Research Article Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 531859, 19 pages
http://dx.doi.org/10.1155/2013/531859
Research Article
Relaxed Viscosity Approximation Methods with
Regularization for Constrained Minimization Problems
Lu-Chuan Ceng,1 Hong-Kun Xu,2,3 and Ching-Feng Wen4
1
Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities,
Shanghai 200234, China
2
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
3
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4
Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Correspondence should be addressed to Ching-Feng Wen; cfwen@kmu.edu.tw
Received 13 January 2013; Accepted 26 March 2013
Academic Editor: Luigi Muglia
Copyright © 2013 Lu-Chuan Ceng et al. 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 a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method
to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex
minimization problem and a suitable equilibrium problem.
1. Introduction
Let đģ be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and
norm ‖ ⋅ ‖, đļ a nonempty closed convex subset of đģ, and 𝑃đļ
the metric projection of đģ, onto đļ. Let 𝑇 : đļ → đļ be selfmapping on đļ. We denote by Fix(𝑇) the set of fixed points of
𝑇 and by R the set of all real numbers. A mapping 𝑇 : đļ → đļ
is called 𝜁-strictly pseudocontractive if there exists a constant
𝜁 ∈ [0, 1) such that
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘‡đ‘Ĩ − 𝑇đ‘Ļķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š + 𝜁ķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝑇) đ‘Ĩ − (đŧ − 𝑇) đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ĩ, đ‘Ļ ∈ đļ.
(1)
In particular, if 𝜁 = 0, then 𝑇 is called a nonexpansive
mapping. A mapping 𝐴 : đļ → đģ is called đ›ŧ-inverse strongly
monotone, if there exists a constant đ›ŧ > 0 such that
ķĩ„Š2
ķĩ„Š
⟨𝐴đ‘Ĩ − 𝐴đ‘Ļ, đ‘Ĩ − đ‘Ļ⟩ ≥ đ›ŧķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ĩ, đ‘Ļ ∈ đļ.
(2)
Let 𝑓 : đļ → R be a convex and a continuous FreĖchet differentiable functional. Consider the minimization problem
(MP) of minimizing 𝑓 over the constraint set đļ
min𝑓 (đ‘Ĩ) ,
đ‘Ĩ∈đļ
(3)
where we assume the existence of minimizers. We denote by
Γ the set of minimizers of (3). The gradient-projection algorithm (GPA) generates a sequence {đ‘Ĩ𝑛 } determined by the
gradient ∇𝑓 and the metric projection 𝑃đļ as follows:
đ‘Ĩ𝑛+1 := 𝑃đļ (đ‘Ĩ𝑛 − 𝜆∇𝑓 (đ‘Ĩ𝑛 )) ,
∀𝑛 ≥ 0,
(4)
∀𝑛 ≥ 0,
(5)
or more generally,
đ‘Ĩ𝑛+1 := 𝑃đļ (đ‘Ĩ𝑛 − 𝜆 𝑛 ∇𝑓 (đ‘Ĩ𝑛 )) ,
where, in both (4) and (5), the initial guess đ‘Ĩ0 is taken from đļ
arbitrarily, the parameters 𝜆 or 𝜆 𝑛 are positive real numbers.
The convergence of algorithms (4) and (5) depends on the
behavior of the gradient ∇𝑓. As a matter of fact, it is known
that if ∇𝑓 is strongly monotone and Lipschitz continuous,
then, for 0 < 𝜆 < 2đ›ŧ/đŋ2 , the operator
𝑆 := 𝑃đļ (đŧ − 𝜆∇𝑓)
(6)
is a contraction. Hence, the sequence {đ‘Ĩ𝑛 } defined by the GPA
(4) converges in norm to the unique solution of (3). More
generally, if the sequence {𝜆 𝑛 } is chosen to satisfy the property
0 < lim inf 𝜆 𝑛 ≤ lim sup𝜆 𝑛 <
𝑛→∞
𝑛→∞
2đ›ŧ
,
đŋ2
(7)
2
Journal of Applied Mathematics
then the sequence {đ‘Ĩ𝑛 } defined by the GPA (5) converges in
norm to the unique minimizer of (3). If the gradient ∇𝑓 is
only assumed to be a Lipschitz continuous, then {đ‘Ĩ𝑛 } can
only be weakly convergent if đģ is infinite dimensional. A
counterexample is given by Xu in [1].
Since the Lipschitz continuity of the gradient ∇𝑓 implies
that it is inverse strongly monotone (ism), it can be expressed
as a proper convex combination of the identity mapping
and a nonexpansive mapping. Consequently, the GPA can be
rewritten as the composite of a projectionand an averaged
mapping which is again an averaged mapping. This shows that
averaged mappings play an important role in the GPA. Very
recently, Xu [1] used averaged mappings to study the convergence analysis of the GPA which is an operator-oriented
approach.
We observe that the regularization, in particular, the
traditional Tikhonov regularization, is usually used to solve
ill-posed optimization problems. Consider the following regularized minimization problem:
đ›ŧ
min𝑓đ›ŧ (đ‘Ĩ) := 𝑓 (đ‘Ĩ) + ‖đ‘Ĩ‖2 ,
(8)
đ‘Ĩ∈đļ
2
where đ›ŧ > 0 is the regularization parameter and again 𝑓 is
convex with an đŋ-Lipschitz continuous gradient ∇𝑓.
The advantage of a regularization method is that it is possible to get strong convergence to the minimum-norm solution of the optimization problem under investigation. The
disadvantage is however its implicity, and hence explicit
iterative methods seem more attractive. See, for example, [1].
Given a mapping 𝐴 : đļ → đģ, the classical variational
inequality problem (VIP) is to find đ‘Ĩ∗ ∈ đļ such that
⟨𝐴đ‘Ĩ∗ , đ‘Ĩ − đ‘Ĩ∗ ⟩ ≥ 0,
∀đ‘Ĩ ∈ đļ.
(9)
The solution set of VIP (9) is denoted by VI(đļ, 𝐴). It is well
known that đ‘Ĩ∗ ∈ VI(đļ, 𝐴) if and only if đ‘Ĩ∗ = 𝑃đļ(đ‘Ĩ∗ − 𝜆𝐴đ‘Ĩ∗ )
for some 𝜆 > 0. The variational inequality was first discussed
by Lions [2] and now is well known. The variational inequality
theory has been studied quite extensively and has emerged
as an important tool in the study of a wide class of obstacle,
unilateral, free, moving, and equilibrium problems arising in
several branches of pure and applied sciences in a unified
and general framework. See, for example, [3–10] and the
references therein.
In this paper, we study the following equilibrium problem
(EP) which is to find đ‘Ĩ∗ ∈ đļ such that
𝐹 (đ‘Ĩ∗ , đ‘Ļ) + ℎ (đ‘Ĩ∗ , đ‘Ļ) ≥ 0,
∀đ‘Ļ ∈ đļ.
(10)
The solution set of EP (10) is denoted by EP(𝐹, ℎ). We will
introduce and consider a relaxed viscosity iterative scheme
with regularization for finding a common element of the
solution set Γ of the minimization problem (3), the solution
set EP(𝐹, ℎ) of the equilibrium problem (10), and the common
fixed point set Fix(𝑇) ∩ (⋂𝑖 Fix(𝑆𝑖 )) of finitely many nonexpansive mappings 𝑆𝑖 : đļ → đļ, 𝑖 = 1, . . . , 𝑁, and a strictly
pseudocontractive mapping 𝑇 in the setting of the infinitedimensional Hilbert space. We will prove that this iterative
scheme converges strongly to a common fixed point of the
mappings 𝑇, 𝑆𝑖 : đļ → đļ, 𝑖 = 1, . . . , 𝑁, which is both a
minimizer of MP (3) and an equilibrium point of EP (10).
2. Preliminaries
Let đģ be a real Hilbert space whose inner product and norm
are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively. Let 𝐾 be a nonempty
closed convex subset of đģ. We write đ‘Ĩ𝑛 ⇀ đ‘Ĩ to indicate
that the sequence {đ‘Ĩ𝑛 } converges weakly to đ‘Ĩ and đ‘Ĩ𝑛 → đ‘Ĩ
to indicate that the sequence {đ‘Ĩ𝑛 } converges strongly to đ‘Ĩ.
Moreover, we use 𝜔𝑤 (đ‘Ĩ𝑛 ) to denote the weak 𝜔-limit set of
the sequence {đ‘Ĩ𝑛 } and 𝜔𝑠 (đ‘Ĩ𝑛 ) to denote the strong 𝜔-limit set
of the sequence {đ‘Ĩ𝑛 }; that is,
𝜔𝑤 (đ‘Ĩ𝑛 ) := {đ‘Ĩ ∈ đģ : đ‘Ĩ𝑛𝑖 ⇀ đ‘Ĩ
for some subsequence {đ‘Ĩ𝑛𝑖 } of {đ‘Ĩ𝑛 }} ,
𝜔𝑠 (đ‘Ĩ𝑛 ) :={đ‘Ĩ ∈ đģ : đ‘Ĩ𝑛𝑖 ķŗ¨€→ đ‘Ĩ
for some subsequence {đ‘Ĩ𝑛𝑖 } of {đ‘Ĩ𝑛 }} .
(11)
The metric (or nearest point) projection from đģ onto 𝐾 is
the mapping 𝑃𝐾 : đģ → 𝐾 which assigns to each point đ‘Ĩ ∈ đģ
the unique point 𝑃𝐾 đ‘Ĩ ∈ 𝐾 satisfying the property
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ − 𝑃𝐾 đ‘Ĩķĩ„Šķĩ„Šķĩ„Š = inf ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š =: 𝑑 (đ‘Ĩ, 𝐾) .
đ‘Ļ∈𝐾
(12)
Some important properties of projections are gathered in
the following.
Proposition 1. For given đ‘Ĩ ∈ đģ and 𝑧 ∈ 𝐾
(i) 𝑧 = 𝑃𝐾 đ‘Ĩ ⇔ ⟨đ‘Ĩ − 𝑧, đ‘Ļ − 𝑧⟩ ≤ 0, for all đ‘Ļ ∈ 𝐾;
(ii) 𝑧 = 𝑃𝐾 đ‘Ĩ ⇔ ‖đ‘Ĩ−𝑧‖2 ≤ ‖đ‘Ĩ−đ‘Ļ‖2 −‖đ‘Ļ−𝑧‖2 , for all đ‘Ļ ∈ 𝐾;
(iii) ⟨𝑃𝐾 đ‘Ĩ − 𝑃𝐾 đ‘Ļ, đ‘Ĩ − đ‘Ļ⟩ ≥ ‖𝑃𝐾 đ‘Ĩ − 𝑃𝐾 đ‘Ļ‖2 , for all đ‘Ļ ∈
đģ, which hence implies that 𝑃𝐾 is nonexpansive and
monotone.
Definition 2. A mapping 𝑇 : đģ → đģ is said to be
(a) nonexpansive if
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘‡đ‘Ĩ − 𝑇đ‘Ļķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ĩ, đ‘Ļ ∈ đģ;
(13)
(b) firmly nonexpansive if 2𝑇 − đŧ is nonexpansive, or
equivalently,
ķĩ„Š2
ķĩ„Š
⟨đ‘Ĩ − đ‘Ļ, 𝑇đ‘Ĩ − 𝑇đ‘Ļ⟩ ≥ ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘Ĩ − 𝑇đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ĩ, đ‘Ļ ∈ đģ;
(14)
alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇
can be expressed as
𝑇=
1
(đŧ + 𝑆) ,
2
(15)
where 𝑆 : đģ → đģ is nonexpansive; projections are
firmly nonexpansive.
Definition 3. Let 𝑇 be a nonlinear operator with domain
𝐷(𝑇) ⊆ đģ and range 𝑅(𝑇) ⊆ đģ.
Journal of Applied Mathematics
3
(a) 𝑇 is said to be monotone if
⟨đ‘Ĩ − đ‘Ļ, 𝑇đ‘Ĩ − 𝑇đ‘Ļ⟩ ≥ 0,
∀đ‘Ĩ, đ‘Ļ ∈ 𝐷 (𝑇) .
(16)
(b) Given a number đ›Ŋ > 0, 𝑇 is said to be đ›Ŋ strongly
monotone if
ķĩ„Š2
ķĩ„Š
(17)
⟨đ‘Ĩ − đ‘Ļ, 𝑇đ‘Ĩ − 𝑇đ‘Ļ⟩ ≥ đ›Ŋķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š , ∀đ‘Ĩ, đ‘Ļ ∈ 𝐷 (𝑇) .
(c) Given a number ] > 0, 𝑇 is said to be ]-inverse
strongly monotone (]-ism) if
ķĩ„Š2
ķĩ„Š
⟨đ‘Ĩ − đ‘Ļ, 𝑇đ‘Ĩ − 𝑇đ‘Ļ⟩ ≥ ]ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘Ĩ − 𝑇đ‘Ļķĩ„Šķĩ„Šķĩ„Š , ∀đ‘Ĩ, đ‘Ļ ∈ 𝐷 (𝑇) . (18)
It can be easily seen that if 𝑇 is nonexpansive, then
đŧ − 𝑇 is monotone. It is also easy to see that a projection
𝑃𝐾 is 1-ism. Inverse strongly monotone (also referred to as
cocoercive) operators have been applied widely in solving
practical problems in various fields.
