Volume 2013, Article ID 616549,14pages http://dx.doi.org/10.1155/2013/616549
Research Article
Mann-Type Extragradient Methods for General Systems of
Variational Inequalities with Multivalued Variational Inclusion
Constraints in Banach Spaces
Lu-Chuan Ceng,
1Abdul Latif,
2and Saleh A. Al-Mezel
21Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities,
Shanghai 200234, China
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Abdul Latif; alatif@kau.edu.sa
Received 13 September 2013; Accepted 30 October 2013 Academic Editor: Jen-Chih Yao
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 Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.
1. Introduction
Let𝑋be a real Banach space whose dual space is denoted by
𝑋∗. The normalized duality mapping𝐽 : 𝑋 → 2𝑋∗
is defined by
𝐽 (𝑥) = {𝑥∗∈ 𝑋∗ : ⟨𝑥, 𝑥∗⟩ = ‖𝑥‖2= 𝑥∗2} , ∀𝑥 ∈ 𝑋,
(1)
where⟨⋅, ⋅⟩denotes the generalized duality pairing. It is an
immediate consequence of the Hahn-Banach theorem that
𝐽(𝑥) is nonempty for each 𝑥 ∈ 𝑋. Let 𝐶 be a nonempty
closed convex subset of𝑋. A mapping𝑇 : 𝐶 → 𝐶is called
nonexpansive if‖𝑇𝑥 − 𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖for every𝑥, 𝑦 ∈ 𝐶.
The set of fixed points of𝑇is denoted by Fix(𝑇). We use the
notation⇀to indicate the weak convergence and the one→
to indicate the strong convergence. A mapping𝐴 : 𝐶 → 𝑋
is said to be as follows:
(i) accretive if for each𝑥, 𝑦 ∈ 𝐶there exists𝑗(𝑥 − 𝑦) ∈
𝐽(𝑥 − 𝑦)such that
⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 0; (2)
(ii)𝛼-strongly accretive if for each𝑥, 𝑦 ∈ 𝐶there exists
𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦)such that
⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝛼𝑥 − 𝑦2, (3)
for some𝛼 ∈ (0, 1);
(iii)𝛽-inverse-strongly-accretive if for each𝑥, 𝑦 ∈ 𝐶there
exists𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦)such that
⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝛽𝐴𝑥 − 𝐴𝑦2, (4)
for some𝛽 > 0;
(iv)𝜆-strictly pseudocontractive if for each𝑥, 𝑦 ∈ 𝐶there
exists𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦)such that
⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 𝑥 − 𝑦2− 𝜆𝑥 − 𝑦 − (𝐴𝑥 − 𝐴𝑦)2,
(5)
Let𝑈 = {𝑥 ∈ 𝑋 : ‖𝑥‖ = 1}denote the unite sphere of𝑋.
A Banach space𝑋is said to be uniformly convex if for each
𝜖 ∈ (0, 2], there exists𝛿 > 0such that for all𝑥, 𝑦 ∈ 𝑈
𝑥 − 𝑦 ≥ 𝜖 ⇒ 𝑥 + 𝑦2 ≤ 1 − 𝛿 . (6)
It is known that a uniformly convex Banach space is reflexive
and strictly convex. A Banach space𝑋is said to be smooth if
the limit
lim
𝑡 → 0𝑥 + 𝑡𝑦 − ‖𝑥‖𝑡 (7)
exists for all𝑥, 𝑦 ∈ 𝑈; in this case,𝑋 is also said to have
a G´ateaux differentiable norm. Moreover, it is said to be
uniformly smooth if this limit is attained uniformly for𝑥, 𝑦 ∈
𝑈. The norm of𝑋is said to be the Fr´echet differential if for
each𝑥 ∈ 𝑈, this limit is attained uniformly for𝑦 ∈ 𝑈. In the
meantime, we define a function𝜌 : [0, ∞) → [0, ∞)called
the modulus of smoothness of𝑋as follows:
𝜌 (𝜏) = sup{1
2(𝑥 + 𝑦 + 𝑥 − 𝑦) − 1 :
𝑥, 𝑦 ∈ 𝑋, ‖𝑥‖ = 1, 𝑦 = 𝜏}. (8)
It is known that 𝑋 is uniformly smooth if and only if
lim𝜏 → 0𝜌(𝜏)/𝜏 = 0. Let𝑞be a fixed real number with1 < 𝑞 ≤
2. Then a Banach space𝑋is said to be𝑞-uniformly smooth if
there exists a constant𝑐 > 0such that𝜌(𝜏) ≤ 𝑐𝜏𝑞for all𝜏 > 0.
As pointed out in [1], no Banach space is𝑞-uniformly smooth
for𝑞 > 2. In addition, it is also known that𝐽is single-valued
if and only if𝑋is smooth, whereas if𝑋is uniformly smooth,
then𝐽is norm-to-norm uniformly continuous on bounded
subsets of𝑋.
Very recently, Cai and Bu [2] considered the following general system of variational inequalities (GSVI) in a real
smooth Banach space𝑋, which involves finding(𝑥∗, 𝑦∗) ∈
𝐶 × 𝐶such that
⟨𝜇1𝐵1𝑦∗+ 𝑥∗− 𝑦∗, 𝐽 (𝑥 − 𝑥∗)⟩ ≥ 0, ∀𝑥 ∈ 𝐶,
⟨𝜇2𝐵2𝑥∗+ 𝑦∗− 𝑥∗, 𝐽 (𝑥 − 𝑦∗)⟩ ≥ 0, ∀𝑥 ∈ 𝐶, (9)
where 𝐶 is a nonempty, closed, and convex subset of 𝑋;
𝐵1, 𝐵2 : 𝐶 → 𝑋are two nonlinear mappings, and𝜇1and𝜇2
are two positive constants. Here the set of solutions of GSVI
(9) is denoted by GSVI(𝐶, 𝐵1, 𝐵2). In particular, if𝑋 = 𝐻,
a real Hilbert space, then GSVI (9) reduces to the following
GSVI of finding(𝑥∗, 𝑦∗) ∈ 𝐶 × 𝐶such that
⟨𝜇1𝐵1𝑦∗+ 𝑥∗− 𝑦∗, 𝑥 − 𝑥∗⟩ ≥ 0, ∀𝑥 ∈ 𝐶,
⟨𝜇2𝐵2𝑥∗+ 𝑦∗− 𝑥∗, 𝑥 − 𝑦∗⟩ ≥ 0, ∀𝑥 ∈ 𝐶, (10)
where 𝜇1 and 𝜇2 are two positive constants. The set of
solutions of problem (10) is still denoted by GSVI(𝐶, 𝐵1, 𝐵2).
Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.
