸 and 𝑥∗ ∈ 𝐸∗ , we denote by
⟨𝑥, 𝑥∗ ⟩ the value of 𝑥∗ at 𝑥. We write 𝑥𝑛 → 𝑥 to indicate that
a sequence {𝑥𝑛 } converges strongly to 𝑥. Similarly, 𝑥𝑛 ⇀ 𝑥
∗
and 𝑥𝑛 ⇀ 𝑥 will symbolize weak and weak∗ convergence,
respectively. We define the modulus 𝛿𝐸 of convexity of 𝐸 as
follows: 𝛿𝐸 is a function of [0, 2] into [0, 1] such that
󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩
󵄩 : 𝑥, 𝑦 ∈ 𝐸, ‖𝑥‖ = 1,
𝛿𝐸 (𝜖) = inf {1 − 󵄩
2
(1)
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑦󵄩󵄩 = 1, 󵄩󵄩𝑥 − 𝑦󵄩󵄩 ≥ 𝜖}
for every 𝜖 ∈ [0, 2]. 𝐸 is called uniformly convex if 𝛿𝐸 (𝜖) > 0
for each 𝜖 > 0. Let 𝑝 > 1. 𝐸 is said to be 𝑝-uniformly convex
if there exists a constant 𝑐 > 0 such that 𝛿𝐸 (𝜖) ≥ 𝑐𝜖𝑝 for every
𝜖 ∈ [0, 2]. It is obvious that a 𝑝-uniformly convex Banach
space is uniformly convex. 𝐸 is said to be strictly convex if
‖𝑥 + 𝑦‖/2 < 1 for all 𝑥, 𝑦 ∈ 𝐸 with ‖𝑥‖ = ‖𝑦‖ = 1 and 𝑥 ≠ 𝑦.
We know that a uniformly convex Banach space is strictly
convex and reflexive. For every 𝑝 > 1, the (generalized)
∗
duality mapping 𝐽𝑝 : 𝐸 → 2𝐸 of 𝐸 is defined by
󵄩 󵄩
𝐽𝑝 𝑥 = {𝑦∗ ∈ 𝐸∗ : ⟨𝑥, 𝑦∗ ⟩ = ‖𝑥‖𝑝 , 󵄩󵄩󵄩𝑦∗ 󵄩󵄩󵄩 = ‖𝑥‖𝑝−1 }
(2)
for all 𝑥 ∈ 𝐸. When 𝑝 = 2, 𝐽2 is called the normalized duality
mapping. We have that for 𝑝, 𝑞 > 1, ‖𝑥‖𝑝 𝐽𝑞 𝑥 = ‖𝑥‖𝑞 𝐽𝑝 𝑥 for
all 𝑥 ∈ 𝐸. 𝐸 is said to be smooth if the limit
󵄩
󵄩󵄩
󵄩𝑥 + 𝑡𝑦󵄩󵄩󵄩 − ‖𝑥‖
(3)
lim 󵄩
𝑡→0
𝑡
exists for every 𝑥, 𝑦 ∈ 𝐸 with ‖𝑥‖ = ‖𝑦‖ = 1. We know that
the duality mapping 𝐽𝑝 of 𝐸 is single valued for each 𝑝 > 1 if 𝐸
is smooth. It is also known that if 𝐸 is strictly convex, then the
duality mapping 𝐽𝑝 of 𝐸 is one to one in the sense that 𝑥 ≠ 𝑦
implies that 𝐽𝑝 𝑥 ∩ 𝐽𝑝 𝑦 = 0 for all 𝑝 > 1. If 𝐸 is reflexive, then
𝐽𝑝 is surjective, and 𝐽𝑝−1 is identical to the duality mapping
𝐽𝑞∗ : 𝐸∗ → 2𝐸 defined by
󵄩 󵄩𝑞
󵄩 󵄩𝑞−1
𝐽𝑞∗ 𝑦∗ = {𝑥 ∈ 𝐸 : ⟨𝑥, 𝑦∗ ⟩ = 󵄩󵄩󵄩𝑦∗ 󵄩󵄩󵄩 , ‖𝑥‖ = 󵄩󵄩󵄩𝑦∗ 󵄩󵄩󵄩 }
(4)
󵄩𝑝
󵄩
𝑐0 = sup {𝑐 > 0 : 󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≥ ‖𝑥‖𝑝 + 𝑝 ⟨𝑦, 𝐽𝑝 𝑥⟩
󵄩 󵄩𝑝
+ 𝑐󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ∀𝑥, 𝑦 ∈ 𝐸} .