Definition 4. A mapping 𝑇 : đģ → đģ is said to be an averaged mapping if it can be written as the average of the identity
đŧ and a nonexpansive mapping; that is,
𝑇 ≡ (1 − đ›ŧ) đŧ + đ›ŧ𝑆,
(19)
where đ›ŧ ∈ (0, 1) and 𝑆 : đģ → đģ is nonexpansive. More
precisely, when the last equality holds, we say that 𝑇 is đ›ŧaveraged. Thus, firmly nonexpansive mappings (in particular,
projections) are (1/2)-averaged maps.
Proposition 5 (see [11]). Let 𝑇 : đģ → đģ be a given mapping.
The notation Fix(𝑇) denotes the set of all fixed points of the
mapping 𝑇, that is, Fix(𝑇) = {đ‘Ĩ ∈ đģ : 𝑇đ‘Ĩ = đ‘Ĩ}.
It is clear that, in a real Hilbert space đģ, 𝑇 : đļ → đļ
is 𝜁-strictly pseudocontractive if and only if there holds the
following inequality:
ķĩ„Š2 1 − 𝜁
ķĩ„Š
⟨𝑇đ‘Ĩ − 𝑇đ‘Ļ, đ‘Ĩ − đ‘Ļ⟩ ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š −
2
ķĩ„Š2
ķĩ„Šķĩ„Š
× ķĩ„Šķĩ„Š(đŧ − 𝑇)đ‘Ĩ − (đŧ − 𝑇)đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ĩ, đ‘Ļ ∈ đļ.
(21)
This immediately implies that if 𝑇 is a 𝜁-strictly pseudocontractive mapping, then đŧ − 𝑇 is ((1 − 𝜁)/2)-inverse
strongly monotone; for further detail, we refer to [12] and
the references therein. It is well known that the class of strict
pseudocontractions strictly includes the class of nonexpansive mappings.
Lemma 7 (see [12, Proposition 2.1]). Let đļ be a nonempty
closed convex subset of a real Hilbert space đģ and 𝑇 : đļ → đļ
be a mapping.
(i) If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then 𝑇
satisfies the Lipschitz condition where
ķĩ„Š
ķĩ„Š 1 + 𝜁 ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
ķĩ„Šķĩ„Šđ‘‡đ‘Ĩ − 𝑇đ‘Ļķĩ„Šķĩ„Šķĩ„Š ≤
1−𝜁ķĩ„Š
∀đ‘Ĩ, đ‘Ļ ∈ đļ.
(22)
(ii) If 𝑇 is a 𝜁-strictly pseudocontractive mapping, then the
mapping đŧ − 𝑇 is semiclosed at 0; that is, if {đ‘Ĩ𝑛 } is a
sequence in đļ such that đ‘Ĩ𝑛 → đ‘ĨĖƒ weakly and (đŧ −
𝑇)đ‘Ĩ𝑛 → 0 strongly, then (đŧ − 𝑇)đ‘ĨĖƒ = 0.
(i) 𝑇 is nonexpansive if and only if the complement đŧ−𝑇 is
(1/2)-ism.
(ii) If 𝑇 is ]-ism, then for 𝛾 > 0, 𝛾𝑇 is (]/𝛾)-ism.
(iii) 𝑇 is averaged if and only if the complement đŧ − 𝑇 is ]ism for some ] > 1/2. Indeed, for đ›ŧ ∈ (0, 1), 𝑇 is đ›ŧaveraged if and only if đŧ − 𝑇 is (1/2đ›ŧ)-ism.
(iii) If 𝑇 is a 𝜁-(quasi-)strict pseudocontraction, then the
fixed point set Fix(𝑇) of 𝑇 is closed and convex so that
the projection 𝑃Fix(𝑇) is well defined.
Proposition 6 (see [11]). Let 𝑆, 𝑇, 𝑉 : đģ → đģ be given
operators.
The following lemma is an immediate consequence of an
inner product.
(i) If 𝑇 = (1 − đ›ŧ)𝑆 + đ›ŧ𝑉 for some đ›ŧ ∈ (0, 1) and if 𝑆 is
averaged and 𝑉 is nonexpansive, then 𝑇 is averaged.
(ii) 𝑇 is firmly nonexpansive if and only if the complement đŧ − 𝑇 is firmly nonexpansive.
(iii) If 𝑇 = (1 − đ›ŧ)𝑆 + đ›ŧ𝑉 for some đ›ŧ ∈ (0, 1) and if 𝑆 is
firmly nonexpansive and 𝑉 is nonexpansive, then 𝑇 is
averaged.
(iv) The composite of finitely many averaged mappings is
averaged. That is, if each of the mappings {𝑇𝑖 }𝑁
𝑖=1 is
averaged, then so is the composite 𝑇1 ⋅ ⋅ ⋅ 𝑇𝑁. In particular, if 𝑇1 is đ›ŧ1 -averaged and 𝑇2 is đ›ŧ2 -averaged,
where đ›ŧ1 , đ›ŧ2 ∈ (0, 1), then the composite 𝑇1 𝑇2 is đ›ŧaveraged, where đ›ŧ = đ›ŧ1 + đ›ŧ2 − đ›ŧ1 đ›ŧ2 .
Lemma 8. In a real Hilbert space đģ, there holds the following
inequality:
(v) If the mappings {𝑇𝑖 }𝑁
𝑖=1 are averaged and have a common fixed point, then
𝑁
⋂ Fix (𝑇𝑖 ) = Fix (𝑇1 ⋅ ⋅ ⋅ 𝑇𝑁) .
𝑖=1
(20)
ķĩ„Š2
ķĩ„Šķĩ„Š
2
ķĩ„Šķĩ„Šđ‘Ĩ + đ‘Ļķĩ„Šķĩ„Šķĩ„Š ≤ ‖đ‘Ĩ‖ + 2 ⟨đ‘Ļ, đ‘Ĩ + đ‘Ļ⟩ ,
∀đ‘Ĩ, đ‘Ļ ∈ đģ.
(23)
The following elementary result on real sequences is quite
well known.
Lemma 9 (see [13]). Let {𝑎𝑛 } be a sequence of nonnegative real
numbers satisfying the property
𝑎𝑛+1 ≤ (1 − 𝑠𝑛 ) 𝑎𝑛 + 𝑠𝑛 𝑡𝑛 + 𝜖𝑛 ,
∀𝑛 ≥ 0,
(24)
where {𝑠𝑛 } ⊂ (0, 1] and {𝑡𝑛 } are the real sequences such that
(i) ∑∞
𝑛=0 𝑠𝑛 = ∞;
(ii) either lim sup𝑛 → ∞ 𝑡𝑛 ≤ 0 or ∑∞
𝑛=0 𝑠𝑛 |𝑡𝑛 | < ∞;
(iii) ∑∞
𝑛=0 𝜖𝑛 < ∞ where 𝜖𝑛 ≥ 0, for all 𝑛 ≥ 0.
Then, lim𝑛 → ∞ 𝑎𝑛 = 0.
4
Journal of Applied Mathematics
Lemma 10 (see [14]). Let đļ be a nonempty closed convex
subset of a real Hilbert space đģ. Let 𝑇 : đļ → đļ be a 𝜁-strictly
pseudocontractive mapping. Let 𝛾 and đ›ŋ be two nonnegative
real numbers such that (𝛾 + đ›ŋ)𝜁 ≤ 𝛾. Then,
ķĩ„Šķĩ„Šķĩ„Šđ›ž (đ‘Ĩ − đ‘Ļ) + đ›ŋ (𝑇đ‘Ĩ − 𝑇đ‘Ļ)ķĩ„Šķĩ„Šķĩ„Š ≤ (𝛾 + đ›ŋ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š , ∀đ‘Ĩ, đ‘Ļ ∈ đļ.
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
(25)
The following lemma appears implicitly in the paper of
Reinermann [15].
Lemma 11 (see [15]). Let đģ be a real Hilbert space. Then, for
all đ‘Ĩ, đ‘Ļ ∈ đģ and 𝜆 ∈ [0, 1],
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđœ†đ‘Ĩ + (1 − 𝜆) đ‘Ļķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š ķĩ„Š2
ķĩ„Š
= 𝜆‖đ‘Ĩ‖2 + (1 − 𝜆) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļķĩ„Šķĩ„Šķĩ„Š − 𝜆 (1 − 𝜆) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š .
(26)
(a) if ‖đ‘Ĩ𝑛 − 𝑇𝑟𝑛 đ‘Ĩ𝑛 ‖ → 0 as 𝑛 → ∞, each weak cluster
point of {đ‘Ĩ𝑛 } satisfies the problem:
𝐹 (đ‘Ĩ, đ‘Ļ) + ℎ (đ‘Ĩ, đ‘Ļ) ≥ 0,
∀đ‘Ļ ∈ đļ,
(29)
that is, 𝜔𝑤 (đ‘Ĩ𝑛 ) ⊆ EP(𝐹, ℎ).
(b) The demiclosedness principle holds in the sense that, if
đ‘Ĩ𝑛 ⇀ đ‘Ĩ∗ and ‖đ‘Ĩ𝑛 − 𝑇𝑟𝑛 đ‘Ĩ𝑛 ‖ → 0 as 𝑛 → ∞, then
(đŧ − 𝑇𝑟𝑘 )đ‘Ĩ∗ = 0 for all 𝑘 ≥ 1.
3. Main Results
Lemma 12 (see [16]). Let đļ be a nonempty closed convex
subset of a real Hilbert space đģ. Let 𝐹 : đļ × đļ → R be a
bifunction such that
(f1) 𝐹(đ‘Ĩ, đ‘Ĩ) = 0 for all đ‘Ĩ ∈ đļ;
(f2) 𝐹 is monotone and upper hemicontinuous in the first
variable;
(f3) 𝐹 is lower semicontinuous and convex in the second
variable.
We now propose the following relaxed viscosity iterative
scheme with regularization:
𝐹 (đ‘ĸ𝑛 , đ‘Ļ) + ℎ (đ‘ĸ𝑛 , đ‘Ļ) +
1
⟨đ‘Ļ − đ‘ĸ𝑛 , đ‘ĸ𝑛 − đ‘Ĩ𝑛 ⟩ ≥ 0,
𝑟𝑛
∀đ‘Ļ ∈ đļ,
đ‘Ļ𝑛,1 = đ›Ŋ𝑛,1 𝑆1 đ‘ĸ𝑛 + (1 − đ›Ŋ𝑛,1 ) đ‘ĸ𝑛 ,
đ‘Ļ𝑛,𝑖 = đ›Ŋ𝑛,𝑖 𝑆𝑖 đ‘ĸ𝑛 + (1 − đ›Ŋ𝑛,𝑖 ) đ‘Ļ𝑛,𝑖−1 ,
Let ℎ : đļ × đļ → R be a bifunction such that
(h1) ℎ(đ‘Ĩ, đ‘Ĩ) = 0 for all đ‘Ĩ ∈ đļ;
(h2) ℎ is monotone and weakly upper semicontinuous in the
first variable;
(h3) ℎ is convex in the second variable.
Moreover, let one suppose that
(H) for fixed 𝑟 > 0 and đ‘Ĩ ∈ đļ, there exists a bounded 𝐾 ⊂ đļ
Ė‚ 𝑧) +
and đ‘ĨĖ‚ ∈ 𝐾 such that for all 𝑧 ∈ đļ \ 𝐾, −𝐹(đ‘Ĩ,
Ė‚ + (1/𝑟)⟨đ‘ĨĖ‚ − 𝑧, 𝑧 − đ‘Ĩ⟩ < 0.
ℎ(𝑧, đ‘Ĩ)
For 𝑟 > 0 and đ‘Ĩ ∈ đģ, let 𝑇𝑟 : đģ → 2đļ be a mapping defined
by
𝑇𝑟 đ‘Ĩ
= {𝑧 ∈ đļ : 𝐹 (𝑧, đ‘Ļ) + ℎ (𝑧, đ‘Ļ)
Lemma 14 (see [17]). Suppose that the hypotheses of
Lemma 12 are satisfied. Let {𝑟𝑛 } be a sequence in (0, ∞) with
lim inf 𝑛 → ∞ 𝑟𝑛 > 0. Suppose that {đ‘Ĩ𝑛 } is a bounded sequence.
Then, the following statements are equivalent and true.
(27)
1
+ ⟨đ‘Ļ − 𝑧, 𝑧 − đ‘Ĩ⟩ ≥ 0, ∀đ‘Ļ ∈ đļ}
𝑟
called the resolvent of 𝐹 and ℎ. Then,
(1) 𝑇𝑟 đ‘Ĩ =Ė¸ 0;
(2) 𝑇𝑟 đ‘Ĩ is a singleton;
(3) 𝑇𝑟 is firmly nonexpansive;
(4) 𝐸𝑃(𝐹, ℎ) = Fix(𝑇𝑟 ) and it is closed and convex.
Lemma 13 (see [16]). Let one suppose that (f1)–(f3), (h1)–(h3)
and (H) hold. Let đ‘Ĩ, đ‘Ļ ∈ đģ, 𝑟1 , 𝑟2 > 0. Then,
ķĩ„¨
ķĩ„¨
ķĩ„Šķĩ„Šķĩ„Šđ‘‡ đ‘Ļ − 𝑇 đ‘Ĩķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ − đ‘Ĩķĩ„Šķĩ„Šķĩ„Š + ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ 𝑟2 − 𝑟1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘‡ đ‘Ļ − đ‘Ļķĩ„Šķĩ„Šķĩ„Š .