Lemma 1 (see [3]). For given𝑥, 𝑦 ∈ 𝐶, (𝑥, 𝑦)is a solution of
problem(10)if and only if𝑥is a fixed point of the mapping
𝐺 : 𝐶 → 𝐶defined by
𝐺 (𝑥) = 𝑃𝐶[𝑃𝐶(𝑥 − 𝜇2𝐵2𝑥) − 𝜇1𝐵1𝑃𝐶(𝑥 − 𝜇2𝐵2𝑥)] ,
∀𝑥 ∈ 𝐶, (11)
where𝑦 = 𝑃𝐶(𝑥−𝜇2𝐵2𝑥)and𝑃𝐶is the the projection of𝐻onto
𝐶.
In particular, if the mappings𝐵𝑖 : 𝐶 → 𝐻is𝛽𝑖-inverse
strongly monotone for𝑖 = 1, 2, then the mapping𝐺is
nonex-pansive provided𝜇𝑖∈ (0, 2𝛽𝑖)for𝑖 = 1, 2.
Define the mapping𝐺 : 𝐶 → 𝐶as follows:
𝐺 (𝑥) := Π𝐶(𝐼 − 𝜇1𝐵1) Π𝐶(𝐼 − 𝜇2𝐵2) 𝑥, ∀𝑥 ∈ 𝐶. (12)
The fixed point set of𝐺is denoted byΩ.
Let 𝐶𝐵(𝑋) be the family of all nonempty, closed and
bounded subsets of a real smooth Banach space𝑋. Also, we
denote by𝐻(⋅, ⋅)the Hausdorff metric on𝐶𝐵(𝑋)defined by
𝐻 (𝐴, 𝐵) := max{sup
𝑥∈𝐵𝑦∈𝐴inf𝑑 (𝑥, 𝑦) ,sup𝑥∈𝐴inf𝑦∈𝐵𝑑 (𝑥, 𝑦)} ,
∀𝐴, 𝐵 ∈ 𝐶𝐵 (𝑋) . (13)
Let𝑇and𝐹 : 𝑋 → 𝐶𝐵(𝑋)be two multivalued mappings,
let𝐴 : 𝐷(𝐴) ⊂ 𝑋 → 2𝑋be an𝑚-accretive mapping, let
𝑔 : 𝑋 → 𝐷(𝐴)be a single-valued mapping, and let𝑁(⋅, ⋅) :
𝑋 × 𝑋 → 𝑋be a nonlinear mapping. Then for any given
V ∈ 𝑋, 𝜆 > 0, Chidume et al. [4] introduced and studied
the multivalued variational inclusion (MVVI) of finding𝑥 ∈
𝐷(𝐴)such that(𝑥, 𝑤, 𝑘)is a solution of the following:
V∈ 𝑁 (𝑤, 𝑘) + 𝜆𝐴 (𝑔 (𝑥)) , ∀𝑤 ∈ 𝑇𝑥, 𝑘 ∈ 𝐹𝑥. (14)
IfV = 0and𝜆 = 1, then the MVVI (14) reduces to the
problem of finding𝑥 ∈ 𝐷(𝐴)such that(𝑥, 𝑤, 𝑘)is a solution
of the following:
0 ∈ 𝑁 (𝑤, 𝑘) + 𝐴 (𝑔 (𝑥)) , ∀𝑤 ∈ 𝑇𝑥, 𝑘 ∈ 𝐹𝑥. (15)
We denote byΓthe set of such solutions𝑥for MVVI (15).
The authors [4] first established an existence theorem for
MVVI (14) in smooth Banach space𝑋and then proved that
the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (15).
Theorem 2 (see [4, Theorem 3.2]). Let 𝑋 be a real smooth
Banach space. Let𝑇and𝐹 : 𝑋 → 𝐶𝐵(𝑋), let𝐴 : 𝐷(𝐴) ⊂
𝑋 → 2𝑋be three multivalued mappings, let𝑔 : 𝑋 → 𝐷(𝐴)
be a single-valued mapping, and let𝑁(⋅, ⋅) : 𝑋 × 𝑋 → 𝑋be a
single-valued continuous mapping satisfying the following con-ditions:
(C1)𝐴 ∘ 𝑔 : 𝑋 → 2𝑋is𝑚-accretive and𝐻-uniformly
con-tinuous;
(C2)𝑇 : 𝑋 → 𝐶𝐵(𝑋)is𝐻-uniformly continuous;
(C4)the mapping𝑥 → 𝑁(𝑥, 𝑦)is𝜙-strongly accretive and
𝜇-𝐻-Lipschitz with respect to the mapping𝑇, where𝜙 :
[0, ∞) → [0, ∞)is a strictly increasing function with
𝜙(0) = 0;
(C5)the mapping 𝑦 → 𝑁(𝑥, 𝑦) is accretive and
𝜉-𝐻-Lipschitz with respect to the mapping𝐹.
For arbitrary𝑥0∈ 𝐷(𝐴)define the sequence{𝑥𝑛}iteratively by
𝑥𝑛+1= 𝑥𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) , 𝑢𝑛∈ 𝐴 (𝑔 (𝑥𝑛)) , (16)
where{𝑢𝑛}is defined by
𝑢𝑛− 𝑢𝑛+1 ≤ (1 + 𝜀)𝐻(𝐴(𝑔(𝑥𝑛+1)) , 𝐴 (𝑔 (𝑥𝑛))) ,
∀𝑛 ≥ 0, (17)
for any𝑤𝑛 ∈ 𝑇𝑥𝑛,𝑘𝑛 ∈ 𝐹𝑥𝑛and some𝜀 > 0, where{𝜎𝑛}is a
positive real sequence such thatlim𝑛 → ∞𝜎𝑛= 0and∑∞𝑛=0𝜎𝑛=
∞. Then, there exists𝑑 > 0such that for0 < 𝜎𝑛 ≤ 𝑑, for all
𝑛 ≥ 0,{𝑥𝑛}converges strongly to𝑥 ∈ Γ; and for any𝑤 ∈ 𝑇𝑥,
𝑘 ∈ 𝐹𝑥,(𝑥, 𝑤, 𝑘)is a solution of the MVVI(15).
Let 𝐶 be a nonempty closed convex subset of a real
smooth Banach space𝑋and letΠ𝐶be a sunny nonexpansive
retraction from 𝑋 onto 𝐶. Motivated and inspired by the
research going on this area, we introduce Mann-type extra-gradient methods for finding solutions of the GSVI (9) which are also ones of the MVVI (15) and common fixed points of a countable family of nonexpansive mappings. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm
in the setting of uniformly convex and2-uniformly smooth
Banach space, and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see for example, [2–10].