(5)
Then, there exists a positive real sequence {𝑐𝑛 } such that
lim𝑛 → ∞ 𝑐𝑛 = 𝑐0 and ‖𝑥 + 𝑦‖𝑝 ≥ ‖𝑥‖𝑝 + 𝑝⟨𝑦, 𝐽𝑝 𝑥⟩ + 𝑐𝑛 ‖𝑦‖𝑝
for all 𝑥, 𝑦 ∈ 𝐸 and 𝑛 ∈ N. So, we get ‖𝑥 + 𝑦‖𝑝 ≥ ‖𝑥‖𝑝 +
𝑝⟨𝑦, 𝐽𝑝 𝑥⟩ + 𝑐0 ‖𝑦‖𝑝 for every 𝑥, 𝑦 ∈ 𝐸. Therefore, 𝑐0 is the
maximum of constants. In the case of Hilbert spaces, the
normalized duality mapping 𝐽2 is the identity mapping and
𝑐0 = 1.
Let 𝐸 be a smooth Banach space and 𝑝 > 1. The function
𝜙𝑝 : 𝐸 × 𝐸 → R is defined by
󵄩 󵄩𝑝
𝜙𝑝 (𝑦, 𝑥) = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 𝑝 ⟨𝑦, 𝐽𝑝 𝑥⟩ + (𝑝 − 1) ‖𝑥‖𝑝
(6)
for every 𝑥, 𝑦 ∈ 𝐸. We have 𝜙𝑝 (𝑥, 𝑦) ≥ 0 for all 𝑥, 𝑦 ∈ 𝐸
and 𝜙𝑝 (𝑧, 𝑥) + 𝜙𝑝 (𝑥, 𝑦) = 𝜙𝑝 (𝑧, 𝑦) + 𝑝⟨𝑥 − 𝑧, 𝐽𝑝 𝑥 − 𝐽𝑝 𝑦⟩ for
every 𝑥, 𝑦, 𝑧 ∈ 𝐸. It is known that if 𝐸 is strictly convex and
smooth, then, for 𝑥, 𝑦 ∈ 𝐸, 𝜙𝑝 (𝑦, 𝑥) = 𝜙𝑝 (𝑥, 𝑦) = 0 if and
only if 𝑥 = 𝑦. Indeed, suppose that 𝜙𝑝 (𝑦, 𝑥) = 𝜙𝑝 (𝑥, 𝑦) = 0.
Then, since
0 = 𝜙𝑝 (𝑦, 𝑥) + 𝜙𝑝 (𝑥, 𝑦)
󵄩 󵄩𝑝
= 𝑝 (‖𝑥‖𝑝 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − ⟨𝑥, 𝐽𝑝 𝑦⟩ − ⟨𝑦, 𝐽𝑝 𝑥⟩)
󵄩 󵄩𝑝
󵄩 󵄩𝑝−1 󵄩 󵄩
≥ 𝑝 (‖𝑥‖𝑝 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − ‖𝑥‖ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ‖𝑥‖𝑝−1 )
(7)
󵄩 󵄩𝑝−1
󵄩 󵄩
= 𝑝 (‖𝑥‖ − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩) (‖𝑥‖𝑝−1 − 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ) ≥ 0,
we have ‖𝑥‖ = ‖𝑦‖. It follows that ⟨𝑦, 𝐽𝑝 𝑥⟩ = 𝑝−1 (‖𝑦‖𝑝 + (𝑝 −
1)‖𝑥‖𝑝 − 𝜙𝑝 (𝑦, 𝑥)) = ‖𝑦‖𝑝 and ‖𝐽𝑝 𝑥‖ = ‖𝑥‖𝑝−1 = ‖𝑦‖𝑝−1 ,
which implies that 𝐽𝑝 𝑦 = 𝐽𝑝 𝑥. Since 𝐽𝑝 is one to one, we
have 𝑥 = 𝑦 (see also [17]). We have the following result from
Theorem 1.
Abstract and Applied Analysis
3
Lemma 3. Let 𝑝 > 1 and 𝐸 be a 𝑝-uniformly convex and
smooth Banach space. Then, for each 𝑥, 𝑦 ∈ 𝐸,
󵄩𝑝
󵄩
𝜙𝑝 (𝑥, 𝑦) ≥ 𝑐0 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
(8)
= −𝜙𝑝 (𝑦𝑛,𝑘 , 𝑥𝑛 ) − 𝑝𝜆 𝑛,𝑘 ⟨𝑦𝑛,𝑘 − 𝑧, 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
Proof. Let 𝑥, 𝑦 ∈ 𝐸. By Theorem 1, we have
(9)
(12)
󵄩󵄩𝑝
󵄩𝑝
󵄩
≥ 𝑐0 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
= −𝜙𝑝 (𝑦𝑛,𝑘 , 𝑥𝑛 ) − 𝑝𝜆 𝑛,𝑘 ⟨𝑦𝑛,𝑘 − 𝑥𝑛 , 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
− 𝑝𝜆 𝑛,𝑘 ⟨𝑥𝑛 − 𝑧, 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
where 𝑐0 is maximum in Remark 2. Hence, we get
󵄩
𝜙𝑝 (𝑥, 𝑦) = ‖𝑥‖𝑝 − 󵄩󵄩󵄩𝑦󵄩󵄩 − 𝑝 ⟨𝑥 − 𝑦, 𝐽𝑝 𝑦⟩
𝜙𝑝 (𝑧, 𝑦𝑛,𝑘 ) − 𝜙𝑝 (𝑧, 𝑥𝑛 )
= −𝜙𝑝 (𝑦𝑛,𝑘 , 𝑥𝑛 ) + 𝑝 ⟨𝑦𝑛,𝑘 − 𝑧, 𝐽𝑝 𝑦𝑛,𝑘 − 𝐽𝑝 𝑥𝑛 ⟩
holds, where 𝑐0 is maximum in Remark 2.