(28)
ķĩ„Š ķĩ„Š
𝑟1 ķĩ„Š
ķĩ„Š ķĩ„¨ķĩ„¨ 𝑟 ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š 𝑟2
ķĩ„Šķĩ„Š 𝑟2
ķĩ„Šķĩ„Š
ķĩ„¨ 2 ķĩ„¨
𝑖 = 2, . . . , 𝑁,
(30)
đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ļ𝑛,𝑁 + (1 − đ›Ŋ𝑛 ) 𝑃đļ (đ‘Ļ𝑛,𝑁 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁)) ,
đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + 𝛾𝑛 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 ))
+ đ›ŋ𝑛 𝑇𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 )) ,
for all 𝑛 ≥ 0, where the mapping 𝑄 : đļ → đļ is a 𝜌contraction; the mapping 𝑇 : đļ → đļ is a 𝜁-strict pseudocontraction; 𝑆𝑖 : đļ → đļ is a nonexpansive mapping for each 𝑖 =
1, . . . , 𝑁; ∇𝑓 : đļ → đģ satisfies the Lipschitz condition (10)
with 0 < 𝜆 < (2/đŋ); 𝐹, ℎ : đļ × đļ → R are two bifunctions
satisfying the hypotheses of Lemma 12; {đ›ŧ𝑛 } is a sequence in
(0, ∞) with ∑∞
𝑛=0 đ›ŧ𝑛 < ∞; {đ›Ŋ𝑛 }, {𝜎𝑛 } are sequences in (0, 1)
with 0 < lim inf 𝑛 → ∞ 𝜎𝑛 ≤ lim sup𝑛 → ∞ 𝜎𝑛 < 1; {𝛾𝑛 }, {đ›ŋ𝑛 } are
sequences in [0, 1] with 𝜎𝑛 + 𝛾𝑛 + đ›ŋ𝑛 = 1, for all 𝑛 ≥ 0; {đ›Ŋ𝑛,𝑖 }𝑁
𝑖=1
are sequences in (0, 1) and (𝛾𝑛 + đ›ŋ𝑛 )𝜁 ≤ 𝛾𝑛 , for all 𝑛 ≥ 0;
{𝑟𝑛 } is a sequence in (0, ∞) with lim inf 𝑛 → ∞ 𝑟𝑛 > 0 and
lim inf 𝑛 → ∞ đ›ŋ𝑛 > 0.
Before stating and proving the main convergence results,
we first establish the following lemmas.
Lemma 15. Let one suppose that Ω = Fix(𝑇) ∩ (⋂𝑖 Fix(𝑆𝑖 )) ∩
EP(𝐹, ℎ) ∩ Γ =Ė¸ 0. Then, the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, {đ‘Ļ𝑛,𝑖 } for all 𝑖,
and {đ‘ĸ𝑛 } are bounded.
Proof. First of all, we can show as in [18] that 𝑃đļ(đŧ − 𝜆∇𝑓đ›ŧ )
is nonexpansive for 𝜆 ∈ (0, 2/(đ›ŧ + đŋ)), and 𝑃đļ(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) is
nonexpansive for all 𝑛 ≥ 0 and 𝜆 ∈ (0, 2/đŋ). We observe that
if 𝑝 ∈ Ω, then
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
(31)
ķĩ„Šķĩ„Šđ‘Ļ𝑛,1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š .
Journal of Applied Mathematics
5
For all, from 𝑖 = 2 to 𝑖 = 𝑁, by induction, one proves that
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛,𝑖 ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛,𝑖 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖−1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š .
(32)
Thus, we obtain that for every 𝑖 = 1, . . . , 𝑁,
ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š .
(33)
For simplicity, put đ‘ĻĖƒđ‘›,𝑁 = 𝑃đļ(đ‘Ļ𝑛,𝑁 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁)) and đ‘ĻĖƒđ‘› =
𝑃đļ(đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 )) for every 𝑛 ≥ 0. Then, đ‘Ļ𝑛 = đ›ŧ𝑛 𝑄đ‘Ļ𝑛,𝑁 +
(1 − đ›ŧ𝑛 )đ‘ĻĖƒđ‘›,𝑁 and đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + 𝛾𝑛 đ‘ĻĖƒđ‘› + đ›ŋ𝑛 𝑇đ‘ĻĖƒđ‘› for every 𝑛 ≥ 0.
Taking into consideration that 𝑇𝑝 = 𝑝 and 𝑃đļ(đŧ − 𝜆∇𝑓)𝑝 = 𝑝
for 𝜆 ∈ (0, 2/đŋ), we have
ķĩ„Š
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − 𝑃đļ (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − 𝑃đļ (đŧ−𝜆∇𝑓đ›ŧ𝑛 ) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) 𝑝 − 𝑃đļ (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) 𝑝 − 𝑃đļ (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š .
(34)
Similarly, we get ‖đ‘ĻĖƒđ‘› − 𝑝‖ ≤ ‖đ‘Ļ𝑛 − 𝑝‖ + 𝜆đ›ŧ𝑛 ‖𝑝‖. Thus, from
(34) we have
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š = ķĩ„Šķĩ„Šķĩ„Šđ›Ŋ𝑛 (𝑄đ‘Ļ𝑛,𝑁 − 𝑝) + (1 − đ›Ŋ𝑛 ) (đ‘ĻĖƒđ‘›,𝑁 − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑄𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š
+ (1 − đ›Ŋ𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 𝜌 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š
+ (1 − đ›Ŋ𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
= (1 − đ›Ŋ𝑛 (1 − 𝜌)) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
= (1 − đ›Ŋ𝑛 (1 − 𝜌)) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
+ đ›Ŋ𝑛 (1 − 𝜌) ķĩ„Š
+ (1 − đ›Ŋ𝑛 ) 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
1−𝜌
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
≤ max {ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
} + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
1−𝜌
Since (𝛾𝑛 + đ›ŋ𝑛 )𝜁 ≤ 𝛾𝑛 for all 𝑛 ≥ 0, utilizing Lemma 10, we
derive from (35)
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„ŠđœŽ (đ‘Ļ𝑛 − 𝑝) + 𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ›žđ‘› (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
≤ max {ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
} + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
1−𝜌
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
= max {ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
} + 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š .
1−𝜌
(36)
By induction, we get
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
}
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ max {ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ0 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
1−𝜌
𝑛
ķĩ„Š ķĩ„Š
+ 2𝜆 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š ⋅ ∑đ›ŧ𝑖 ,
(37)
∀𝑛 ≥ 0.
𝑖=0
This implies that {đ‘Ĩ𝑛 } is bounded and so are {đ‘Ļ𝑛 }, {đ‘ĸ𝑛 }, and
{đ‘Ļ𝑛,𝑖 } for each 𝑖 = 1, . . . , 𝑁. It is clear that both {đ‘ĻĖƒđ‘›,𝑁} and {đ‘ĻĖƒđ‘› }
are also bounded. Since ‖𝑇đ‘ĻĖƒđ‘› − 𝑝‖ ≤ ((1 + 𝜁)/(1 − 𝜁))‖đ‘ĻĖƒđ‘› − 𝑝‖,
{𝑇đ‘ĻĖƒđ‘› } is also bounded.
Lemma 16. Let one suppose that Ω =Ė¸ 0. Moreover, let one
suppose that the following hold:
(H1) lim𝑛 → ∞ đ›Ŋ𝑛 = 0 and ∑∞
𝑛=0 đ›Ŋ𝑛 = ∞;
(H2) ∑∞
𝑛=1 |đ›Ŋ𝑛 − đ›Ŋ𝑛−1 | < ∞ or lim𝑛 → ∞ (|đ›Ŋ𝑛 − đ›Ŋ𝑛−1 |/đ›Ŋ𝑛 ) = 0;
(H3) ∑∞
𝑛=1 |đ›Ŋ𝑛,𝑖 −đ›Ŋ𝑛−1,𝑖 | < ∞ or lim𝑛 → ∞ (|đ›Ŋ𝑛,𝑖 −đ›Ŋ𝑛−1,𝑖 |/đ›Ŋ𝑛 ) =
0 for each 𝑖 = 1, . . . , 𝑁;
(H4) ∑∞
𝑛=1 |𝑟𝑛 − 𝑟𝑛−1 | < ∞ or lim𝑛 → ∞ (|𝑟𝑛 − 𝑟𝑛−1 |/đ›Ŋ𝑛 ) = 0;
(H5) ∑∞
𝑛=1 |𝜎𝑛 − 𝜎𝑛−1 | < ∞ or lim𝑛 → ∞ (|𝜎𝑛 − 𝜎𝑛−1 |/đ›Ŋ𝑛 ) = 0;
(H6) ∑∞
< ∞ or
𝑛=1 |𝛾𝑛 /(1 − 𝜎𝑛 ) − 𝛾𝑛−1 /(1 − 𝜎𝑛−1 )|
lim𝑛 → ∞ (1/đ›Ŋ𝑛 )|𝛾𝑛 /(1 − 𝜎𝑛 ) − 𝛾𝑛−1 /(1 − 𝜎𝑛−1 )| = 0.
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
≤ max {ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
} + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
1−𝜌
Then, lim𝑛 → ∞ ‖đ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ‖ = 0, that is, {đ‘Ĩ𝑛 } is asymptotically
regular.
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘„đ‘ − 𝑝ķĩ„Šķĩ„Šķĩ„Š
≤ max {ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š , ķĩ„Š
} + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š .
1−𝜌
(35)
≤
Proof. Taking into account 0 < lim inf 𝑛 → ∞ 𝜎𝑛
lim sup𝑛 → ∞ 𝜎𝑛 < 1, we may assume, without loss of generality, that {𝜎𝑛 } ⊂ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1). First, we write
6
Journal of Applied Mathematics
đ‘Ĩ𝑛 = 𝜎𝑛−1 đ‘Ļ𝑛−1 + (1 − 𝜎𝑛−1 )V𝑛−1 , for all 𝑛 ≥ 1, where V𝑛−1 =
(đ‘Ĩ𝑛 − 𝜎𝑛−1 đ‘Ļ𝑛−1 )/(1 − 𝜎𝑛−1 ). It follows that for all 𝑛 ≥ 1
V𝑛 − V𝑛−1 =
đ‘Ĩ𝑛+1 − 𝜎𝑛 đ‘Ļ𝑛 đ‘Ĩ𝑛 − 𝜎𝑛−1 đ‘Ļ𝑛−1
−
1 − 𝜎𝑛
1 − 𝜎𝑛−1
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 −đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛−1,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁ķĩ„Šķĩ„Šķĩ„Š +đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑄đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − đ›Ŋ𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š +𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š)
𝛾𝑛 (đ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑇đ‘ĻĖƒđ‘›−1 )
1 − 𝜎𝑛
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛−1,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
𝛾
𝛾𝑛−1
+( 𝑛 −
) đ‘ĻĖƒ
1 − 𝜎𝑛 1 − 𝜎𝑛−1 𝑛−1
+(
đ›ŋ𝑛−1
đ›ŋ𝑛
−
) 𝑇đ‘ĻĖƒđ‘›−1 .
1 − 𝜎𝑛 1 − 𝜎𝑛−1
ķĩ„Š
ķĩ„Š
+ đ›Ŋ𝑛 𝜌 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
(38)
Since (𝛾𝑛 + đ›ŋ𝑛 )𝜁 ≤ 𝛾𝑛 for all 𝑛 ≥ 0, utilizing Lemma 10, we
have
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ›žđ‘› (đ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑇đ‘ĻĖƒđ‘›−1 )ķĩ„Šķĩ„Š
(39)
ķĩ„Š
ķĩ„Š
≤ (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ķĩ„Šķĩ„Šķĩ„Š .
Next, we estimate ‖đ‘Ļ𝑛 −đ‘Ļ𝑛−1 ‖. Observe that for every 𝑛 ≥ 1
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − 𝑃đļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛−1,𝑁 − 𝑃đļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛−1 ) đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛−1,𝑁 − 𝑃đļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛−1 ) đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛−1,𝑁 − (đŧ − 𝜆∇𝑓đ›ŧ𝑛−1 ) đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđœ†∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛−1,𝑁) − 𝜆∇𝑓đ›ŧ𝑛−1 (đ‘Ļ𝑛−1,𝑁)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š
ķĩ„Š
ķĩ„¨ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š + 𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š ,
(40)
and similarly,
ķĩ„Šķĩ„Š Ėƒ
ķĩ„¨
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„¨ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘ĻĖƒđ‘›−1 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š .
Also, from (30), we have
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
𝛾 đ‘ĻĖƒ + đ›ŋ𝑛 𝑇đ‘ĻĖƒđ‘› 𝛾𝑛−1 đ‘ĻĖƒđ‘›−1 + đ›ŋ𝑛−1 𝑇đ‘ĻĖƒđ‘›−1
= 𝑛 𝑛
−
1 − 𝜎𝑛
1 − 𝜎𝑛−1
=
Then, passing to the norm we get from (40) that
(41)
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
+ 𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛−1,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ļ𝑛−1,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀1 (ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨) ,
where 𝜆‖đ‘Ļ𝑛,𝑁‖ + ‖𝑄đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁‖ ≤ 𝑀1 , for all 𝑛 ≥ 0 for some
𝑀1 ≥ 0. Furthermore, by the definition of đ‘Ļ𝑛,𝑖 one obtains
that, for all 𝑖 = 𝑁, . . . , 2
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖 − đ‘Ļ𝑛−1,𝑖 ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛,𝑖 ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑖 − đ›Ŋ𝑛−1,𝑖 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„Š
ķĩ„Š
+ (1 − đ›Ŋ𝑛,𝑖 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖−1 − đ‘Ļ𝑛−1,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š .
(42)
∀𝑛 ≥ 1.