2. Preliminaries
Let𝑋be a real Banach space with dual𝑋∗. We denote by𝐽
the normalized duality mapping from𝑋to2𝑋∗defined by
𝐽 (𝑥) = {𝑥∗∈ 𝑋∗: ⟨𝑥, 𝑥∗⟩ = ‖𝑥‖2= 𝑥∗2} , (18)
where⟨⋅, ⋅⟩denotes the generalized duality pairing.
Through-out this paper the single-valued normalized duality map is
still denoted by𝐽. Unless otherwise stated, we assume that𝑋
is a smooth Banach space with dual𝑋∗.
A multivalued mapping𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 2𝑋is said to
be as follows: (i) accretive, if
⟨𝑢 −V, 𝐽 (𝑥 − 𝑦)⟩ ≥ 0, ∀𝑢 ∈ 𝐴𝑥, V∈ 𝐴𝑦; (19)
(ii)𝑚-accretive, if𝐴is accretive and(𝐼 + 𝑟𝐴)(𝐷(𝐴)) = 𝑋,
for all𝑟 > 0, where𝐼is the identity mapping;
(iii)𝜁-inverse strongly accretive, if there exists a constant
𝜁 > 0such that
⟨𝑢 −V, 𝐽 (𝑥 − 𝑦)⟩ ≥ 𝜁‖𝑢 −V‖2, ∀𝑢 ∈ 𝐴𝑥, V∈ 𝐴𝑦; (20)
(iv)𝜙-strongly accretive, if there exists a strictly increasing
continuous function 𝜙 : [0, ∞) → [0, ∞) with
𝜙(0) = 0such that
⟨𝑢 −V, 𝐽 (𝑥 − 𝑦)⟩ ≥ 𝜙 (𝑥 − 𝑦)𝑥 − 𝑦,
∀𝑢 ∈ 𝐴𝑥, V∈ 𝐴𝑦; (21)
(v)𝜙-expansive, if
‖𝑢 −V‖ ≥ 𝜙 (𝑥 − 𝑦), ∀𝑢 ∈ 𝐴𝑥, V∈ 𝐴𝑦. (22)
It is easy to see that if𝐴is𝜙-strongly accretive, then𝐴is
𝜙-expansive.
A mapping𝑇 : 𝑋 → 𝐶𝐵(𝑋)is said to be𝐻-uniformly
continuous, if for any given𝜀 > 0there exists a𝛿 > 0such
that whenever‖𝑥 − 𝑦‖ < 𝛿then𝐻(𝑇𝑥, 𝑇𝑦) < 𝜀.
A mapping𝑁 : 𝑋 × 𝑋 → 𝑋is𝜙-strongly accretive, with
respect to𝑇 : 𝑋 → 𝐶𝐵(𝑋), in the first argument if
⟨𝑁 (𝑢, 𝑧) − 𝑁 (V, 𝑧) , 𝐽 (𝑥 − 𝑦)⟩
≥ 𝜙 (𝑥 − 𝑦)𝑥 − 𝑦, ∀𝑢 ∈ 𝑇𝑥, V∈ 𝑇𝑦. (23)
A mapping𝑆 : 𝑋 → 2𝑋is called lower semicontinuous if
𝑆−1(𝑉) := {𝑥 ∈ 𝑋 : 𝑆𝑥 ∩ 𝑉 ̸= 0}is open in𝑋whenever𝑉 ⊂ 𝑌
is open.
We list some propositions and lemmas that will be used in the sequel.
Proposition 3 (see [11]). Let {𝜆𝑛} and{𝑏𝑛}be sequences of
nonnegative numbers and{𝛼𝑛} ⊂ (0, 1)a sequence satisfying
the conditions that{𝜆𝑛}is bounded,∑∞𝑛=0𝛼𝑛 = ∞, and𝑏𝑛 →
0, as𝑛 → ∞. Let the recursive inequality
𝜆2𝑛+1≤ 𝜆2𝑛− 2𝛼𝑛𝜓 (𝜆𝑛+1) + 2𝛼𝑛𝑏𝑛𝜆𝑛+1, ∀𝑛 ≥ 0, (24)
be given where𝜓 : [0, ∞) → [0, ∞)is a strictly increasing
function such that it is positive on(0, ∞)and𝜓(0) = 0. Then
𝜆𝑛 → 0, as𝑛 → ∞.
Proposition 4 (see [12]). Let𝑋be a real smooth Banach space.
Let𝑇and𝐹 : 𝑋 → 2𝑋be two multivalued mappings, and let
𝑁(⋅, ⋅) : 𝑋 × 𝑋 → 𝑋be a nonlinear mapping satisfying the
following conditions:
(i) the mapping𝑥 → 𝑁(𝑥, 𝑦)is𝜙-strongly accretive with
respect to the mapping𝑇;
(ii) the mapping𝑦 → 𝑁(𝑥, 𝑦)is accretive with respect to
the mapping𝐹.
Then the mapping𝑆 : 𝑋 → 2𝑋defined by𝑆𝑥 = 𝑁(𝑇𝑥, 𝐹𝑥)is
Proposition 5 (see [13]). Let𝑋be a real Banach space and let
𝑆 : 𝑋 → 2𝑋\ {0}be a lower semicontinuous and𝜙-strongly
accretive mapping; then for any𝑥 ∈ 𝑋, 𝑆𝑥is a one-point set;
that is,𝑆is a single-valued mapping.
Recall that a Banach space 𝑋 is said to satisfy Opial’s
condition, if whenever {𝑥𝑛} is a sequence in 𝑋 which
converges weakly to𝑥as𝑛 → ∞, then
lim sup
𝑛 → ∞ 𝑥𝑛− 𝑥 <lim sup𝑛 → ∞ 𝑥𝑛− 𝑦 ,
∀𝑦 ∈ 𝑋with𝑥 ̸= 𝑦.
(25)
Lemma 6 (Demiclosedness principle; see [14, Lemma 2]). Let
𝐶be a nonempty closed convex subset of a reflexive Banach
space𝑋that satisfies Opial’s condition and suppose that𝑇 :
𝐶 → 𝑋is nonexpansive. Then the mapping𝐼−𝑇is demiclosed
at zero, that is,𝑥𝑛⇀ 𝑥and𝑥𝑛− 𝑇𝑥𝑛 → 0imply𝑥 = 𝑇𝑥; that
is,𝑥 ∈Fix(𝑇).
The following lemma is an immediate consequence of the
subdifferential inequality of the function(1/2)‖ ⋅ ‖2.