󵄩𝑝
󵄩
󵄩 󵄩𝑝
‖𝑥‖𝑝 ≥ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 + 𝑝 ⟨𝑥 − 𝑦, 𝐽𝑝 𝑦⟩ + 𝑐0 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
Proof. Let 𝑦𝑛,𝑘 = 𝐽𝑞∗ (𝐽𝑝 𝑥𝑛 − 𝜆 𝑛,𝑘 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )) for 𝑛 ∈ N and
𝑘 = 1, 2, . . . , 𝑛. Then, for 𝑧 ∈ ⋂𝑛∈N 𝐶𝑛 , we obtain
(10)
for all 𝑛 ∈ N and 𝑘 = 1, 2, . . . , 𝑛. Using Remark 5 with that
𝑧 ∈ 𝐶𝑘 , we get
⟨𝑥𝑛 − 𝑧, 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
= ⟨𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 , 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
which is the desired result.
Let 𝐶 be a nonempty closed convex subset of a strictly
convex and reflexive Banach space 𝐸, and let 𝑥 ∈ 𝐸. Then,
there exists a unique element 𝑥0 ∈ 𝐶 such that ‖𝑥0 − 𝑥‖ =
inf 𝑦∈𝐶‖𝑦 − 𝑥‖. Putting 𝑥0 = 𝑃𝐶𝑥, we call 𝑃𝐶 the metric
projection onto 𝐶 (see [24]). We have the following result [25,
p. 196] for the metric projection.
Lemma 4. Let 𝐶 be a nonempty closed convex subset of a
strictly convex, reflexive, and smooth Banach space 𝐸, and let
𝑥 ∈ 𝐸. Then, 𝑦 = 𝑃𝐶𝑥 if and only if ⟨𝑦 − 𝑧, 𝐽2 (𝑥 − 𝑦)⟩ ≥ 0 for
all 𝑧 ∈ 𝐶, where 𝑃𝐶 is the metric projection onto 𝐶.
Remark 5. For 𝑝 > 1, it holds that ‖𝑥‖𝐽𝑝 𝑥 = ‖𝑥‖𝑝−1 𝐽2 𝑥
for every 𝑥 ∈ 𝐸. Therefore, under the same assumption as
Lemma 4, we have that 𝑦 = 𝑃𝐶𝑥 if and only if ⟨𝑦 − 𝑧, 𝐽𝑝 (𝑥 −
𝑦)⟩ ≥ 0 for all 𝑧 ∈ 𝐶.
3. Main Results
+ ⟨𝑃𝐶𝑘 𝑥𝑛 − 𝑧, 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
󵄩𝑝
󵄩
≥ 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
for every 𝑛 ∈ N and 𝑘 = 1, 2 . . . , 𝑛. Thus, by Lemma 3 we have
𝜙𝑝 (𝑧, 𝑦𝑛,𝑘 ) − 𝜙𝑝 (𝑧, 𝑥𝑛 )
󵄩
󵄩𝑝
≤ −𝑐0 󵄩󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩
− 𝑝𝜆 𝑛,𝑘 ⟨𝑦𝑛,𝑘 − 𝑥𝑛 , 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )⟩
󵄩
󵄩𝑝
− 𝑝𝜆 𝑛,𝑘 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩𝑝−1
󵄩
󵄩𝑝
󵄩
󵄩󵄩
≤ −𝑐0 󵄩󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩 + 𝑝𝜆 𝑛,𝑘 󵄩󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
󵄩
󵄩𝑝
− 𝑝𝜆 𝑛,𝑘 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
Theorem 6. Let 𝑝, 𝑞 > 1 be such that 1/𝑝 + 1/𝑞 = 1. Let
{𝐶𝑛 }𝑛∈N be a family of nonempty closed convex subsets of a 𝑝uniformly convex and smooth Banach space 𝐸 whose duality
mapping 𝐽𝑝 is weakly sequentially continuous. Suppose that
⋂𝑛∈N 𝐶𝑛 ≠ 0. Let 𝜆 𝑛,𝑘 ∈]0, (1+1/(𝑝−1))𝑝−1 𝑐0 [ and 𝛼𝑛,𝑘 ∈ [0, 1]
for all 𝑛 ∈ N and 𝑘 = 1, 2, . . . , 𝑛 with ∑𝑛𝑘=1 𝛼𝑛,𝑘 = 1 for
every 𝑛 ∈ N, where 𝑐0 is maximum in Remark 2. Let {𝑥𝑛 } be
a sequence generated by 𝑥1 = 𝑥 ∈ 𝐸 and
𝑥𝑛+1 =
𝑛
( ∑ 𝛼𝑛,𝑘 (𝐽𝑝 𝑥𝑛 − 𝜆 𝑛,𝑘 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 )))
(14)
for each 𝑛 ∈ N and 𝑘 = 1, 2, . . . , 𝑛. Since it holds that
Firstly, we consider the iteration of Crombez’s type and get
the following result.