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,1 − đ‘Ļ𝑛−1,1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛,1 ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„Š
ķĩ„Š
+ (1 − đ›Ŋ𝑛,1 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
= ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨ .
Substituting (46) in all (45) type one obtains for 𝑖 = 2, . . . , 𝑁
𝑖
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ∑ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘˜ đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑘−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
đ‘Ļ𝑛 − đ‘Ļ𝑛−1 = (1 − đ›Ŋ𝑛 ) (đ‘ĻĖƒđ‘›,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁)
+ đ›Ŋ𝑛 (𝑄đ‘Ļ𝑛,𝑁 − 𝑄đ‘Ļ𝑛−1,𝑁) .
(46)
ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖 − đ‘Ļ𝑛−1,𝑖 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
Simple calculations show that
+ (đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ) (𝑄đ‘Ļ𝑛−1,𝑁 − đ‘ĻĖƒđ‘›−1,𝑁)
(45)
In the case of 𝑖 = 1, we have
đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ļ𝑛,𝑁 + (1 − đ›Ŋ𝑛 ) đ‘ĻĖƒđ‘›,𝑁,
đ‘Ļ𝑛−1 = đ›Ŋ𝑛−1 𝑄đ‘Ļ𝑛−1,𝑁+ (1 − đ›Ŋ𝑛−1 ) đ‘ĻĖƒđ‘›−1,𝑁,
(44)
𝑘=2
(43)
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨ .
(47)
Journal of Applied Mathematics
7
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ 𝑀2 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨
𝑟𝑛
𝑘=2
This together with (44) implies that
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ]
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 )
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀2 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„Š
ķĩ„Š
× [ ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
𝑁
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ∑ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘˜ đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑘−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ] ,
𝑘=2
ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨
ķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨]
ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀1 (ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨)
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
𝑁
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ∑ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘˜ đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑘−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑘=2
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀1 (ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨) .
By Lemma 13, we know that
ķĩ„¨ķĩ„¨
𝑟 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜅 ķĩ„¨ķĩ„¨ķĩ„¨1 − 𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ,
𝑟𝑛 ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨
(49)
where 𝜅 = sup𝑛≥0 ‖đ‘ĸ𝑛 − đ‘Ĩ𝑛 ‖. So, substituting (49) in (48) we
obtain
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨
𝑟 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜅 ķĩ„¨ķĩ„¨ķĩ„¨1 − 𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨)
ķĩ„¨ķĩ„¨
𝑟𝑛 ķĩ„¨ķĩ„¨
𝑁
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ∑ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘˜ đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑘−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑘=2
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀1 (ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨)
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜅 ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨
𝑟
𝑛
𝑁
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ∑ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘˜ đ‘ĸ𝑛−1 − đ‘Ļ𝑛−1,𝑘−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑘=2
ķĩ„Š
ķĩ„Šķĩ„¨
ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛−1 − đ‘ĸ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,1 − đ›Ŋ𝑛−1,1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀1 (ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨)
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
(50)
(48)
where 𝑏 > 0 is a minorant for {𝑟𝑛 } and 𝜅 + 𝑀1 + ∑𝑁
𝑘=2 ‖𝑆𝑘 đ‘ĸ𝑛 −
đ‘Ļ𝑛,𝑘−1 ‖ + ‖𝑆1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ ≤ 𝑀2 , for all 𝑛 ≥ 0 for some 𝑀2 ≥ 0.
This together with (38)-(39), implies that
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ›žđ‘› (đ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑇đ‘ĻĖƒđ‘›−1 )ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„ŠV𝑛 − V𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Š
1 − 𝜎𝑛
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ ķĩ„Šđ‘ĻĖƒ ķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„¨ķĩ„¨ đ›ŋ
đ›ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒđ‘›−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨
ķĩ„Š
ķĩ„Š
(𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ķĩ„Šķĩ„Šķĩ„Š
≤
1 − 𝜎𝑛
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ ķĩ„Šđ‘ĻĖƒ ķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ ķĩ„Šđ‘‡đ‘ĻĖƒ ķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ (ķĩ„Šđ‘ĻĖƒ ķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒ ķĩ„Šķĩ„Š)
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„¨
ķĩ„Š
ķĩ„Š
ķĩ„¨ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ (ķĩ„Šđ‘ĻĖƒ ķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒ ķĩ„Šķĩ„Š)
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀2 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ]
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
+ 𝜆 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„Š ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ (ķĩ„Šđ‘ĻĖƒ ķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒ ķĩ„Šķĩ„Š)
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Š 𝑛−1 ķĩ„Š ķĩ„Š 𝑛−1 ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
8
Journal of Applied Mathematics
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀3 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨ķĩ„Š
ķĩ„Š
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 − V𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ 𝜎𝑛 { (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ]
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨
ķĩ„¨
+ 𝑀3 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + 𝑀3 ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„Š
ķĩ„Š
= (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨đ‘Ÿđ‘› − 𝑟𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨ķĩ„¨
ķĩ„¨
+ 𝑀3 [
+ ∑ ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ 2 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨],
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
where 𝑀2 + 𝜆‖đ‘Ļ𝑛 ‖ + ‖đ‘ĻĖƒđ‘› ‖ + ‖𝑇đ‘ĻĖƒđ‘› ‖ ≤ 𝑀3 , for all 𝑛 ≥ 0 for some
𝑀3 ≥ 0.
Further, we observe that
đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + (1 − 𝜎𝑛 ) V𝑛 ,
(52)
đ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 = (1 − 𝜎𝑛 ) (V𝑛 − V𝑛−1 )
Then, passing to the norm, we get from (51)
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − 𝜎𝑛 ) ķĩ„Šķĩ„Šķĩ„ŠV𝑛 − V𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 − V𝑛−1 ķĩ„Šķĩ„Šķĩ„Š + 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
≤ (1 − 𝜎𝑛 )
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀3 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨]
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„Š
ķĩ„Š
+ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛−1 − V𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ ] + 2𝑀 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ,
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
Simple calculations show that
+ 𝜎𝑛 (đ‘Ļ𝑛 − đ‘Ļ𝑛−1 ) .
ķĩ„Š
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
∀𝑛 ≥ 1.
+ (𝜎𝑛 − 𝜎𝑛−1 ) (đ‘Ļ𝑛−1 − V𝑛−1 )
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ]}
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ 2 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
(51)
đ‘Ĩ𝑛 = 𝜎𝑛−1 đ‘Ļ𝑛−1 + (1 − đ›Ŋ𝑛−1 ) V𝑛−1 ,
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨đ‘Ÿđ‘› − 𝑟𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨ķĩ„¨
ķĩ„¨
+ 𝑀2 [
+ ∑ ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
(54)
(53)
where 𝑀3 + ‖đ‘Ļ𝑛 − V𝑛 ‖ ≤ 𝑀, for all 𝑛 ≥ 0 for some 𝑀 ≥ 0. By
hypotheses (H1)–(H6) and Lemma 9, from ∑∞
𝑛=0 đ›ŧ𝑛 < ∞, we
obtain the claim.
Lemma 17. Let one suppose that Ω =Ė¸ 0. Let one suppose that
{đ‘Ĩ𝑛 } is asymptotically regular. Then, ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0 and ‖đ‘Ĩ𝑛 −
đ‘ĸ𝑛 ‖ = ‖đ‘Ĩ𝑛 − 𝑇𝑟𝑛 đ‘Ĩ𝑛 ‖ → 0 as 𝑛 → ∞.
Proof. We recall that, by the firm nonexpansivity of 𝑇𝑟𝑛 , a
standard calculation (see [17]) shows that if V ∈ EP(𝐹, ℎ), then
ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Šķĩ„Šđ‘ĸ𝑛 − Vķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − Vķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š .
ķĩ„Š
ķĩ„Š
× { (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ 𝑁 ķĩ„¨
ķĩ„¨
+ 𝑀3 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + ∑ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
𝑏
𝑘=1
ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
+ 2 ķĩ„¨ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ + ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ +
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
+ ķĩ„¨ķĩ„¨ķĩ„¨ 𝑛 −
ķĩ„¨ ]}
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
(55)
Let 𝑝 ∈ Ω. Then by Lemma 11, we have from (33)–(34) the
following
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š = ķĩ„Šķĩ„Šķĩ„Šđ›Ŋ𝑛 (𝑄đ‘Ļ𝑛,𝑁 − 𝑝) + (1 − đ›Ŋ𝑛 ) (đ‘ĻĖƒđ‘›,𝑁 − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + [ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š]
Journal of Applied Mathematics
9
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
= đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) ,
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
= 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − 𝑝ķĩ„Šķĩ„Šķĩ„Š
−
ķĩ„Š
ķĩ„Š2
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 )
(56)
ķĩ„Š2
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š2 ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š = ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛 − 𝑃đļ (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š
ķĩ„Š
≤ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
(57)
Since (𝛾𝑛 + đ›ŋ𝑛 )𝜁 ≤ 𝛾𝑛 for all 𝑛 ≥ 0, utilizing Lemma 10, we
have
ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= ķĩ„Šķĩ„Šķĩ„ŠđœŽđ‘› (đ‘Ļ𝑛 − 𝑝) + 𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
1
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„ŠđœŽđ‘› (đ‘Ļ𝑛 − 𝑝) + (𝛾𝑛 + đ›ŋ𝑛 )
ķĩ„Šķĩ„Š
𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š2
ķĩ„Š
× [𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)] ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 )
ķĩ„Šķĩ„Š2
ķĩ„Šķĩ„Šķĩ„Š 1
ķĩ„Š
[𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)]ķĩ„Šķĩ„Šķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š 𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š
1
ķĩ„Š
− 𝜎𝑛 (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Š(đ‘Ļ𝑛 − 𝑝) −
ķĩ„Šķĩ„Š
𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š2
ķĩ„Š
× [𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)] ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 )
ķĩ„Šķĩ„Š2
ķĩ„Šķĩ„Š 1
ķĩ„Š
ķĩ„Š
[𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)]ķĩ„Šķĩ„Šķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š 𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š 1
ķĩ„Šķĩ„Š2
ķĩ„Š
ķĩ„Š
− 𝜎𝑛 (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Š
[𝛾𝑛 (đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 )]ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š 𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 )
ķĩ„Šķĩ„Š 1
ķĩ„Šķĩ„Š2
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Š
[𝛾𝑛 (đ‘ĻĖƒđ‘› − 𝑝) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑝)]ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š 𝛾𝑛 + đ›ŋ𝑛
ķĩ„Šķĩ„Š
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
−
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
𝛾𝑛 + đ›ŋ𝑛 ķĩ„Š 𝑛+1
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − 𝑝ķĩ„Šķĩ„Šķĩ„Š
−
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
𝛾𝑛 + đ›ŋ𝑛 ķĩ„Š 𝑛+1
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
ķĩ„Š2
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
× [ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)]
−
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
−
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
−
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
+ 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
−
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š .
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
(58)
Taking into account 0 < lim inf 𝑛 → ∞ 𝜎𝑛 ≤ lim sup𝑛 → ∞ 𝜎𝑛 <
1, we may assume that {𝜎𝑛 } ⊂ [𝑐, 𝑑] for some 𝑐, 𝑑 ∈ (0, 1). So,
we deduce that
𝑐 ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝑐 ķĩ„Š 𝑛+1
𝜎𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2
≤
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1 − 𝜎𝑛 ķĩ„Š 𝑛+1
ķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
ķĩ„Š2
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
≤ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛+1 ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
(59)
Since đ›ŧ𝑛 → 0, đ›Ŋ𝑛 → 0 and ‖đ‘Ĩ𝑛 − đ‘Ĩ𝑛+1 ‖ → 0 as 𝑛 → ∞, we
conclude from the boundedness of {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘Ļ𝑛,𝑁} that
‖đ‘Ĩ𝑛+1 −đ‘Ļ𝑛 ‖ → 0 as 𝑛 → ∞. This together with ‖đ‘Ĩ𝑛 −đ‘Ĩ𝑛+1 ‖ →
0, implies that
ķĩ„Šķĩ„Š
lim ķĩ„Šđ‘Ĩ
𝑛→∞ ķĩ„Š 𝑛
ķĩ„Š
− đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š = 0.
(60)
10
Journal of Applied Mathematics
Furthermore, from (33), (55), and (56), we have
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
(61)
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š2 ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) ,
which hence implies that
ķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2
(62)
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
+ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
(H8) there exists a constant 𝜏 > 0 such that (1/đ›Ŋ𝑛 )|1/đ›Ŋ𝑛,𝑘0 −
1/đ›Ŋ𝑛−1,𝑘0 | < 𝜏 for all 𝑛 ≥ 1. Then,
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
lim ķĩ„Š 𝑛+1
= 0.
𝑛→∞
đ›Ŋ𝑛,𝑘0
Proof. We start by (54). Dividing both the terms by đ›Ŋ𝑛,𝑘0 we
have
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
≤ [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Š 𝑛
đ›Ŋ𝑛,𝑘0
ķĩ„¨
𝑁 ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨đ‘Ÿđ‘› − 𝑟𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ∑𝑘=1 ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
+𝑀 [
+
𝑏đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„¨ ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
2 ķĩ„¨ķĩ„¨đ›ŧ − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„¨ 𝑛
+
+
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
Since đ›ŧ𝑛 → 0, đ›Ŋ𝑛 → 0 and ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0 as 𝑛 → ∞, we
deduce from the boundedness of {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘Ļ𝑛,𝑁} that
ķĩ„Šķĩ„Š
lim ķĩ„Šđ‘Ĩ
𝑛→∞ ķĩ„Š 𝑛
ķĩ„Š
− đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š = 0.