Lemma 7. In a real smooth Banach space𝑋, there holds the
inequality
𝑥 + 𝑦2≤ ‖𝑥‖2+ 2⟨𝑦, 𝐽 (𝑥 + 𝑦)⟩, ∀𝑥, 𝑦 ∈ 𝑋. (26)
Let𝐷be a subset of𝐶and letΠbe a mapping of𝐶into𝐷.
ThenΠis said to be sunny if
Π [Π (𝑥) + 𝑡 (𝑥 − Π (𝑥))] = Π (𝑥) , (27)
wheneverΠ(𝑥) + 𝑡(𝑥 − Π(𝑥)) ∈ 𝐶for𝑥 ∈ 𝐶and𝑡 ≥ 0. A
mappingΠof𝐶into itself is called a retraction ifΠ2 = Π. If
a mappingΠof𝐶into itself is a retraction, thenΠ(𝑧) = 𝑧for
every𝑧 ∈ 𝑅(Π), where𝑅(Π)is the range ofΠ. A subset𝐷of
𝐶is called a sunny nonexpansive retract of𝐶if there exists a
sunny nonexpansive retraction from𝐶onto𝐷. The following
lemma concerns the sunny nonexpansive retraction.
Lemma 8 (see [15]). Let𝐶be a nonempty closed convex subset
of a real smooth Banach space𝑋. Let𝐷be a nonempty subset
of𝐶. LetΠbe a retraction of𝐶onto𝐷. Then the following are
equivalent:
(i)Πis sunny and nonexpansive;
(ii)‖Π(𝑥) − Π(𝑦)‖2 ≤ ⟨𝑥 − 𝑦, 𝐽(Π(𝑥) − Π(𝑦))⟩, for all
𝑥, 𝑦 ∈ 𝐶;
(iii)⟨𝑥 − Π(𝑥), 𝐽(𝑦 − Π(𝑥))⟩ ≤ 0, for all𝑥 ∈ 𝐶,𝑦 ∈ 𝐷.
It is well known that if 𝑋 = 𝐻 a Hilbert space, then
a sunny nonexpansive retractionΠ𝐶 is coincident with the
metric projection from𝑋 onto 𝐶; that is, Π𝐶 = 𝑃𝐶. If 𝐶
is a nonempty closed convex subset of a strictly convex and
uniformly smooth Banach space𝑋and if𝑇 : 𝐶 → 𝐶 is
a nonexpansive mapping with the fixed point set Fix(𝑇) ̸= 0,
then the set Fix(𝑇)is a sunny nonexpansive retract of𝐶. The
following result is an easy consequence ofLemma 8.
Lemma 9. Let 𝐶 be a nonempty closed convex subset of a
smooth Banach space 𝑋. Let Π𝐶 be a sunny nonexpansive
retraction from𝑋onto𝐶and let𝐵1, 𝐵2: 𝐶 → 𝑋be nonlinear
mappings. For given𝑥∗, 𝑦∗∈ 𝐶, (𝑥∗, 𝑦∗)is a solution of GSVI
(9)if and only if𝑥∗= Π𝐶(𝑦∗− 𝜇1𝐵1𝑦∗), where𝑦∗= Π𝐶(𝑥∗−
𝜇2𝐵2𝑥∗).
In terms ofLemma 9, we observe that
𝑥∗ = Π𝐶[Π𝐶(𝑥∗− 𝜇2𝐵2𝑥∗) − 𝜇1𝐵1Π𝐶(𝑥∗− 𝜇2𝐵2𝑥∗)] ,
(28)
which implies that 𝑥∗ is a fixed point of the mapping 𝐺.
Throughout this paper, the set of fixed points of the mapping
𝐺is denoted byΩ.
Lemma 10 (see [16]). Given a number𝑟 > 0. A real Banach
space 𝑋 is uniformly convex if and only if there exists a
continuous strictly increasing function𝜑 : [0, ∞) → [0, ∞),
𝜑(0) = 0, such that
𝜆𝑥 + (1 − 𝜆)𝑦2≤ 𝜆‖𝑥‖2+ (1 − 𝜆) 𝑦2
− 𝜆 (1 − 𝜆) 𝜑 (𝑥 − 𝑦), (29)
for all𝜆 ∈ [0, 1], and𝑥, 𝑦 ∈ 𝑋such that‖𝑥‖ ≤ 𝑟and‖𝑦‖ ≤ 𝑟.
Lemma 11 (see [17]). Let𝐶be a nonempty closed convex subset
of a Banach space𝑋. Let𝑆0, 𝑆1, . . .be a sequence of mappings
of𝐶into itself. Suppose that∑∞𝑛=1sup{‖𝑆𝑛𝑥−𝑆𝑛−1𝑥‖ : 𝑥 ∈ 𝐶} <
∞. Then for each𝑦 ∈ 𝐶,{𝑆𝑛𝑦}converges strongly to some point
of𝐶. Moreover, let𝑆be a mapping of𝐶into itself defined by
𝑆𝑦 =lim𝑛 → ∞𝑆𝑛𝑦for all𝑦 ∈ 𝐶. Thenlim𝑛 → ∞sup{‖𝑆𝑥−𝑆𝑛𝑥‖ : 𝑥 ∈ 𝐶} = 0.
3. Mann-Type Extragradient Algorithms in
Uniformly Convex and 2-Uniformly Smooth
Banach Spaces
In this section, we introduce Mann-type extragradient algo-rithms in uniformly convex and 2-uniformly smooth Banach spaces and show weak and strong convergence theorems. We will use some useful lemmas in the sequel.
Lemma 12 (see [2, Lemma 2.8]). Let𝐶be a nonempty closed
convex subset of a real2-uniformly smooth Banach space𝑋.
Let the mapping𝐵𝑖 : 𝐶 → 𝑋be𝛼𝑖-inverse-strongly accretive.
Then, we have (𝐼 − 𝜇𝑖𝐵𝑖) 𝑥 − (𝐼 − 𝜇𝑖𝐵𝑖) 𝑦2 ≤ 𝑥 − 𝑦2+ 2𝜇 𝑖(𝜇𝑖𝜅2− 𝛼𝑖) 𝐵𝑖𝑥 − 𝐵𝑖𝑦2, ∀𝑥, 𝑦 ∈ 𝐶, (30)
for𝑖 = 1, 2, where𝜇𝑖> 0. In particular, if0 < 𝜇𝑖 ≤ 𝛼𝑖/𝜅2, then
𝐼 − 𝜇𝑖𝐵𝑖is nonexpansive for𝑖 = 1, 2.