𝐽𝑞∗
(13)
(11)
𝑠𝑡 ≤
(15)
for 𝑠, 𝑡 ≥ 0, 𝑝, 𝑞 > 1 with 1/𝑝 + 1/𝑞 = 1, and 𝛽 > 0, we have
󵄩𝑝−1
󵄩󵄩
󵄩󵄩
󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
≤
1 󵄩󵄩
󵄩𝑝
󵄩𝑦 − 𝑥𝑛 󵄩󵄩󵄩
𝛽𝑘 𝑝 󵄩 𝑛,𝑘
1/(𝑝−1) 𝑝
+ 𝛽𝑘
(16)
− 1 󵄩󵄩
󵄩𝑝
󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩
󵄩
󵄩
𝑝
for every 𝑘 ∈ N, 𝛽𝑘 > 0 and 𝑛 ≥ 𝑘. Therefore, it follows that
𝜙𝑝 (𝑧, 𝑦𝑛,𝑘 ) − 𝜙𝑝 (𝑧, 𝑥𝑛 )
𝑘=1
for every 𝑛 ∈ N. If 0 < lim inf 𝑛 → ∞ 𝜆 𝑛,𝑘 ≤ lim sup𝑛 → ∞ 𝜆 𝑛,𝑘 <
(1 + 1/(𝑝 − 1))𝑝−1 𝑐0 and lim inf 𝑛 → ∞ 𝛼𝑛,𝑘 > 0 for each 𝑘 ∈ N,
then {𝑥𝑛 } converges weakly to a point in ⋂∞
𝑛=1 𝐶𝑛 .
𝑡𝑞
1 𝑠𝑝
+ 𝛽𝑞−1
𝛽𝑝
𝑞
≤(
𝜆 𝑛,𝑘
󵄩𝑝
󵄩
− 𝑐0 ) 󵄩󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩
𝛽𝑘
1/(𝑝−1)
+ 𝜆 𝑛,𝑘 ((𝑝 − 1) 𝛽𝑘
󵄩𝑝
󵄩
− 𝑝) 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
(17)
4
Abstract and Applied Analysis
for every 𝑛 ∈ N, 𝑘 = 1, 2, . . . , 𝑛, and 𝛽𝑘 > 0. Since
𝑢2 ∈ 𝐶𝑘 for every 𝑘 ∈ N. Let lim𝑛 → ∞ 𝜙𝑝 (𝑢1 , 𝑥𝑛 ) = 𝜇1 and
lim𝑛 → ∞ 𝜙𝑝 (𝑢2 , 𝑥𝑛 ) = 𝜇2 . Since
𝑛
𝜙𝑝 (𝑧, 𝑥𝑛+1 ) = ‖𝑧‖𝑝 − 𝑝 ⟨𝑧, ∑ 𝛼𝑛,𝑘 𝐽𝑝 𝑦𝑛,𝑘 ⟩
𝜇1 − 𝜇2 = lim (𝜙𝑝 (𝑢1 , 𝑥𝑛𝑖 ) − 𝜙𝑝 (𝑢2 , 𝑥𝑛𝑖 ))
𝑖→∞
𝑘=1
󵄩 󵄩𝑝 󵄩 󵄩𝑝
= 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩 + 𝑝 lim ⟨𝑢2 − 𝑢1 , 𝐽𝑝 𝑥𝑛𝑖 ⟩
󵄩󵄩𝑝/(𝑝−1)
󵄩󵄩 𝑛
󵄩󵄩
󵄩󵄩
󵄩
+ (𝑝 − 1) 󵄩󵄩 ∑ 𝛼𝑛,𝑘 𝐽𝑝 𝑦𝑛,𝑘 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩𝑘=1
󵄩
󵄩
𝑖→∞
and 𝐽𝑝 is weakly sequentially continuous, we have
𝑛
≤ ‖𝑧‖𝑝 − 𝑝 ∑ 𝛼𝑛,𝑘 ⟨𝑧, 𝐽𝑝 𝑦𝑛,𝑘 ⟩
(18)
𝑘=1
󵄩 󵄩𝑝 󵄩 󵄩𝑝
𝜇1 − 𝜇2 = 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩 + 𝑝 ⟨𝑢2 − 𝑢1 , 𝐽𝑝 𝑢1 ⟩
= −𝜙𝑝 (𝑢2 , 𝑢1 ) .