(66)
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›ž / (1 − 𝜎𝑛 ) − 𝛾𝑛−1 / (1 − 𝜎𝑛−1 )ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
].
đ›Ŋ𝑛,𝑘0
(63)
(67)
So, by (H8) we have
Remark 18. By the last lemma we have 𝜔𝑤 (đ‘Ĩ𝑛 ) = 𝜔𝑤 (đ‘ĸ𝑛 ) and
𝜔𝑠 (đ‘Ĩ𝑛 ) = 𝜔𝑠 (đ‘ĸ𝑛 ); that is, the sets of strong/weak cluster points
of {đ‘Ĩ𝑛 } and {đ‘ĸ𝑛 } coincide.
Of course, if đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0, as 𝑛 → ∞, for all index 𝑖, the
assumptions of Lemma 16 are enough to assure that
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
= 0,
lim ķĩ„Š 𝑛+1
𝑛→∞
đ›Ŋ𝑛,𝑖
∀𝑖 ∈ {1, . . . , 𝑁} .
(64)
In the next lemma, we examine the case in which at least one
sequence {đ›Ŋ𝑛,𝑘0 } is a null sequence.
Lemma 19. Let one suppose that Ω =Ė¸ 0. Let one suppose
that (H1) holds. Moreover, for an index 𝑘0 ∈ {1, . . . , 𝑁},
lim𝑛 → ∞ đ›Ŋ𝑛,𝑘0 = 0, and the following hold:
(H7) for all 𝑖,
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑖 − đ›Ŋ𝑛−1,𝑖 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨đ›ŧ𝑛 − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
= lim
= lim
lim
𝑛→∞
𝑛→∞ đ›Ŋ đ›Ŋ
𝑛→∞ đ›Ŋ đ›Ŋ
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
𝑛 𝑛,𝑘0
𝑛 𝑛,𝑘0
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đœŽ − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨
= lim ķĩ„¨ 𝑛
= lim ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨
𝑛→∞ đ›Ŋ đ›Ŋ
𝑛→∞ đ›Ŋ đ›Ŋ
𝑛 𝑛,𝑘0
𝑛 𝑛,𝑘0
1
𝑛→∞đ›Ŋ đ›Ŋ
𝑛 𝑛,𝑘0
= lim
ķĩ„¨ķĩ„¨ 𝛾
𝛾𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨ 𝑛
−
ķĩ„¨ķĩ„¨
ķĩ„¨ = 0;
ķĩ„¨ķĩ„¨ 1 − 𝜎𝑛 1 − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
(65)
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
≤ [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Š 𝑛
đ›Ŋ𝑛−1,𝑘0
ķĩ„¨
ķĩ„¨
1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„Š ķĩ„¨ķĩ„¨ 1
ķĩ„Š
ķĩ„¨ķĩ„¨
+ [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
−
ķĩ„¨ķĩ„¨ đ›Ŋ𝑛,𝑘0 đ›Ŋ𝑛−1,𝑘0 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
𝑁 ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ ∑ ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
+ 𝑀 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + 𝑘=1 ķĩ„¨
𝑏đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›ž / (1 − 𝜎𝑛 ) − 𝛾𝑛−1 / (1 − 𝜎𝑛−1 )ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
đ›Ŋ𝑛,𝑘0
ķĩ„¨ ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
2 ķĩ„¨ķĩ„¨đ›ŧ − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„¨ 𝑛
+
+
]
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
≤ [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Š 𝑛
đ›Ŋ𝑛−1,𝑘0
ķĩ„¨
ķĩ„¨
1 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„Š ķĩ„¨ķĩ„¨ 1
ķĩ„Š
ķĩ„¨ķĩ„¨
+ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„¨ķĩ„¨ķĩ„¨ķĩ„¨
−
ķĩ„¨ķĩ„¨ đ›Ŋ𝑛,𝑘0 đ›Ŋ𝑛−1,𝑘0 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨ķĩ„¨ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ķĩ„¨ ∑𝑁 ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
ķĩ„¨
+ 𝑀 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + 𝑘=1 ķĩ„¨
𝑏đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
Journal of Applied Mathematics
11
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›ž / (1 − 𝜎𝑛 ) − 𝛾𝑛−1 / (1 − 𝜎𝑛−1 )ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
đ›Ŋ𝑛,𝑘0
Proof. Let 𝑝 ∈ Ω. Then, by Lemma 11 we have
ķĩ„¨
ķĩ„¨
2 ķĩ„¨ķĩ„¨đ›ŧ − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„¨ 𝑛
đ›Ŋ𝑛,𝑘0
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= ķĩ„Šķĩ„Šķĩ„Šđ›Ŋ𝑛 (𝑄đ‘Ļ𝑛,𝑁 − 𝑝) + (1 − đ›Ŋ𝑛 ) (đ‘ĻĖƒđ‘›,𝑁 − 𝑝)ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 )
ķĩ„Š
ķĩ„Š2
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − 𝑃đļ(đŧ − 𝜆∇𝑓)𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 )
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›Ŋ − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
+
]
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ đ›Ŋ𝑛 𝜏 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
≤ [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Š 𝑛
đ›Ŋ𝑛−1,𝑘0
ķĩ„¨
𝑁 ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ ∑ ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
+ 𝑀 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + 𝑘=1 ķĩ„¨
𝑏đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Š2
× ķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓)𝑝 − 𝜆đ›ŧ𝑛 đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›ž / (1 − 𝜎𝑛 ) − 𝛾𝑛−1 / (1 − 𝜎𝑛−1 )ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
đ›Ŋ𝑛,𝑘0
ķĩ„¨ ķĩ„¨
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
2 ķĩ„¨ķĩ„¨đ›ŧ − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đœŽđ‘› − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„¨ 𝑛
+
+
]
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
1
+ (1 − 𝜌) đ›Ŋ𝑛 ⋅
= [1 − (1 − 𝜌) đ›Ŋ𝑛 ] ķĩ„Š 𝑛
đ›Ŋ𝑛−1,𝑘0
1−𝜌
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 )
ķĩ„Š2
ķĩ„Š
× [ķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Š
−2𝜆đ›ŧ𝑛 ⟨đ‘Ļ𝑛,𝑁, (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝⟩ ]
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 )
2 ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
× [ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆 (𝜆 − ) ķĩ„Šķĩ„Šķĩ„Š∇𝑓(đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)ķĩ„Šķĩ„Šķĩ„Š
đŋ
ķĩ„Š
ķĩ„Š
× {𝜏 ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
+2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š ]
ķĩ„¨
𝑁 ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨
ķĩ„¨đ‘Ÿ − 𝑟 ķĩ„¨ķĩ„¨ ∑ ķĩ„¨ķĩ„¨đ›Ŋ𝑛,𝑘 − đ›Ŋ𝑛−1,𝑘 ķĩ„¨ķĩ„¨ķĩ„¨
+ 𝑀 [ ķĩ„¨ 𝑛 𝑛−1 ķĩ„¨ + 𝑘=1 ķĩ„¨
𝑏đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 )
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đ›ž / (1 − 𝜎𝑛 ) − 𝛾𝑛−1 / (1 − 𝜎𝑛−1 )ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
2 ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
× [ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆 (𝜆 − ) ķĩ„Šķĩ„Šķĩ„Š∇𝑓(đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)ķĩ„Šķĩ„Šķĩ„Š
đŋ
ķĩ„¨ ķĩ„¨
ķĩ„¨
ķĩ„¨
2 ķĩ„¨ķĩ„¨đ›ŧ − đ›ŧ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨ ķĩ„¨ķĩ„¨ķĩ„¨đ›Ŋ𝑛 − đ›Ŋ𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ ķĩ„¨ 𝑛
+
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
+2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š ] .
(71)
ķĩ„¨
ķĩ„¨ķĩ„¨
ķĩ„¨đœŽ − 𝜎𝑛−1 ķĩ„¨ķĩ„¨ķĩ„¨
+ķĩ„¨ 𝑛
]} .
đ›Ŋ𝑛 đ›Ŋ𝑛,𝑘0
So, we obtain
(68)
Therefore, utilizing Lemma 9, from (H1), (H7), and the
asymptotical regularity of {đ‘Ĩ𝑛 } (due to Lemma 16), we deduce
that
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
= 0.
lim
𝑛→∞
đ›Ŋ𝑛,𝑘0
(69)
Lemma 20. Let one suppose that Ω =Ė¸ 0. Let one suppose that
(H1)–(H6) hold. Then,
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
lim ķĩ„Šķĩ„Šđ‘ĻĖƒ − đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š = lim ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š = 0.
𝑛 → ∞ ķĩ„Š 𝑛,𝑁
𝑛→∞
(70)
2
ķĩ„Š2
ķĩ„Š
− 𝜆) ķĩ„Šķĩ„Šķĩ„Š∇𝑓(đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)ķĩ„Šķĩ„Šķĩ„Š
đŋ
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ (1 − đ›Ŋ𝑛 ) 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
× (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
+ 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š)
(1 − đ›Ŋ𝑛 ) 2𝜆 (
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
+ 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š .
(72)
12
Journal of Applied Mathematics
Since đ›ŧ𝑛 → 0, đ›Ŋ𝑛 → 0, ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0, and 0 < 𝜆 < 2/đŋ,
from the boundedness of {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘Ļ𝑛,𝑁} it follows that
lim𝑛 → ∞ ‖∇𝑓(đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)‖ = 0, and hence
ķĩ„Š
ķĩ„Š
lim ķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š = 0.
(73)
𝑛→∞ ķĩ„Š
Moreover, from the firm nonexpansiveness of 𝑃đļ we obtain
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
= ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ(đŧ − 𝜆∇𝑓đ›ŧ𝑛 )đ‘Ļ𝑛,𝑁 − 𝑃đļ(đŧ − 𝜆∇𝑓)𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
1 ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š2
≤ {ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
2
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
− (1 − đ›Ŋ𝑛 ) 𝜆2 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š2
− ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
− (1 − đ›Ŋ𝑛 ) 𝜆2 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
2ķĩ„Š
ķĩ„Š
(74)
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
+ 2𝜆 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š .
and so
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
(77)
Since đ›Ŋ𝑛 → 0, ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0, and ‖∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)‖ →
0, from the boundedness of {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, {đ‘Ļ𝑛,𝑁}, and {đ‘ĻĖƒđ‘›,𝑁}, it
follows that lim𝑛 → ∞ ‖đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁‖ = 0. Observe that
+ 2𝜆 ⟨đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁, ∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)⟩
(75)
Thus, we have
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
− (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š = đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š ķŗ¨€→ 0 as 𝑛 ķŗ¨€→ ∞, (78)
and hence
ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šķĩ„Šđ‘ƒđļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛 −𝑃đļ (đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
= 2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − đ‘Ļ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Š ķŗ¨€→ 0 as 𝑛 ķŗ¨€→ ∞.
(79)
Thus, lim𝑛 → ∞ ‖đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ‖ = 0.
+ 2 (1−đ›Ŋ𝑛 ) 𝜆 ⟨đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁, ∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁)−∇𝑓 (𝑝)⟩
ķĩ„Š
ķĩ„Š2
− (1 − đ›Ŋ𝑛 ) 𝜆2 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š ,
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š2
− ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
+ 2 (1 − đ›Ŋ𝑛 ) 𝜆 ⟨đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁, ∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)⟩
+ 2𝜆 ⟨đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁, ∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)⟩
ķĩ„Š
ķĩ„Š2
− 𝜆2 ķĩ„Šķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š .
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
+ 2 (1 − đ›Ŋ𝑛 ) 𝜆⟨đ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁, ∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)⟩
1 ķĩ„Šķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
{ķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
2 ķĩ„Š
ķĩ„Š
ķĩ„Š2
− ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝 − (đ‘ĻĖƒđ‘›,𝑁 − 𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š }
ķĩ„Š2
− 𝜆 ķĩ„Šķĩ„Šķĩ„Š∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓 (𝑝)ķĩ„Šķĩ„Šķĩ„Šķĩ„Š } ,
ķĩ„Š2
ķĩ„Š
(1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘ĻĖƒđ‘›,𝑁ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
× ķĩ„Šķĩ„Šķĩ„Šķĩ„Š(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝ķĩ„Šķĩ„Šķĩ„Šķĩ„Š
≤ ⟨(đŧ − 𝜆∇𝑓đ›ŧ𝑛 ) đ‘Ļ𝑛,𝑁 − (đŧ − 𝜆∇𝑓) 𝑝, đ‘ĻĖƒđ‘›,𝑁 − 𝑝⟩
=
which implies that
(76)
Lemma 21. Let one suppose that Ω =Ė¸ 0. Let one suppose that
0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤ lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1 for each 𝑖 =
1, . . . , 𝑁. Moreover, suppose that (H1)–(H6) are satisfied. Then,
lim𝑛 → ∞ ‖𝑆𝑖 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ = 0 for each 𝑖 = 1, . . . , 𝑁.
Journal of Applied Mathematics
13
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
− đ›Ŋ𝑛,𝑁 (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑁−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
Proof. First of all, observe that
đ‘Ĩ𝑛+1 − đ‘Ļ𝑛 = 𝛾𝑛 (đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 )
= 𝛾𝑛 (đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ) + đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑇đ‘Ļ𝑛 ) + đ›ŋ𝑛 (𝑇đ‘Ļ𝑛 − đ‘Ļ𝑛 ) .