Lemma 13 (see [2, Lemma 2.9]). Let𝐶be a nonempty closed
LetΠ𝐶be a sunny nonexpansive retraction from𝑋onto𝐶. Let
the mapping𝐵𝑖 : 𝐶 → 𝑋be𝛼𝑖-inverse-strongly accretive for
𝑖 = 1, 2. Let𝐺 : 𝐶 → 𝐶be the mapping defined by
𝐺𝑥 = Π𝐶[Π𝐶(𝑥 − 𝜇2𝐵2𝑥) − 𝜇1𝐵1Π𝐶(𝑥 − 𝜇2𝐵2𝑥)] ,
∀𝑥 ∈ 𝐶. (31)
If0 < 𝜇𝑖≤ 𝛼𝑖/𝜅2for𝑖 = 1, 2, then𝐺 : 𝐶 → 𝐶is nonexpansive.
Theorem 14. Let 𝑋 be a uniformly convex and2-uniformly
smooth Banach space satisfying Opial’s condition and let𝐶be a
nonempty closed convex subset of𝑋such that𝐶±𝐶 ⊂ 𝐶. LetΠ𝐶
be a sunny nonexpansive retraction from𝑋onto𝐶. Let𝑇and
𝐹 : 𝑋 → CB(𝑋)and let𝐴 : 𝐶 → 2𝐶be three multivalued
mappings, let𝑔 : 𝑋 → 𝐶be a single-valued mapping, and let
𝑁(⋅, ⋅) : 𝑋 × 𝑋 → 𝐶be a single-valued continuous mapping
satisfying conditions(𝐶1)–(𝐶5)inTheorem 2and
(C6)𝑁(𝑇𝑥, 𝐹𝑥) + 𝐴(𝑔(𝑥)) : 𝑋 → 2𝐶 \ {0}is𝜁-inverse
strongly accretive with𝜁 ≥ 𝜅2.
Let𝐵𝑖 : 𝐶 → 𝑋be𝛼𝑖-inverse strongly accretive for𝑖 = 1, 2.
Let{𝑆𝑖}∞𝑖=0be a countable family of nonexpansive mappings of𝐶
into itself such thatΔ := ⋂∞𝑖=0Fix(𝑆𝑖)∩Ω∩Γ ̸= 0, whereΩis the
fixed point set of the mapping𝐺 = Π𝐶(𝐼 − 𝜇1𝐵1)Π𝐶(𝐼 − 𝜇2𝐵2)
with0 < 𝜇𝑖 < 𝛼𝑖/𝜅2for𝑖 = 1, 2. Assume that{𝛼𝑛},{𝛽𝑛}, and
{𝜎𝑛}are sequences in[0, 1]such that
(i)0 <lim inf𝑛 → ∞𝛼𝑛 ≤lim sup𝑛 → ∞𝛼𝑛< 1;
(ii)0 <lim inf𝑛 → ∞𝛽𝑛≤lim sup𝑛 → ∞𝛽𝑛 < 1;
(iii)0 <lim inf𝑛 → ∞𝜎𝑛≤lim sup𝑛 → ∞𝜎𝑛 < 1.
For arbitrary𝑥0∈ 𝐶define the sequence{𝑥𝑛}iteratively by
𝑦𝑛= 𝛽𝑛𝑆𝑛𝑥𝑛+ (1 − 𝛽𝑛) Π𝐶(𝐼 − 𝜇1𝐵1) × Π𝐶(𝐼 − 𝜇2𝐵2) 𝑥𝑛, 𝑥𝑛+1= 𝛼𝑛[𝑥𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)] + (1 − 𝛼𝑛) 𝑦𝑛, 𝑢𝑛 ∈ 𝐴 (𝑔 (𝑥𝑛)) , ∀𝑛 ≥ 0, (32) where{𝑢𝑛}is defined by 𝑢𝑛− 𝑢𝑛+1 ≤ (1 + 𝜀)𝐻(𝐴(𝑔(𝑥𝑛+1)) , 𝐴 (𝑔 (𝑥𝑛))) , ∀𝑛 ≥ 0, (33)
for any𝑤𝑛 ∈ 𝑇𝑥𝑛,𝑘𝑛 ∈ 𝐹𝑥𝑛 and some 𝜀 > 0. Assume
that∑∞𝑛=0sup𝑥∈𝐷‖𝑆𝑛+1𝑥 − 𝑆𝑛𝑥‖ < ∞for any bounded subset
𝐷 of 𝐶 and let𝑆 be a mapping of𝐶 into itself defined by
𝑆𝑥 = lim𝑛 → ∞𝑆𝑛𝑥for all𝑥 ∈ 𝐶and suppose thatFix(𝑆) =
⋂∞𝑖=0Fix(𝑆𝑖). Then{𝑥𝑛}converges weakly to some𝑥 ∈ Δ, and
for any𝑤 ∈ 𝑇𝑥,𝑘 ∈ 𝐹𝑥, (𝑥, 𝑤, 𝑘)is a solution of the MVVI
(15).
Proof. First of all, let us show that for anyV ∈ 𝐶,𝜆 > 0,
there exists a point ̃𝑥 ∈ 𝐶such that(̃𝑥, 𝑤, 𝑘)is a solution
of the MVVI (14), for any𝑤 ∈ 𝑇̃𝑥and 𝑘 ∈ 𝐹̃𝑥. Indeed,
following the argument idea in the proof of Chidume et al.
[4, Theorem 3.1], we put 𝑉𝑥 := 𝑁(𝑇𝑥, 𝐹𝑥)for all𝑥 ∈ 𝑋.
Then by Proposition 4, 𝑉 is 𝜙-strongly accretive. Since 𝑇
and𝐹are𝐻-uniformly continuous and𝑁(⋅, ⋅)is continuous,
𝑉𝑥is continuous and hence lower semicontinuous. Thus, by
Proposition 5𝑉𝑥 is single-valued. Moreover, since 𝑉is
𝜙-strongly accretive and by assumption 𝐴 ∘ 𝑔 : 𝑋 → 2𝐶
is 𝑚-accretive, we have that𝑉 + 𝜆𝐴 ∘ 𝑔is an 𝑚-accretive
and𝜙-strongly accretive mapping, and hence by Cioranescu
[18, page 184] for any𝑥 ∈ 𝑋we have(𝑉 + 𝜆𝐴 ∘ 𝑔)(𝑥) is
closed and bounded. Therefore, by Morales [19],𝑉 + 𝜆𝐴 ∘ 𝑔
is surjective. Hence, for anyV ∈ 𝑋and 𝜆 > 0there exists
̃𝑥 ∈ 𝐷(𝐴) = 𝐶such thatV ∈ 𝑉̃𝑥 + 𝜆𝐴(𝑔(̃𝑥)) = 𝑁(𝑤, 𝑘) +
𝜆𝐴(𝑔(̃𝑥)), where𝑤 ∈ 𝑇̃𝑥and𝑘 ∈ 𝐹̃𝑥. In addition, in terms
ofProposition 5we know that𝑉 + 𝜆𝐴 ∘ 𝑔is a single-valued
mapping. Assume that𝑁(𝑇𝑥, 𝐹𝑥) + 𝜆𝐴(𝑔(𝑥)) : 𝑋 → 𝐶is
𝜁-inverse strongly accretive with𝜁 ≥ 𝜅2. Then by Lemma12, we
conclude that the mapping𝑥 → 𝑥 − (𝑁(𝑇𝑥, 𝐹𝑥) + 𝜆𝐴(𝑔(𝑥)))
is nonexpansive. Meantime, byLemma 13we know that𝐺 :
𝐶 → 𝐶is also nonexpansive.