𝑛
󵄩 󵄩𝑝
+ (𝑝 − 1) ∑ 𝛼𝑛,𝑘 󵄩󵄩󵄩𝑦𝑛,𝑘 󵄩󵄩󵄩
𝑛
= ∑ 𝛼𝑛,𝑘 𝜙𝑝 (𝑧, 𝑦𝑛,𝑘 )
𝑘=1
Using the idea of [9, p. 256], we also have the following
result by the iteration of Bregman’s type.
for every 𝑛 ∈ N, we have
𝜙𝑝 (𝑧, 𝑥𝑛+1 ) − 𝜙𝑝 (𝑧, 𝑥𝑛 )
𝑛
𝑘=1
𝜆 𝑛,𝑘
󵄩𝑝
󵄩
− 𝑐0 ) 󵄩󵄩󵄩𝑦𝑛,𝑘 − 𝑥𝑛 󵄩󵄩󵄩
𝛽𝑘
𝑛
1/(𝑝−1)
+ ∑ 𝛼𝑛,𝑘 𝜆 𝑛,𝑘 ((𝑝 − 1) 𝛽𝑘
𝑘=1
󵄩𝑝
󵄩
− 𝑝) 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩
(19)
for all 𝑛 ∈ N and 𝛽1 , 𝛽2 , . . . , 𝛽𝑛 > 0. Since 𝜆 𝑛,𝑘 ∈]0, (1 + 1/(𝑝 −
1))𝑝−1 𝑐0 [, 𝛼𝑛,𝑘 ∈ [0, 1] for all 𝑛 ∈ N and 𝑘 = 1, 2, . . . , 𝑛,
𝑝−1
0 < lim inf 𝜆 𝑛,𝑘 ≤ lim sup 𝜆 𝑛,𝑘 < (1 +
𝑛→∞
𝑛→∞
1
)
(𝑝 − 1)
𝑐0 ,
(20)
𝑛→∞
for each 𝑘 ∈ N, we can choose 𝛽𝑘 > 0 for every 𝑘 ∈ N such
1/(𝑝−1)
− 𝑝) ≤ 0
that 𝛼𝑛,𝑘 (𝜆 𝑛,𝑘 /𝛽𝑘 − 𝑐0 ) ≤ 0, 𝛼𝑛,𝑘 𝜆 𝑛,𝑘 ((𝑝 − 1)𝛽𝑘
for all 𝑛 ≥ 𝑘 and
𝑛→∞
𝜆 𝑛,𝑘
− 𝑐0 ) < 0,
𝛽𝑘
lim sup 𝛼𝑛,𝑘 𝜆 𝑛,𝑘 ((𝑝 −
𝑛→∞
1/(𝑝−1)
1) 𝛽𝑘
󵄩󵄩
𝑥𝑛+1 = 𝐽𝑞∗ (𝐽𝑝 𝑥𝑛 − 𝜆 𝑛 𝐽𝑝 (𝑥𝑛 − 𝑃𝐶𝑖(𝑛) 𝑥𝑛 ))
(25)
for every 𝑛 ∈ N, where the index mapping 𝑖 : N → 𝐼 satisfies
that, for every 𝑗 ∈ 𝐼, there exists 𝑀𝑗 ∈ N such that 𝑗 ∈
{𝑖(𝑛), . . . , 𝑖(𝑛 + 𝑀𝑗 − 1)} for each 𝑛 ∈ N. If 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤
lim sup𝑛 → ∞ 𝜆 𝑛 < (1 + 1/(𝑝 − 1))𝑝−1 𝑐0 , then, {𝑥𝑛 } converges
weakly to a point in ⋂𝑗∈𝐼 𝐶𝑗 .