(80)
By Lemmas 16 and 20, we know that ‖đ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ‖ → 0 and
‖đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ‖ → 0 as 𝑛 → ∞. Hence, utilizing Lemma 7(i), we
have
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š 1 + 𝜁 ķĩ„Šķĩ„Š Ėƒ
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š +
ķĩ„Šđ‘Ļ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1−𝜁ķĩ„Š 𝑛
2 ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šđ‘ĻĖƒ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š ,
1−𝜁ķĩ„Š 𝑛
(81)
which together with ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0 implies that
lim𝑛 → ∞ ‖đ›ŋ𝑛 (𝑇đ‘Ļ𝑛 − đ‘Ļ𝑛 )‖ = 0. Taking into account
lim inf 𝑛 → ∞ đ›ŋ𝑛 > 0, we have
ķĩ„Š
ķĩ„Š
lim ķĩ„Šķĩ„Šđ‘‡đ‘Ļ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š = 0.
𝑛→∞ ķĩ„Š
So, we have
ķĩ„Š2
ķĩ„Š
đ›Ŋ𝑛,𝑁 (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑁−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ›ŋ𝑛 (𝑇đ‘Ļ𝑛 − đ‘Ļ𝑛 )ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ļ𝑛 − 𝛾𝑛 (đ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ) − đ›ŋ𝑛 (𝑇đ‘ĻĖƒđ‘› − 𝑇đ‘Ļ𝑛 )ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + 𝛾𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + đ›ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒđ‘› − 𝑇đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘ĻĖƒđ‘› − 𝑇đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š +
(83)
(82)
Let us show that for each 𝑖 ∈ {1, . . . , 𝑁}, one has ‖𝑆𝑖 đ‘ĸ𝑛 −
đ‘Ļ𝑛,𝑖−1 ‖ → 0 as 𝑛 → ∞. Let 𝑝 ∈ Ω. When 𝑖 = 𝑁, by
Lemma 11, we have from (33)-(34) the following:
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
= đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
= đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
+ (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁−1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
− đ›Ŋ𝑛,𝑁 (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑁−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
− đ›Ŋ𝑛,𝑁 (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑁−1 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
(84)
Since đ›ŧ𝑛 → 0, đ›Ŋ𝑛 → 0, 0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑁 ≤
lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑁 < 1, and lim𝑛 → ∞ ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ = 0, it is known
that {‖𝑆𝑁đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑁−1 ‖} is a null sequence.
Let 𝑖 ∈ {1, . . . , 𝑁 − 1}. Then, one has
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘›,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š2 ķĩ„Š
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
= đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘đ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
+ (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁−1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
+ (1 − đ›Ŋ𝑛,𝑁) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁−1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2
+ đ›Ŋ𝑛,𝑁ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
+ (1 − đ›Ŋ𝑛,𝑁) [đ›Ŋ𝑛,𝑁−1 ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘−1 đ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
+ (1 − đ›Ŋ𝑛,𝑁−1 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁−2 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ]
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
14
Journal of Applied Mathematics
ķĩ„Š2
ķĩ„Š
+ (đ›Ŋ𝑛,𝑁 + (1 − đ›Ŋ𝑛,𝑁) đ›Ŋ𝑛,𝑁−1 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
To conclude, we have that
𝑁
ķĩ„Š
ķĩ„Š2
+ ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁−2 − 𝑝ķĩ„Šķĩ„Šķĩ„Š ,
𝑘=𝑁−1
(85)
and so, after (𝑁 − 𝑖 + 1) iterations,
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑁
𝑁
𝑗=𝑖+2
𝑙=𝑗
+ (đ›Ŋ𝑛,𝑁 + ∑ (∏ (1 − đ›Ŋ𝑛,𝑙 )) đ›Ŋ𝑛,𝑗−1 )
𝑁
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
× ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
𝑘=𝑖+1
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑁
𝑁
𝑗=𝑖+2
𝑙=𝑗
+ (đ›Ŋ𝑛,𝑁 + ∑ (∏ (1 − đ›Ŋ𝑛,𝑙 )) đ›Ŋ𝑛,𝑗−1 )
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘†2 đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘†2 đ‘ĸ𝑛 − đ‘Ļ𝑛,1 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,1 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘†2 đ‘ĸ𝑛 − đ‘Ļ𝑛,1 ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛,1 ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š ,
(89)
from which ‖𝑆2 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ → 0. Thus, by induction ‖𝑆𝑖 đ‘ĸ𝑛 −
đ‘ĸ𝑛 ‖ → 0 for all 𝑖 = 2, . . . , 𝑁 since it is enough to observe that
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖−1 − 𝑆𝑖−1 đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘–−1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š
(90)
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛,𝑖−1 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘–−1 đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−2 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘–−1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š .
Remark 22. As an example, we consider 𝑁 = 2 and the following sequences:
(a) 𝜎𝑛 = 1/2 + 2/𝑛, 𝛾𝑛 = đ›ŋ𝑛 = 1/4 − 1/𝑛 for all 𝑛 > 4;
(b) đ›Ŋ𝑛 = 1/√𝑛, 𝑟𝑛 = 2 − 1/𝑛, for all 𝑛 > 1;
(c) đ›Ŋ𝑛,1 = 1/2 − 1/𝑛, đ›Ŋ𝑛,2 = 1/2 − 1/𝑛2 , for all 𝑛 > 2.
𝑁
ķĩ„Š
ķĩ„Š2
× ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ∏ (1 − đ›Ŋ𝑛,𝑘 )
Then, they satisfy the hypotheses on the parameter sequences
in Lemma 21.
𝑘=𝑖+1
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
× [đ›Ŋ𝑛,𝑖 ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛,𝑖 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑖−1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š2
ķĩ„Š
−đ›Ŋ𝑛,𝑖 (1 − đ›Ŋ𝑛,𝑖 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š ]
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑁
Lemma 23. Let one suppose that Ω =Ė¸ 0 and đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 for all
𝑖 as 𝑛 → ∞. Suppose there exists 𝑘 ∈ {1, . . . , 𝑁} such that
đ›Ŋ𝑛,𝑘 → 0 as 𝑛 → ∞. Let 𝑘0 ∈ {1, . . . , 𝑁} be the largest index
such that đ›Ŋ𝑛,𝑘0 → 0 as 𝑛 → ∞. Suppose that
(i) (đ›ŧ𝑛 + đ›Ŋ𝑛 )/đ›Ŋ𝑛,𝑘0 → 0 as 𝑛 → ∞;
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − đ›Ŋ𝑛,𝑖 ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š .
𝑘=𝑖
(86)
Again, we obtain that
(ii) if 𝑖 ≤ 𝑘0 and đ›Ŋ𝑛,𝑖 → 0, then đ›Ŋ𝑛,𝑘0 /đ›Ŋ𝑛,𝑖 → 0 as 𝑛 →
∞;
(iii) if đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0, then đ›Ŋ𝑖 lies in (0, 1).
𝑁
ķĩ„Š2
ķĩ„Š
đ›Ŋ𝑛,𝑖 ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š
Moreover, suppose that (H1), (H7), and (H8) hold. Then,
lim𝑛 → ∞ ‖𝑆𝑖 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ = 0 for each 𝑖 = 1, . . . , 𝑁.
𝑘=𝑖
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2 ķĩ„Š
ķĩ„Š2
+ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
+ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š .
(87)
Since đ›ŧ𝑛 → 0, đ›Ŋ𝑛 → 0, 0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤
lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1 for each 𝑖 = 1, . . . , 𝑁 − 1, and
lim𝑛 → ∞ ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ = 0, it is known that
ķĩ„Šķĩ„Š
lim ķĩ„Šđ‘† đ‘ĸ
𝑛→∞ ķĩ„Š 𝑖 𝑛
ķĩ„Š
− đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š = 0.
Obviously, for 𝑖 = 1, we have ‖𝑆1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ → 0.
(88)
Proof. First of all, we note that if (H7) holds, then also (H2)–
(H6) are satisfied. So {đ‘Ĩ𝑛 } is asymptotically regular. Let 𝑘0
be as in the hypotheses. As in Lemma 21, for every index
𝑖 ∈ {1, . . . , 𝑁} such that đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0 (which leads to
0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤ lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1), one has ‖𝑆𝑖 đ‘ĸ𝑛 −
đ‘Ļ𝑛,𝑖−1 ‖ → 0 as 𝑛 → ∞.
For all the other indexes 𝑖 ≤ 𝑘0 , we can prove that ‖𝑆𝑖 đ‘ĸ𝑛 −
đ‘Ļ𝑛,𝑖−1 ‖ → 0 as 𝑛 → ∞ in a similar manner. By the following
relation (due to (86)):
ķĩ„Š2
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (𝛾𝑛 + đ›ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − 𝑝ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š2
= 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − 𝑝ķĩ„Šķĩ„Šķĩ„Š
Journal of Applied Mathematics
15
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
Proof. By Remark 18, we have 𝜔𝑤 (đ‘Ĩ𝑛 ) = 𝜔𝑤 (đ‘ĸ𝑛 ) and 𝜔𝑠 (đ‘Ĩ𝑛 ) =
𝜔𝑠 (đ‘ĸ𝑛 ). Observe that
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š2
≤ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛,1 ķĩ„Šķĩ„Šķĩ„Š ≤ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,1 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š + đ›Ŋ𝑛,1 ķĩ„Šķĩ„Šķĩ„Šđ‘†1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ķĩ„Šķĩ„Šķĩ„Š .
ķĩ„Š
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
ķĩ„Š
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑁
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − đ›Ŋ𝑛,𝑖 ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š
By Lemmas 17 and 21, ‖đ‘Ĩ𝑛 − đ‘ĸ𝑛 ‖ → 0 and ‖𝑆1 đ‘ĸ𝑛 − đ‘ĸ𝑛 ‖ → 0
as 𝑛 → ∞, and hence
𝑘=𝑖
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Šķĩ„Š
lim ķĩ„Šđ‘Ĩ
𝑛→∞ ķĩ„Š 𝑛
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š2
≤ đ›Ŋ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 2𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
× (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑁
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
+ ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š − đ›Ŋ𝑛,𝑖 ∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š ,
𝑘=𝑖
(91)
we immediately obtain that
ķĩ„Š2
ķĩ„Š
∏ (1 − đ›Ŋ𝑛,𝑘 ) ķĩ„Šķĩ„Šķĩ„Šđ‘†đ‘– đ‘ĸ𝑛 − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š
(96)
So, we get 𝜔𝑤 (đ‘Ĩ𝑛 ) = 𝜔𝑤 (đ‘Ļ𝑛,1 ) and 𝜔𝑠 (đ‘Ĩ𝑛 ) = 𝜔𝑠 (đ‘Ļ𝑛,1 ).
Let 𝑝 ∈ 𝜔𝑤 (đ‘Ĩ𝑛 ). Since 𝑝 ∈ 𝜔𝑤 (đ‘ĸ𝑛 ), by Lemma 21
and Lemma 7(ii) (demiclosedness principle), we have 𝑝 ∈
Fix(𝑆𝑖 ) for all index 𝑖, that is, 𝑝 ∈ ⋂𝑖 Fix(𝑆𝑖 ). Taking
into consideration that 𝑇 is 𝜁-strictly pseudocontractive, by
Lemma 7(i), we get
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
≤ ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘Ĩ𝑛 − 𝑇đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘‡đ‘Ļ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
𝑘=𝑖
đ›Ŋ𝑛 ķĩ„Šķĩ„Š
ķĩ„Š2 đ›ŧ
ķĩ„Š ķĩ„Š
ķĩ„Šķĩ„Šđ‘„đ‘Ļ𝑛,𝑁 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝑛 2𝜆 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š
đ›Ŋ𝑛,𝑖
đ›Ŋ𝑛,𝑖
ķĩ„Š
− đ‘Ļ𝑛,1 ķĩ„Šķĩ„Šķĩ„Š = 0.
ķĩ„Š
ķĩ„Šķĩ„Š
ķĩ„Šķĩ„Šđ‘‡đ‘Ĩ𝑛 − đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š
𝑁
≤
(95)
(92)
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
× (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Šķĩ„Šđ‘Ĩ − đ‘Ĩ𝑛+1 ķĩ„Šķĩ„Šķĩ„Š
+ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − 𝑝ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − 𝑝ķĩ„Šķĩ„Šķĩ„Š) ķĩ„Š 𝑛
.
đ›Ŋ𝑛,𝑖
By Lemma 19 or by hypothesis (ii) of the sequences, we have
ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛+1 ķĩ„Šķĩ„Šķĩ„Š ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ𝑛+1 ķĩ„Šķĩ„Šķĩ„Š đ›Ŋ𝑛,𝑘0
=
⋅
ķŗ¨€→ 0.
(93)
đ›Ŋ𝑛,𝑖
đ›Ŋ𝑛,𝑘0
đ›Ŋ𝑛,𝑖
So, the thesis follows.
Remark 24. Let us consider 𝑁 = 3 and the following
sequences:
(a) đ›ŧ𝑛 = 1/𝑛5/4 , đ›Ŋ𝑛 = 1/𝑛1/2 , 𝑟𝑛 = 2 − 1/𝑛2 , for all 𝑛 > 1;
(b) 𝜎𝑛 = 1/2 + 2/𝑛2 , 𝛾𝑛 = đ›ŋ𝑛 = 1/4 − 1/𝑛2 , for all 𝑛 > 2;
(c) đ›Ŋ𝑛,1 = 1/𝑛1/4 , đ›Ŋ𝑛,2 = 1/2 − 1/𝑛2 , đ›Ŋ𝑛,3 = 1/𝑛1/3 , for all
𝑛 > 1.
It is easy to see that all hypotheses (i)–(iii), (H1), (H7), and
(H8) of Lemma 23 are satisfied.