Without loss of generality we may assume thatV= 0and
𝜆 = 1. Let𝑝 ∈ Δand let𝑟 > 0be sufficiently large such that
𝑥0∈ 𝐵𝑟(𝑝) =: 𝐵. Then𝑝 ∈ 𝐷(𝐴) = 𝐶such that0 ∈ 𝑁(𝑤, 𝑘) +
𝐴 ∘ 𝑔(𝑝)for any𝑤 ∈ 𝑇𝑝and𝑘 ∈ 𝐹𝑝. Let𝑀 := sup{‖𝑢‖ :
𝑢 ∈ 𝑁(𝑤, 𝑘) + 𝐴(𝑔(𝑥)), 𝑥 ∈ 𝐵, 𝑤 ∈ 𝑇𝑥, 𝑘 ∈ 𝐹𝑥}. Then as
𝐴 ∘ 𝑔, 𝑇and𝐹are𝐻-uniformly continuous on𝑋, for𝜀1 :=
𝜙(𝑟)/8(1 + 𝜀), and𝜀2 := 𝜙(𝑟)/8𝜇(1 + 𝜀),𝜀3 := 𝜙(𝑟)/8𝜉(1 + 𝜀),
there exist𝛿1, 𝛿2, 𝛿3> 0such that for any𝑥, 𝑦 ∈ 𝑋,‖𝑥 − 𝑦‖ <
𝛿1,‖𝑥−𝑦‖ < 𝛿2and‖𝑥−𝑦‖ < 𝛿3imply𝐻(𝐴∘𝑔(𝑥), 𝐴∘𝑔(𝑦)) <
𝜀1,𝐻(𝑇𝑥, 𝑇𝑦) < 𝜀2and𝐻(𝐹𝑥, 𝐹𝑦) < 𝜀3, respectively.
Let us show that𝑥𝑛 ∈ 𝐵for all𝑛 ≥ 0. We show this by
induction. First,𝑥0∈ 𝐵by construction. Assume that𝑥𝑛 ∈ 𝐵.
We show that𝑥𝑛+1 ∈ 𝐵. If possible we assume that𝑥𝑛+1 ∉ 𝐵,
then‖𝑥𝑛+1− 𝑝‖ > 𝑟. Further from (32) it follows that
𝑦𝑛− 𝑝 = 𝛽𝑛𝑆𝑛𝑥𝑛+ (1 − 𝛽𝑛) Π𝐶(𝐼 − 𝜇1𝐵1) Π𝐶 × (𝐼 − 𝜇2𝐵2) 𝑥𝑛− 𝑝 ≤ 𝛽𝑛𝑆𝑛𝑥𝑛− 𝑝 + (1 − 𝛽𝑛) 𝐺𝑥𝑛− 𝑝 ≤ 𝛽𝑛𝑥𝑛− 𝑝 + (1 − 𝛽𝑛) 𝑥𝑛− 𝑝 = 𝑥𝑛− 𝑝 , (34) and hence 𝑥𝑛+1− 𝑝2 = ⟨𝛼𝑛[𝑥𝑛− 𝑝 − 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)] + (1 − 𝛼𝑛) (𝑦𝑛− 𝑝) , 𝐽 (𝑥𝑛+1− 𝑝)⟩ = ⟨𝛼𝑛(𝑥𝑛− 𝑝) + (1 − 𝛼𝑛) × (𝑦𝑛− 𝑝) , 𝐽 (𝑥𝑛+1− 𝑝)⟩ − 𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩
≤ 𝛼𝑛(𝑥𝑛− 𝑝) + (1 − 𝛼𝑛) (𝑦𝑛− 𝑝) × 𝑥𝑛+1− 𝑝 − 𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩ ≤ (𝛼𝑛𝑥𝑛− 𝑝 + (1 − 𝛼𝑛) 𝑥𝑛− 𝑝) × 𝑥𝑛+1− 𝑝 − 𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩ = 𝑥𝑛− 𝑝𝑥𝑛+1− 𝑝 − 𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩ ≤ 12(𝑥𝑛− 𝑝2+ 𝑥𝑛+1− 𝑝2) − 𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩ , (35)
which immediately yields
𝑥𝑛+1− 𝑝2 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛, 𝐽 (𝑥𝑛+1− 𝑝)⟩ = 𝑥𝑛− 𝑝2 − 2𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛+1, 𝑘𝑛+1) + 𝑢𝑛+1, 𝐽 (𝑥𝑛+1− 𝑝)⟩ − 2𝛼𝑛𝜎𝑛⟨𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛− (𝑁 (𝑤𝑛+1, 𝑘𝑛+1) + 𝑢𝑛+1) , 𝐽 (𝑥𝑛+1− 𝑝)⟩ . (36)
Since 𝑁(⋅, ⋅) is 𝜙-strongly accretive with respect to 𝑇 and
𝐴(𝑔(⋅))is accretive, we deduce from (36) that
𝑥𝑛+1− 𝑝2 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑥𝑛+1− 𝑝)𝑥𝑛+1− 𝑝 + 2𝛼𝑛𝜎𝑛[𝑁 (𝑤𝑛+1, 𝑘𝑛+1) − 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛+1− 𝑢𝑛]𝑥𝑛+1− 𝑝 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑥𝑛+1− 𝑝)𝑥𝑛+1− 𝑝 + 2𝛼𝑛𝜎𝑛[𝑁 (𝑤𝑛+1, 𝑘𝑛+1) − 𝑁 (𝑤𝑛+1, 𝑘𝑛) + 𝑁 (𝑤𝑛+1, 𝑘𝑛) − 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛+1− 𝑢𝑛]𝑥𝑛+1− 𝑝 . (37)
Again from (32) we have that
𝑥𝑛+1− 𝑝 ≤ 𝛼𝑛𝑥𝑛− 𝑝 − 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) + (1 − 𝛼𝑛) 𝑦𝑛− 𝑝 ≤ 𝛼𝑛[𝑥𝑛− 𝑝 + 𝜎𝑛𝑁(𝑤𝑛, 𝑘𝑛) + 𝑢𝑛] + (1 − 𝛼𝑛) 𝑥𝑛− 𝑝 ≤ 𝛼𝑛[𝑟 + 𝜎𝑛𝑀] + (1 − 𝛼𝑛) 𝑟 ≤ 2𝑟. (38)
Also, fromProposition 5,𝑉𝑥 = 𝑁(𝑇𝑥, 𝐹𝑥)is a single-valued
mapping; that is, for any𝑘, 𝑘 ∈ 𝐹𝑥and𝑤, 𝑤 ∈ 𝑇𝑥we have
𝑁(𝑤, 𝑘) = 𝑁(𝑤, 𝑘)and𝑁(𝑤, 𝑘) = 𝑁(𝑤, 𝑘). On the other
hand, it follows from Nadler [20] that, for𝑘𝑛+1 ∈ 𝐹𝑥𝑛+1and
𝑤𝑛+1∈ 𝑇𝑥𝑛+1, there exist𝑘𝑛 ∈ 𝐹𝑥𝑛and𝑤𝑛∈ 𝑇𝑥𝑛such that
𝑘𝑛+1− 𝑘𝑛 ≤ (1 + 𝜀) 𝐻 (𝐹𝑥𝑛+1, 𝐹𝑥𝑛) ,
𝑤𝑛+1− 𝑤𝑛 ≤ (1 + 𝜀) 𝐻 (𝑇𝑥𝑛+1, 𝑇𝑥𝑛) ,
(39)
respectively. Therefore, from (37) and (33), we have that