(21)
− 𝑝) < 0
󵄩
󵄩
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = lim 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 󵄩󵄩󵄩󵄩 = 0
𝑛→∞
𝜙𝑝 (𝑧, 𝑥𝑛+1 ) − 𝜙𝑝 (𝑧, 𝑥𝑛 )
≤(
𝜆𝑛
󵄩𝑝
󵄩
− 𝑐0 ) 󵄩󵄩󵄩𝑥𝑛+1 − 𝑥𝑛 󵄩󵄩󵄩
𝛽
(26)
󵄩𝑝
󵄩
+ 𝜆 𝑛 ((𝑝 − 1) 𝛽1/(𝑝−1) − 𝑝) 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑖(𝑛) 𝑥𝑛 󵄩󵄩󵄩󵄩
for each 𝑘 ∈ N. Hence, there exists lim𝑛 → ∞ 𝜙𝑝 (𝑧, 𝑥𝑛 ) for every
𝑧 ∈ ⋂𝑛∈N 𝐶𝑛 and
lim 󵄩𝑦
𝑛 → ∞ 󵄩 𝑛,𝑘
Theorem 7. Let 𝑝, 𝑞 > 1 be such that 1/𝑝 + 1/𝑞 = 1. Let 𝐼 be
a countable set and {𝐶𝑗 }𝑗∈𝐼 a family of nonempty closed convex
subsets of a 𝑝-uniformly convex and smooth Banach space 𝐸
whose duality mapping 𝐽𝑝 is weakly sequentially continuous.
Suppose that ⋂𝑗∈𝐼 𝐶𝑗 ≠ 0. Let 𝜆 𝑛 ∈]0, (1 + 1/(𝑝 − 1))𝑝−1 𝑐0 [ for
all 𝑛 ∈ N, where 𝑐0 is maximum in Remark 2, and let {𝑥𝑛 } be a
sequence generated by 𝑥1 = 𝑥 ∈ 𝐸 and
Proof. Let 𝑧 ∈ ⋂𝑗∈𝐼 𝐶𝑗 . As in the proof of Theorem 6, we have
lim inf 𝛼𝑛,𝑘 > 0
lim sup 𝛼𝑛,𝑘 (
(24)
Similarly, we obtain 𝜇2 − 𝜇1 = −𝜙𝑝 (𝑢1 , 𝑢2 ). So, we get
𝜙𝑝 (𝑢1 , 𝑢2 ) + 𝜙𝑝 (𝑢2 , 𝑢1 ) = 0, that is, 𝑢1 = 𝑢2 . Therefore, {𝑥𝑛 }
converges weakly to a point in ⋂𝑛∈N 𝐶𝑛 .
𝑘=1
≤ ∑ 𝛼𝑛,𝑘 (
(23)
(22)
for all 𝑘 ∈ N. It follows from Lemma 3 that {𝑥𝑛 } is bounded.
Let {𝑥𝑛𝑖 } and {𝑥𝑚𝑗 } be subsequences of {𝑥𝑛 } such that 𝑥𝑛𝑖 ⇀ 𝑢1
and 𝑥𝑚𝑗 ⇀ 𝑢2 . Then, we get ‖𝑥𝑛𝑖 −𝑃𝐶𝑘 𝑥𝑛𝑖 ‖ → 0 which implies
that 𝑢1 ∈ 𝐶𝑘 for every 𝑘 ∈ N. In the same way, we also have
for all 𝑛 ∈ N and 𝛽 > 0. Since 𝜆 𝑛 ∈]0, (1 + 1/(𝑝 − 1))𝑝−1 𝑐0 [
for all 𝑛 ∈ N and 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤ lim sup𝑛 → ∞ 𝜆 𝑛 <
(1 + 1/(𝑝 − 1))𝑝−1 𝑐0 , we can find that 𝛽 > 0 such that
lim sup (
𝑛→∞
𝜆𝑛
− 𝑐0 ) < 0,
𝛽
(27)
lim sup 𝜆 𝑛 ((𝑝 − 1) 𝛽1/(𝑝−1) − 𝑝) < 0.
𝑛→∞
Then, there exists lim𝑛 → ∞ 𝜙𝑝 (𝑧, 𝑥𝑛 ) for every 𝑧 ∈ ⋂𝑖∈𝐼 𝐶𝑖 and
󵄩󵄩
lim 󵄩𝑥
𝑛 → ∞ 󵄩 𝑛+1
󵄩
󵄩
󵄩
− 𝑥𝑛 󵄩󵄩󵄩 = lim 󵄩󵄩󵄩󵄩𝑥𝑛 − 𝑃𝐶𝑖(𝑛) 𝑥𝑛 󵄩󵄩󵄩󵄩 = 0.