Remark 25. Under the hypotheses of Lemma 23, analogously
to Lemma 21, one can see that
ķĩ„Š
ķĩ„Š
lim ķĩ„Šķĩ„Šđ‘† đ‘ĸ − đ‘Ļ𝑛,𝑖−1 ķĩ„Šķĩ„Šķĩ„Š = 0, ∀𝑖 ∈ {2, . . . , 𝑁} .
(94)
𝑛→∞ ķĩ„Š 𝑖 𝑛
Corollary 26. Let one suppose that the hypotheses of either
Lemma 21 or Lemma 23 are satisfied. Then, 𝜔𝑤 (đ‘Ĩ𝑛 ) =
𝜔𝑤 (đ‘ĸ𝑛 ) = 𝜔𝑤 (đ‘Ļ𝑛,1 ), 𝜔𝑠 (đ‘Ĩ𝑛 ) = 𝜔𝑠 (đ‘ĸ𝑛 ) = 𝜔𝑠 (đ‘Ļ𝑛,1 ), and
𝜔𝑤 (đ‘Ĩ𝑛 ) ⊂ Ω.
≤
1 + 𝜁 ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑇đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š
1−𝜁ķĩ„Š 𝑛
=
2 ķĩ„Šķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Šđ‘Ĩ − đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š + ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − 𝑇đ‘Ļ𝑛 ķĩ„Šķĩ„Šķĩ„Š ,
1−𝜁ķĩ„Š 𝑛
(97)
which together with ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0 (by Lemma 17) and ‖đ‘Ļ𝑛 −
𝑇đ‘Ļ𝑛 ‖ → 0 (by (82)) implies that
ķĩ„Šķĩ„Š
lim ķĩ„Šđ‘Ĩ
𝑛→∞ ķĩ„Š 𝑛
ķĩ„Š
− 𝑇đ‘Ĩ𝑛 ķĩ„Šķĩ„Šķĩ„Š = 0.
(98)
Utilizing Lemma 7(ii) (demiclosedness principle), we
have 𝑝 ∈ Fix(𝑇). Furthermore, by Lemmas 14 and 17, we
know that 𝑝 ∈ EP(𝐹, ℎ). Finally, by similar argument as in
[18], we can show that 𝑝 ∈ Γ, and as a result 𝑝 ∈ Ω.
Theorem 27. Let one suppose that Ω =Ė¸ 0. Let {đ›Ŋ𝑛 }, {đ›Ŋ𝑛,𝑖 }, 𝑖 =
1, . . . , 𝑁, be sequences in (0, 1) such that 0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤
lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1 for all index 𝑖. Moreover, Let one suppose
that (H1)–(H6) hold. Then, the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘ĸ𝑛 },
explicitly defined by scheme (30), all converge strongly to the
unique solution đ‘Ĩ∗ ∈ Ω of the following variational inequality:
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
∀𝑧 ∈ Ω.
(99)
Proof. Since the mapping đ‘ƒΩ đ‘„ is a 𝜌-contraction, it has a
unique fixed point đ‘Ĩ∗ ; it is the unique solution of (99). Since
(H1)–(H6) hold, the sequence {đ‘Ĩ𝑛 } is asymptotically regular
(by Lemma 16). In terms of Lemma 17, ‖đ‘Ĩ𝑛 − đ‘Ļ𝑛 ‖ → 0 and
16
Journal of Applied Mathematics
‖đ‘Ĩ𝑛 − đ‘ĸ𝑛 ‖ → 0 as 𝑛 → ∞. Moreover, utilizing Lemmas 8
and 10, we have from (33)-(34) the following:
we know that 𝜔𝑤 (đ‘Ĩ𝑛 ) ⊂ Ω, and hence 𝑧 ∈ Ω. Taking into
consideration that đ‘Ĩ∗ = đ‘ƒΩ đ‘„đ‘Ĩ∗ we obtain from (101) that
lim sup ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
ķĩ„Šķĩ„Š
∗ ķĩ„Š2
ķĩ„Šķĩ„Šđ‘Ĩ𝑛+1 − đ‘Ĩ ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š2
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĻĖƒđ‘› − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š
≤ 𝜎𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + (1 − 𝜎𝑛 ) (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
𝑛→∞
= lim sup [⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ĩ𝑛 − đ‘Ĩ∗ ⟩ + ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ𝑛 ⟩]
𝑛→∞
= lim sup ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ĩ𝑛 − đ‘Ĩ∗ ⟩
𝑛→∞
ķĩ„Š
ķĩ„Š ķĩ„Š2
ķĩ„Š
≤ (ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
= lim ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ĩ𝑛𝑘 − đ‘Ĩ∗ ⟩
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
ķĩ„Š
= ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ ķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š
ķĩ„Š2
≤ ķĩ„Šķĩ„Šķĩ„Šđ›Ŋ𝑛 (𝑄đ‘Ļ𝑛,𝑁 − 𝑄đ‘Ĩ∗ ) + (1 − đ›Ŋ𝑛 ) (đ‘ĻĖƒđ‘›,𝑁 − đ‘Ĩ∗ )ķĩ„Šķĩ„Šķĩ„Š
= ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0.
𝑘→∞
∗ķĩ„Š
ķĩ„Š2
∗
∗
(102)
∞
Since ∑∞
𝑛=0 đ›ŧ𝑛 < ∞ and ∑𝑛=0 đ›Ŋ𝑛 = ∞, we deduce that
∞
∗
∑𝑛=0 𝜆đ›ŧ𝑛 ‖𝑝‖(2‖đ‘Ļ𝑛 − đ‘Ĩ ‖ + 𝜆đ›ŧ𝑛 ‖𝑝‖) < ∞ and ∑∞
𝑛=0 (1 − 𝜌)đ›Ŋ𝑛 =
∞. In terms of Lemma 9 we derive đ‘Ĩ𝑛 → đ‘Ĩ∗ as 𝑛 → ∞.
∗
+ 2đ›Ŋ𝑛 ⟨𝑄đ‘Ĩ − đ‘Ĩ , đ‘Ļ𝑛 − đ‘Ĩ ⟩
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
In a similar way, we can derive the following result.
ķĩ„Š
ķĩ„Š2
ķĩ„Š
ķĩ„Š
≤ (đ›Ŋ𝑛 𝜌 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + (1 − đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š)
+ 2đ›Ŋ𝑛 ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
2ķĩ„Š
ķĩ„Š2
= (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛,𝑁 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š
(100)
+ 2đ›Ŋ𝑛 ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
(i) (đ›ŧ𝑛 + đ›Ŋ𝑛 )/đ›Ŋ𝑛,𝑘0 → 0 as 𝑛 → ∞;
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š2
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘ĸ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š
(ii) if 𝑖 ≤ 𝑘0 and đ›Ŋ𝑛,𝑖 → 0, then đ›Ŋ𝑛,𝑘0 /đ›Ŋ𝑛,𝑖 → 0 as 𝑛 →
∞;
(iii) if đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0, then đ›Ŋ𝑖 lies in (0, 1).
+ 2đ›Ŋ𝑛 ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
Then, the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘ĸ𝑛 } explicitly defined by
scheme (30) all converge strongly to the unique solution đ‘Ĩ∗ ∈ Ω
of the following variational inequality:
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š2
ķĩ„Š
≤ (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
+ 2đ›Ŋ𝑛 ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
∀𝑧 ∈ Ω.
(103)
Remark 29. According to the above argument processes for
Theorems 27 and 28, we can readily see that if in scheme (30),
the iterative step đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ļ𝑛,𝑁 +(1−đ›Ŋ𝑛 )𝑃đļ(đ‘Ļ𝑛,𝑁 −𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁))
is replaced by the iterative one đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ĩ𝑛 +(1−đ›Ŋ𝑛 )𝑃đļ(đ‘Ļ𝑛,𝑁 −
𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁)), then Theorems 27 and 28 remain valid.
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š)
ķĩ„Š2
ķĩ„Š
= (1 − (1 − 𝜌) đ›Ŋ𝑛 ) ķĩ„Šķĩ„Šķĩ„Šđ‘Ĩ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š
2
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ļ𝑛 − đ‘Ĩ∗ ⟩
1−𝜌
ķĩ„Š ķĩ„Š ķĩ„Š
ķĩ„Š
ķĩ„Š ķĩ„Š
+ 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š (2 ķĩ„Šķĩ„Šķĩ„Šđ‘Ļ𝑛 − đ‘Ĩ∗ ķĩ„Šķĩ„Šķĩ„Š + 𝜆đ›ŧ𝑛 ķĩ„Šķĩ„Šķĩ„Šđ‘ķĩ„Šķĩ„Šķĩ„Š) .
+ (1 − 𝜌) đ›Ŋ𝑛 ⋅
Remark 30. Theorems 27 and 28 improve, extend, supplement, and develop [17, Theorems 3.12 and 3.13] and [1,
Theorems 5.2 and 6.1] in the following aspects:
Now, let {đ‘Ĩ𝑛𝑘 } be a subsequence of {đ‘Ĩ𝑛 } such that
lim sup ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ĩ𝑛 − đ‘Ĩ∗ ⟩ = lim ⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , đ‘Ĩ𝑛𝑘 − đ‘Ĩ∗ ⟩ .
𝑛→∞
Theorem 28. Let one suppose that Ω =Ė¸ 0. Let {đ›Ŋ𝑛 }, {đ›Ŋ𝑛,𝑖 }, 𝑖 =
1, . . . , 𝑁, be sequences in (0, 1) such that đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 for all 𝑖 as
𝑛 → ∞. Suppose that there exists 𝑘 ∈ {1, . . . , 𝑁} for which
đ›Ŋ𝑛,𝑘 → 0 as 𝑛 → ∞. Let 𝑘0 ∈ {1, . . . , 𝑁} the largest index for
which đ›Ŋ𝑛,𝑘0 → 0. Moreover, let one suppose that (H1), (H7),
and (H8) hold, and
𝑘→∞
(101)
By the boundedness of {đ‘Ĩ𝑛 }, we may assume, without loss of
generality, that đ‘Ĩ𝑛𝑘 ⇀ 𝑧 ∈ 𝜔𝑤 (đ‘Ĩ𝑛 ). According to Corollary 26,
(a) the multistep iterative scheme (30) of [17] is extended
to develop our relaxed viscosity iterative scheme (30)
with regularization for MP (3), EP (10), and strict
pseudocontraction 𝑇 by virtue of Xu iterative schemes
in [1];
(b) the argument techniques in Theorems 27 and 28 are
very different from the ones in [17, Theorems 3.12 and
3.13] and the ones in [1, Theorems 5.2 and 6.1] because
we use the properties of strict pseudocontractive
mappings and maximal monotone mappings (see,
e.g., Lemmas 7 and 10);
Journal of Applied Mathematics
17
(c) compared with the proof of Theorems 5.2 and 6.1
in [1], the proof of Theorems 27 and 28 shows that
lim𝑛 → ∞ ‖đ‘Ļ𝑛,𝑁 − 𝑃đļ(đŧ − 𝜆∇𝑓đ›ŧ𝑛 )đ‘Ļ𝑛,𝑁‖ = lim𝑛 → ∞ ‖đ‘Ļ𝑛 −
𝑃đļ(đŧ − 𝜆∇𝑓đ›ŧ𝑛 )đ‘Ļ𝑛 ‖ = 0 via the argument of
lim𝑛 → ∞ ‖∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁) − ∇𝑓(𝑝)‖ = 0, for all 𝑝 ∈ Ω (see
Lemma 20 and its proof);
(d) the problem of finding an element of Fix(𝑇) ∩
(⋂𝑖 Fix(𝑆𝑖 )) ∩ EP(𝐹, ℎ) ∩ Γ in Theorems 27 and 28 is
more general than the one of finding an element of
Fix(𝑇) ∩ (⋂𝑖 Fix(𝑆𝑖 )) ∩ EP(𝐹, ℎ) in [17, Theorems 3.12
and 3.13] and the one of finding an element of Γ in [1,
Theorems 5.2 and 6.1].
us consider the following mixed problem of finding đ‘Ĩ∗ ∈
Fix(𝑇) ∩ EP(𝐹, ℎ) ∩ Γ such that
⟨(đŧ − 𝑆Ėƒ1 ) đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
∀đ‘Ļ ∈ Fix (𝑇) ∩ EP (𝐹, ℎ) ∩ Γ,
⟨(đŧ − 𝑆Ėƒ2 ) đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
∀đ‘Ļ ∈ Fix (𝑇) ∩ EP (𝐹, ℎ) ∩ Γ,
..
.
⟨(đŧ − 𝑆Ėƒđ‘€) đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
⟨𝐴 1 đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
∀đ‘Ļ ∈ đļ,
⟨𝐴 2 đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
∀đ‘Ļ ∈ đļ,
..
.
4. Applications
For a given nonlinear mapping 𝐴 : đļ → đģ, we consider
again the variational inequality problem (VIP) of finding đ‘Ĩ ∈
đļ such that
⟨𝐴đ‘Ĩ, đ‘Ļ − đ‘Ĩ⟩ ≥ 0,
∀đ‘Ļ ∈ Fix (𝑇) ∩ EP (𝐹, ℎ) ∩ Γ,
∀đ‘Ļ ∈ đļ.