𝑥𝑛+1− 𝑝2 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑟) 𝑟 + 2𝛼𝑛𝜎𝑛[𝑁(𝑤𝑛+1, 𝑘𝑛+1) − 𝑁 (𝑤𝑛+1, 𝑘𝑛) + 𝑁(𝑤𝑛+1, 𝑘𝑛) − 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛+1− 𝑢𝑛]2𝑟 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑟) 𝑟 + 2𝛼𝑛𝜎𝑛[𝜉 (1 + 𝜀) 𝐻 (𝐹𝑥𝑛+1, 𝐹𝑥𝑛) + 𝜇 (1 + 𝜀) 𝐻 (𝑇𝑥𝑛+1, 𝑇𝑥𝑛) + (1 + 𝜀) 𝐻 (𝐴 (𝑔 (𝑥𝑛+1)) , 𝐴 (𝑔 (𝑥𝑛)))] 2𝑟 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑟) 𝑟 + 2𝛼𝑛𝜎𝑛[𝜙 (𝑟) 8 + 𝜙 (𝑟) 8 + 𝜙 (𝑟) 8 ] 2𝑟 = 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑟) 𝑟 + 𝛼𝑛𝜎𝑛32𝜙 (𝑟) 𝑟 ≤ 𝑥𝑛− 𝑝2. (40)
So, we get‖𝑥𝑛+1− 𝑝‖ ≤ 𝑟, a contradiction. Therefore,{𝑥𝑛}is
bounded.
Let us show that lim𝑛 → ∞‖𝑥𝑛− 𝑦𝑛‖ = 0and lim𝑛 → ∞‖𝑥𝑛−
Indeed, utilizingLemma 10and the nonexpansivity of the
mapping𝑥 → 𝑥−(𝑁(𝑇𝑥, 𝐹𝑥)+𝐴(𝑔(𝑥))), we obtain from (32)
that for all𝑛 ≥ 0
𝑥𝑛+1− 𝑝2 = 𝛼𝑛[𝑥𝑛− 𝑝 − 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)] + (1 − 𝛼𝑛) (𝑦𝑛− 𝑝)2 ≤ 𝛼𝑛𝑥𝑛− 𝑝 − 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)2 + (1 − 𝛼𝑛) 𝑦𝑛− 𝑝2− 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) = 𝛼𝑛(1 − 𝜎𝑛) (𝑥𝑛− 𝑝) + 𝜎𝑛[𝑥𝑛− (𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) − 𝑝]2 + (1 − 𝛼𝑛) 𝑦𝑛− 𝑝2− 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) ≤ 𝛼𝑛[(1 − 𝜎𝑛) 𝑥𝑛− 𝑝2 + 𝜎𝑛𝑥𝑛− (𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) − 𝑝2 − 𝜎𝑛(1 − 𝜎𝑛) 𝜑1(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)] + (1 − 𝛼𝑛) 𝑥𝑛− 𝑝2− 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) ≤ 𝛼𝑛[(1 − 𝜎𝑛) 𝑥𝑛− 𝑝2+ 𝜎𝑛𝑥𝑛− 𝑝2 − 𝜎𝑛(1 − 𝜎𝑛) 𝜑1(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)] + (1 − 𝛼𝑛) 𝑥𝑛− 𝑝2− 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) = 𝛼𝑛𝑥𝑛− 𝑝2− 𝛼𝑛𝜎𝑛(1 − 𝜎𝑛) 𝜑1 × (𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) + (1 − 𝛼𝑛) 𝑥𝑛− 𝑝2− 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) = 𝑥𝑛− 𝑝2 − 𝛼𝑛𝜎𝑛(1 − 𝜎𝑛) 𝜑1(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) − 𝛼𝑛(1 − 𝛼𝑛) 𝜑 × (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) ≤ 𝑥𝑛− 𝑝2. (41)
It is easy to see that the limit lim𝑛 → ∞‖𝑥𝑛 − 𝑝‖ exists.
Meantime, it can be readily seen from (41) that
𝛼𝑛𝜎𝑛(1 − 𝜎𝑛) 𝜑1(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)
+ 𝛼𝑛(1 − 𝛼𝑛) 𝜑
× (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛))
≤ 𝑥𝑛− 𝑝2− 𝑥𝑛+1− 𝑝2,
(42)
which together with conditions (i) and (iii) and the existence
of lim𝑛 → ∞‖𝑥𝑛− 𝑝‖, implies that
lim
𝑛 → ∞𝜑1(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) = 0,
lim
𝑛 → ∞𝜑 (𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛)) = 0.
(43)
Utilizing the properties of𝜑and𝜑1, we get
lim 𝑛 → ∞𝑁(𝑤𝑛, 𝑘𝑛) + 𝑢𝑛 = 0, lim 𝑛 → ∞𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) = 0. (44) Note that 𝑥𝑛− 𝑦𝑛 ≤ 𝑥𝑛− 𝑦𝑛− 𝜎𝑛(𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛) + 𝜎𝑛𝑁(𝑤𝑛, 𝑘𝑛) + 𝑢𝑛. (45) So, we have lim 𝑛 → ∞𝑥𝑛− 𝑦𝑛 = 0. (46)
Also, observe that
𝑥𝑛+1− 𝑥𝑛 ≤ 𝛼𝑛𝜎𝑛𝑁(𝑤𝑛, 𝑘𝑛) + 𝑢𝑛
+ (1 − 𝛼𝑛) 𝑥𝑛− 𝑦𝑛
≤ 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛 + 𝑥𝑛− 𝑦𝑛.