𝑛→∞
(28)
Abstract and Applied Analysis
5
So, we have that {𝑥𝑛 } is bounded from Lemma 3. Let {𝑥𝑛𝑘 } be
a subsequence of {𝑥𝑛 } such that 𝑥𝑛𝑘 ⇀ 𝑢. For fixed 𝑗 ∈ 𝐼,
there exists a strictly increasing sequence {𝑚𝑘 } ⊂ N such that
𝑛𝑘 ≤ 𝑚𝑘 ≤ 𝑛𝑘 + 𝑀𝑗 − 1 and 𝑖(𝑚𝑘 ) = 𝑗 for every 𝑘 ∈ N. It
follows that
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑚𝑘 − 𝑥𝑛𝑘 󵄩󵄩󵄩 ≤
󵄩
󵄩
𝑛𝑘 +𝑀𝑗 −1
∑
𝑙=𝑛𝑘
󵄩󵄩
󵄩
󵄩󵄩𝑥𝑙+1 − 𝑥𝑙 󵄩󵄩󵄩
(29)
𝜆 𝑛 ∈]0, 2[ for all 𝑛 ∈ N, and let {𝑥𝑛 } be a sequence generated
by 𝑥1 = 𝑥 ∈ 𝐻 and
𝑥𝑛+1 = 𝑥𝑛 − 𝜆 𝑛 (𝑥𝑛 − 𝑃𝐶𝑖(𝑛) 𝑥𝑛 )
for every 𝑛 ∈ N, where the index mapping 𝑖 : N → 𝐼 satisfies
that, for every 𝑗 ∈ 𝐼, there exists 𝑀𝑗 ∈ N such that 𝑗 ∈
{𝑖(𝑛), . . . , 𝑖(𝑛 + 𝑀𝑗 − 1)} for each 𝑛 ∈ N. If 0 < lim inf 𝑛 → ∞ 𝜆 𝑛 ≤
lim sup𝑛 → ∞ 𝜆 𝑛 < 2, then, {𝑥𝑛 } converges weakly to a point in
⋂𝑗∈𝐼 𝐶𝑗 .
for all 𝑘 ∈ N which implies that 𝑥𝑚𝑘 ⇀ 𝑢. Since
lim𝑘 → ∞ ‖𝑥𝑚𝑘 − 𝑃𝐶𝑗 𝑥𝑚𝑘 ‖ = 0, 𝑢 ∈ 𝐶𝑗 for every 𝑗 ∈ 𝐼. So, we
get 𝑢 ∈ ⋂𝑗∈𝐼 𝐶𝑗 . As in the proof of Theorem 6, using that 𝐽𝑝
is weakly sequentially continuous, we get that {𝑥𝑛 } converges
weakly to a point in ⋂𝑗∈𝐼 𝐶𝑗 .
Acknowledgment
Suppose that the index set 𝐼 is a finite set {0, 1, 2, . . . , 𝑁 −
1}. For the cyclic iteration, the index mapping 𝑖 is defined
by 𝑖(𝑗) = 𝑗 mod 𝑁 for each 𝑗 ∈ 𝐼. Clearly it satisfies the
assumption in Theorem 7. In the case where the index set 𝐼 is
countably infinite, that is, 𝐼 = N, one of the simplest examples
of 𝑖 : N → N can be defined as follows:
References
1
{
{
{
{
2
{
{
{
{
{3
𝑖 (𝑛) = {
{. . . ,
{
{
{
{
𝑘
{
{
{
{. . . .
(𝑛 = 2𝑚 − 1 for some 𝑚 ∈ N) ,
(𝑛 = 2 (2𝑚 − 1) for some 𝑚 ∈ N) ,
(𝑛 = 4 (2𝑚 − 1) for some 𝑚 ∈ N) ,
(𝑛 = 2𝑘−1 (2𝑚 − 1) for some 𝑚 ∈ N) ,
(30)
Then, the assumption in Theorem 7 is satisfied by letting
𝑀𝑗 = 2𝑗 for each 𝑗 ∈ 𝐼 = N.
4. Deduced Results
Since a real Hilbert space 𝐻 is 2-uniformly convex and
the maximum 𝑐0 in Remark 2 is equal to 1, we get the
following results. At first, we have the following theorem
which generalizes the results of [2] by Theorem 6.
Theorem 8. Let {𝐶𝑛 }𝑛∈N be a family of nonempty closed convex
subsets of 𝐻 such that ⋂𝑛∈N 𝐶𝑛 ≠ 0. Let 𝜆 𝑛,𝑘 ∈]0, 2[ and 𝛼𝑛,𝑘 ∈
[0, 1] for all 𝑛 ∈ N and 𝑘 = 1, 2, . . . , 𝑛 with ∑𝑛𝑘=1 𝛼𝑛,𝑘 = 1 for
every 𝑛 ∈ N. Let {𝑥𝑛 } be a sequence generated by 𝑥1 = 𝑥 ∈ 𝐻
and
𝑛
𝑥𝑛+1 = ∑ 𝛼𝑛,𝑘 (𝑥𝑛 − 𝜆 𝑛,𝑘 (𝑥𝑛 − 𝑃𝐶𝑘 𝑥𝑛 ))
(31)
𝑘=1
for every 𝑛 ∈ N. If it holds that 0 < lim inf 𝑛 → ∞ 𝜆 𝑛,𝑘 ≤
lim sup𝑛 → ∞ 𝜆 𝑛,𝑘 < 2 and lim inf 𝑛 → ∞ 𝛼𝑛,𝑘 > 0 for each 𝑘 ∈ N,
then, {𝑥𝑛 } converges weakly to a point in ⋂∞
𝑛=1 𝐶𝑛 .