(104)
Recall that if đ‘ĸ is a point in đļ, then the following relation
holds:
đ‘ĸ ∈ VI (đļ, 𝐴) ⇐⇒ đ‘ĸ = 𝑃đļ (đŧ − 𝜆𝐴) đ‘ĸ,
for some 𝜆 > 0,
(105)
from which we have the following relation:
⟨𝐴 𝑁đ‘Ĩ∗ , đ‘Ļ − đ‘Ĩ∗ ⟩ ≥ 0,
(108)
We denote by (SVI) the set of solutions of the above (𝑁 +
𝑀) system. This problem is equivalent to finding a common
𝑀
fixed point of 𝑇, {𝑃Fix(𝑇)∩EP(𝐹,ℎ)∩Γ đ‘†Ėƒđ‘– }𝑁
𝑖=1 , {𝑃đļ (đŧ − 𝜇𝐴 𝑖 )}𝑖=1 . The
following results are then consequences of Theorems 27 and
28.
Theorem 31. Let one suppose that Ω = Fix(𝑇) ∩ (SVI) ∩
EP(F, h) ∩ Γ =Ė¸ 0. Fix 𝜇 ∈ (0, 2đ›ŧ], and 𝜆 ∈ (0, 2/đŋ). Let {đ›ŧ𝑛 },
{đ›Ŋ𝑛,𝑖 }, 𝑖 = 1, . . . , (𝑀 + 𝑁), be sequences in (0, 1) such that
0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤ lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1 for all index
𝑖. Moreover, Let one suppose that (H1)–(H6) hold. Then the
sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘ĸ𝑛 } explicitly defined by the following
scheme:
𝐹 (đ‘ĸ𝑛 , đ‘Ļ) + ℎ (đ‘ĸ𝑛 , đ‘Ļ) +
đ‘ĸ ∈ Γ ⇐⇒ đ‘ĸ ∈ VI (đļ, ∇𝑓) ⇐⇒ đ‘ĸ = 𝑃đļ (đŧ − 𝜆∇𝑓) đ‘ĸ,
for some 𝜆 > 0.
(106)
∀đ‘Ĩ, đ‘Ļ ∈ đļ.
1
⟨đ‘Ļ − đ‘ĸ𝑛 , đ‘ĸ𝑛 − đ‘Ĩ𝑛 ⟩ ≥ 0,
𝑟𝑛
∀đ‘Ļ ∈ đļ,
đ‘Ļ𝑛,1 = đ›Ŋ𝑛,1 𝑃Fix(𝑇)∩EP(𝐹,ℎ)∩Γ đ‘†Ėƒ1 đ‘ĸ𝑛 + (1 − đ›Ŋ𝑛,1 ) đ‘ĸ𝑛 ,
đ‘Ļ𝑛,𝑖 = đ›Ŋ𝑛,𝑖 𝑃Fix(𝑇)∩EP(𝐹,ℎ)∩Γ đ‘†Ėƒđ‘– đ‘ĸ𝑛
An operator 𝐴 : đļ → đģ is said to be an đ›ŧ-inverse
strongly monotone operator if there exists a constant đ›ŧ > 0
such that
ķĩ„Š2
ķĩ„Š
⟨𝐴đ‘Ĩ − 𝐴đ‘Ļ, đ‘Ĩ − đ‘Ļ⟩ ≥ đ›ŧķĩ„Šķĩ„Šķĩ„Šđ´đ‘Ĩ − 𝐴đ‘Ļķĩ„Šķĩ„Šķĩ„Š ,
∀đ‘Ļ ∈ đļ.
(107)
As an example, we recall that the đ›ŧ-inverse strongly
monotone operators are firmly nonexpansive mappings if đ›ŧ ≥
1 and that every đ›ŧ-inverse strongly monotone operator is also
a (1/đ›ŧ) Lipschitz continuous (see [19]). We observe that, if 𝐴
is đ›ŧ-inverse strongly monotone, the mapping 𝑃đļ(đŧ − 𝜇𝐴) is
nonexpansive for all 𝜇 ∈ (0, 2đ›ŧ] since they are compositions
of nonexpansive mappings (see [19, page 419]).
Let us consider 𝑆Ėƒ1 , . . . , 𝑆Ėƒđ‘€ a finite number of nonexpansive self-mappings on đļ and 𝐴 1 , . . . , 𝐴 𝑁 be a finite number
of đ›ŧ-inverse strongly monotone operators. Let 𝑇 : đļ → đļ
be a 𝜁-strict pseudocontraction on đļ with fixed points. Let
+ (1 − đ›Ŋ𝑛,𝑖 ) đ‘Ļ𝑛,𝑖−1 ,
𝑖 = 2, . . . , 𝑀,
đ‘Ļ𝑛,𝑀+𝑗 = đ›Ŋ𝑛,𝑀+𝑗 𝑃đļ (đŧ − 𝜇𝐴 𝑗 ) đ‘ĸ𝑛
+ (1 − đ›Ŋ𝑛,𝑀+𝑗 ) đ‘Ļ𝑛,𝑀+𝑗−1 ,
𝑗 = 1, . . . , 𝑁,
đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ļ𝑛,𝑀+𝑁 + (1 − đ›Ŋ𝑛 )
× đ‘ƒđļ (đ‘Ļ𝑛,𝑀+𝑁 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑀+𝑁)) ,
đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + 𝛾𝑛 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 ))
+ đ›ŋ𝑛 𝑇𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 )) ,
∀𝑛 ≥ 0,
(109)
all converge strongly to the unique solution đ‘Ĩ∗ ∈ Ω of the
following variational inequality:
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
∀𝑧 ∈ Ω.
(110)
18
Journal of Applied Mathematics
Theorem 32. Let one suppose that Ω =Ė¸ 0. Fix 𝜇 ∈ (0, 2đ›ŧ] and
𝜆 ∈ (0, 2/đŋ). Let {đ›Ŋ𝑛 }, {đ›Ŋ𝑛,𝑖 }, 𝑖 = 1, . . . , (𝑀 + 𝑁), be sequences
in (0, 1) and đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 for all 𝑖 as 𝑛 → ∞. Suppose that there
exists 𝑘 ∈ {1, . . . , 𝑀 + 𝑁} such that đ›Ŋ𝑛,𝑘 → 0 as 𝑛 → ∞. Let
𝑘0 ∈ {1, . . . , 𝑀 + 𝑁} be the largest index for which đ›Ŋ𝑛,𝑘0 → 0.
Moreover, let one suppose that (H1), (H7), and (H8) hold, and
(⋂𝑖 Fix(𝑆𝑖 )) ∩ EP(𝐹, ℎ) ∩ Γ = (⋂𝑖 Fix(𝑆𝑖 )) ∩ EP(𝐹, ℎ) ∩ Γ =Ė¸ 0,
(𝛾𝑛 + đ›ŋ𝑛 )𝜁 ≤ 𝛾𝑛 , for all 𝑛 ≥ 0, and
đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + 𝛾𝑛 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 ))
+ đ›ŋ𝑛 𝑇𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 ))
(i) (đ›ŧ𝑛 + đ›Ŋ𝑛 )/đ›Ŋ𝑛,𝑘0 → 0 as 𝑛 → ∞;
= 𝜎𝑛 đ‘Ļ𝑛 + đ›ŋ𝑛 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 ))
(ii) if 𝑖 ≤ 𝑘0 and đ›Ŋ𝑛,𝑖 → 0, then đ›Ŋ𝑛,𝑘0 /đ›Ŋ𝑛,𝑖 → 0 as 𝑛 →
∞;
= 𝜎𝑛 đ‘Ļ𝑛 + (1 − 𝜎𝑛 ) 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 )) .
(iii) if đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0, then đ›Ŋ𝑖 lies in (0, 1).
Then, the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘ĸ𝑛 } explicitly defined by
scheme (109) all converge strongly to the unique solution đ‘Ĩ∗ ∈
Ω of the following variational inequality:
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
∀𝑧 ∈ Ω.
(111)
Remark 33. If we choose ∇𝑓 = 𝐴 1 = ⋅ ⋅ ⋅ = 𝐴 𝑁 = 0 in system
(108), we obtain a system of hierarchical fixed point problems
introduced by Moudafi and MaingeĖ [20, 21].
Further, utilizing Theorems 27 and 28, we again give
the following strong convergence theorems for finding a
common element of the solution set Γ of MP (3), the solution
set EP(𝐹, ℎ) of EP (10), and the common fixed point set
(⋂𝑖 Fix(𝑆𝑖 )) of a finite family of nonexpansive mappings 𝑆𝑖 :
đļ → đļ, 𝑖 = 1, . . . , 𝑁.
Theorem 34. Let one suppose that Ω = (⋂𝑖 Fix(𝑆𝑖 )) ∩
EP(𝐹, ℎ) ∩ Γ =Ė¸ 0. Let {đ›Ŋ𝑛 }, {đ›Ŋ𝑛,𝑖 }, 𝑖 = 1, . . . , 𝑁, be sequences
in (0, 1) such that 0 < lim inf 𝑛 → ∞ đ›Ŋ𝑛,𝑖 ≤ lim sup𝑛 → ∞ đ›Ŋ𝑛,𝑖 < 1
for all index 𝑖. Moreover, Let one suppose that there hold (H1)–
(H6) with 𝛾𝑛 = 0, for all 𝑛 ≥ 0. Then, the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 },
and {đ‘ĸ𝑛 } generated explicitly by
𝐹 (đ‘ĸ𝑛 , đ‘Ļ) + ℎ (đ‘ĸ𝑛 , đ‘Ļ) +
1
⟨đ‘Ļ − đ‘ĸ𝑛 , đ‘ĸ𝑛 − đ‘Ĩ𝑛 ⟩ ≥ 0,
𝑟𝑛
∀đ‘Ļ ∈ đļ,
đ‘Ļ𝑛,1 = đ›Ŋ𝑛,1 𝑆1 đ‘ĸ𝑛 + (1 − đ›Ŋ𝑛,1 ) đ‘ĸ𝑛 ,
đ‘Ļ𝑛,𝑖 = đ›Ŋ𝑛,𝑖 𝑆𝑖 đ‘ĸ𝑛 + (1 − đ›Ŋ𝑛,𝑖 ) đ‘Ļ𝑛,𝑖−1 ,
đ‘Ļ𝑛 = đ›Ŋ𝑛 𝑄đ‘Ļ𝑛,𝑁 + (1 − đ›Ŋ𝑛 ) 𝑃đļ (đ‘Ļ𝑛,𝑁 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛,𝑁)) ,
∀𝑛 ≥ 0,
(112)
all converge strongly to the unique solution đ‘Ĩ∗ ∈ Ω of the
following variational inequality:
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
∀𝑧 ∈ Ω.
Thus, the conditions in Theorem 27 are all satisfied. and from
which we obtain the desired result.
Theorem 35. Let one suppose that Ω = (⋂𝑖 Fix(𝑆𝑖 )) ∩
EP(𝐹, ℎ) ∩ Γ =Ė¸ 0. Let {đ›Ŋ𝑛 }, {đ›Ŋ𝑛,𝑖 }, 𝑖 = 1, . . . , 𝑁, be sequences
in (0, 1) such that đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 for all 𝑖 as 𝑛 → ∞. Suppose that
there exists 𝑘 ∈ {1, . . . , 𝑁} for which đ›Ŋ𝑛,𝑘 → 0 as 𝑛 → ∞.
Let 𝑘0 ∈ {1, . . . , 𝑁} be the largest index for which đ›Ŋ𝑛,𝑘0 → 0.
Moreover, let one suppose that there hold (H1), (H7), and (H8)
with 𝛾𝑛 = 0, for all 𝑛 ≥ 0, and
(i) (đ›ŧ𝑛 + đ›Ŋ𝑛 )/đ›Ŋ𝑛,𝑘0 → 0 as 𝑛 → ∞;
(ii) if 𝑖 ≤ 𝑘0 and đ›Ŋ𝑛,𝑖 → 0, then đ›Ŋ𝑛,𝑘0 /đ›Ŋ𝑛,𝑖 → 0 as 𝑛 →
∞;
(iii) if đ›Ŋ𝑛,𝑖 → đ›Ŋ𝑖 =Ė¸ 0, then đ›Ŋ𝑖 lies in (0, 1).
Then the sequences {đ‘Ĩ𝑛 }, {đ‘Ļ𝑛 }, and {đ‘ĸ𝑛 } generated explicitly by
(112) all converge strongly to the unique solution đ‘Ĩ∗ ∈ Ω of the
following variational inequality:
⟨𝑄đ‘Ĩ∗ − đ‘Ĩ∗ , 𝑧 − đ‘Ĩ∗ ⟩ ≤ 0,
∀𝑧 ∈ Ω.
(115)
Acknowledgments
This research was partially supported by the National Science
Foundation of China (11071169), Innovation Program of
Shanghai Municipal Education Commission (09ZZ133), and
Leading Academic Discipline Project of Shanghai Normal
University (DZL707). This research was partially supported
by the Grant from the NSC 101-2115-M-037-001.
References
𝑖 = 2, . . . , 𝑁,
đ‘Ĩ𝑛+1 = 𝜎𝑛 đ‘Ļ𝑛 + (1 − 𝜎𝑛 ) 𝑃đļ (đ‘Ļ𝑛 − 𝜆∇𝑓đ›ŧ𝑛 (đ‘Ļ𝑛 )) ,
(114)
(113)
Proof. In Theorems 27, put 𝑇 = đŧ the identity mapping and
𝛾𝑛 = 0, for all 𝑛 ≥ 0. Then, 𝑇 is a 𝜁-strictly pseudocontractive
mapping with 𝜁 = 0. Hence, we deduce that Ω = Fix(𝑇) ∩
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Volume 2014
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
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