Next, we have the following theorem which extends the
result of [1] by Theorem 7.
Theorem 9. Let 𝐼 be a countable set and {𝐶𝑗 }𝑗∈𝐼 a family of
nonempty closed convex subsets of 𝐻 such that ⋂𝑗∈𝐼 𝐶𝑗 ≠ 0. Let
(32)
The first author was supported by the Grant-in-Aid for
Scientific Research no. 22540175 from the Japan Society for
the Promotion of Science.
[1] L. M. Bregman, “The method of successive projection for
finding a common point of convexsets,” Soviet Mathematics, vol.
6, pp. 688–692, 1965.
[2] G. Crombez, “Image recovery by convex combinations of
projections,” Journal of Mathematical Analysis and Applications,
vol. 155, no. 2, pp. 413–419, 1991.
[3] S. Kitahara and W. Takahashi, “Image recovery by convex
combinations of sunny nonexpansive retractions,” Topological
Methods in Nonlinear Analysis, vol. 2, no. 2, pp. 333–342, 1993.
[4] W. Takahashi and T. Tamura, “Limit theorems of operators by
convex combinations of nonexpansive retractions in Banach
spaces,” Journal of Approximation Theory, vol. 91, no. 3, pp. 386–
397, 1997.
[5] Y. I. Alber, “Metric and generalized projections operators
in Banach spaces: properties andapplications,” in Theory and
Applications of Nonlinear Operators of Accretive and Monotone
Type, A. G. Kartasatos, Ed., vol. 178 of Lecture Notes in Pure and
Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY,
USA, 1996.
[6] Y. I. Alber and S. Reich, “An iterative method for solving
a class of nonlinear operator equations in Banach spaces,”
Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54,
1994.
[7] S. Kamimura and W. Takahashi, “Strong convergence of a
proximal-type algorithm in a Banach space,” SIAM Journal on
Optimization, vol. 13, no. 3, pp. 938–945, 2002.
[8] Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, [Thése], These, Universit é de
Paris, Paris, France, 1968.
[9] H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,”
Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264,
2001.
[10] K. Nakajo and W. Takahashi, “Strong convergence theorems
for nonexpansive mappings and nonexpansive semigroups,”
Journal of Mathematical Analysis and Applications, vol. 279, no.
2, pp. 372–379, 2003.
[11] S. Ohsawa and W. Takahashi, “Strong convergence theorems for
resolvents of maximal monotone operators in Banach spaces,”
Archiv der Mathematik, vol. 81, no. 4, pp. 439–445, 2003.
6
[12] W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence
theorems by hybrid methods for families of nonexpansive
mappings in Hilbert spaces,” Journal of Mathematical Analysis
and Applications, vol. 341, no. 1, pp. 276–286, 2008.
[13] Y. Kimura and W. Takahashi, “On a hybrid method for a family
of relatively nonexpansive mappings in a Banach space,” Journal
of Mathematical Analysis and Applications, vol. 357, no. 2, pp.
356–363, 2009.
[14] K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence
theorems by the hybrid method for families of mappings
in Banach spaces,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 71, no. 3-4, pp. 812–818, 2009.
[15] Y. Kimura, K. Nakajo, and W. Takahashi, “Strongly convergent
iterative schemes for a sequence of nonlinear mappings,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 3, pp. 407–416,
2008.
[16] F. Kohsaka and W. Takahashi, “Iterative scheme for finding a
common point of infinitely many convex sets in a Banach space,”
Journal of Nonlinear and Convex Analysis, vol. 5, no. 3, pp. 407–
414, 2004.
[17] S.-y. Matsushita and W. Takahashi, “A strong convergence
theorem for relatively nonexpansive mappings in a Banach
space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–
266, 2005.
[18] S.-y. Matsushita, K. Nakajo, and W. Takahashi, “Strong convergence theorems obtained by a generalized projections hybrid
method for families of mappings in Banach spaces,” Nonlinear
Analysis: Theory, Methods & Applications A, vol. 73, no. 6, pp.
1466–1480, 2010.
[19] X. Liu and L. Huang, “Total bounded variation-based Poissonian images recovery by split Bregman iteration,” Mathematical
Methods in the Applied Sciences, vol. 35, no. 5, pp. 520–529, 2012.
[20] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings
and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
[21] J. Diestel, Geometry of Banach Spaces, Selected Topics, vol. 485 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
[22] H. K. Xu, “Inequalities in Banach spaces with applications,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no.
12, pp. 1127–1138, 1991.
[23] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, vol. 68 of North-Holland Mathematics Studies, NorthHolland Publishing, Amsterdam, The Netherlands, 2nd edition,
1985.
[24] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Pure and Applied
Mathematics, Marcel Dekker, New York, NY, USA, 1984.
[25] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.
Abstract and Applied Analysis
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