(47)
Thus, from (44) and (46) it follows that lim
𝑛 → ∞𝑥𝑛+1− 𝑥𝑛 = 0. (48)
Let us show that lim𝑛 → ∞‖𝑥𝑛 − 𝐺𝑥𝑛‖ = lim𝑛 → ∞‖𝑥𝑛 −
Indeed, for simplicity, put 𝑞 = Π𝐶(𝑝 − 𝜇2𝐵2𝑝), 𝑢𝑛 =
Π𝐶(𝑥𝑛− 𝜇2𝐵2𝑥𝑛)andV𝑛 = Π𝐶(𝑢𝑛− 𝜇1𝐵1𝑢𝑛). ThenV𝑛= 𝐺𝑥𝑛
for all𝑛 ≥ 0. FromLemma 12we have
𝑢𝑛− 𝑞2 = Π𝐶(𝑥𝑛− 𝜇2𝐵2𝑥𝑛) − Π𝐶(𝑝 − 𝜇2𝐵2𝑝)2 ≤ 𝑥𝑛− 𝑝 − 𝜇2(𝐵2𝑥𝑛− 𝐵2𝑝)2 ≤ 𝑥𝑛− 𝑝2− 2𝜇2(𝛼2− 𝜅2𝜇2) × 𝐵2𝑥𝑛− 𝐵2𝑝2, (49) V𝑛− 𝑝2 = Π𝐶(𝑢𝑛− 𝜇1𝐵1𝑢𝑛) − Π𝐶(𝑞 − 𝜇1𝐵1𝑞)2 ≤ 𝑢𝑛− 𝑞 − 𝜇1(𝐵1𝑢𝑛− 𝐵1𝑞)2 ≤ 𝑢𝑛− 𝑞2− 2𝜇1(𝛼1− 𝜅2𝜇1) × 𝐵1𝑢𝑛− 𝐵1𝑞2. (50)
Substituting (49) for (50), we obtain
V𝑛− 𝑝2
≤ 𝑥𝑛− 𝑝2
− 2𝜇2(𝛼2− 𝜅2𝜇2) 𝐵2𝑥𝑛− 𝐵2𝑝2
− 2𝜇1(𝛼1− 𝜅2𝜇1) 𝐵1𝑢𝑛− 𝐵1𝑞2.
(51)
Utilizing [21, Proposition 1] andLemma 10, from (32) and (51)
we have 𝑦𝑛− 𝑝2 ≤ 𝛽𝑛𝑆𝑛𝑥𝑛− 𝑝2+ (1 − 𝛽𝑛) 𝐺𝑥𝑛− 𝑝2 − 𝛽𝑛(1 − 𝛽𝑛) 𝜑2(𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛) ≤ 𝛽𝑛𝑥𝑛− 𝑝2+ (1 − 𝛽𝑛) V𝑛− 𝑝2 − 𝛽𝑛(1 − 𝛽𝑛) 𝜑2(𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛) ≤ 𝛽𝑛𝑥𝑛− 𝑝2 − 𝛽𝑛(1 − 𝛽𝑛) 𝜑2(𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛) + (1 − 𝛽𝑛) [𝑥𝑛− 𝑝2 − 2𝜇2(𝛼2− 𝜅2𝜇2) 𝐵2𝑥𝑛− 𝐵2𝑝2 − 2𝜇1(𝛼1− 𝜅2𝜇1) 𝐵1𝑢𝑛− 𝐵1𝑞2] = 𝑥𝑛− 𝑝2 − 𝛽𝑛(1 − 𝛽𝑛) 𝜑2(𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛) − 2 (1 − 𝛽𝑛) [𝜇2(𝛼2− 𝜅2𝜇2) 𝐵2𝑥𝑛− 𝐵2𝑝2 + 2𝜇1(𝛼1− 𝜅2𝜇1) 𝐵1𝑢𝑛− 𝐵1𝑞2] , (52) which hence implies that
𝛽𝑛(1 − 𝛽𝑛) 𝜑2(𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛) + 2 (1 − 𝛽𝑛) [𝜇2(𝛼2− 𝜅2𝜇2) × 𝐵2𝑥𝑛− 𝐵2𝑝2 + 𝜇1(𝛼1− 𝜅2𝜇1) × 𝐵1𝑢𝑛− 𝐵1𝑞2] ≤ 𝑥𝑛− 𝑝2− 𝑦𝑛− 𝑝2 ≤ (𝑥𝑛− 𝑝 +𝑦𝑛− 𝑝)𝑥𝑛− 𝑦𝑛. (53)
Since0 < 𝜇𝑖 < 𝛼𝑖/𝜅2for𝑖 = 1, 2and{𝑥𝑛},{𝑦𝑛}are bounded,
we obtain from (46), (53), condition (ii) and the properties of
𝜑2that lim 𝑛 → ∞𝐵2𝑥𝑛− 𝐵2𝑝 =𝑛 → ∞lim 𝐵1𝑢𝑛− 𝐵1𝑞 = 0, lim 𝑛 → ∞𝑆𝑛𝑥𝑛− 𝐺𝑥𝑛 = 0. (54)
UtilizingProposition 3andLemma 8, we have
𝑢𝑛− 𝑞2 = Π𝐶(𝑥𝑛− 𝜇2𝐵2𝑥𝑛) − Π𝐶(𝑝 − 𝜇2𝐵2𝑝)2 ≤ ⟨𝑥𝑛− 𝜇2𝐵2𝑥𝑛− (𝑝 − 𝜇2𝐵2𝑝) , 𝐽 (𝑢𝑛− 𝑞)⟩ = ⟨𝑥𝑛− 𝑝, 𝐽 (𝑢𝑛− 𝑞)⟩ + 𝜇2⟨𝐵2𝑝 − 𝐵2𝑥𝑛, 𝐽 (𝑢𝑛− 𝑞)⟩ ≤ 12[𝑥𝑛− 𝑝2+ 𝑢𝑛− 𝑞2 − 𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) ] + 𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 , (55)
which implies that
𝑢𝑛− 𝑞2≤ 𝑥𝑛− 𝑝2
− 𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞))
+ 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 .
In the same way, we derive V𝑛− 𝑝2 = Π𝐶(𝑢𝑛− 𝜇1𝐵1𝑢𝑛) − Π𝐶(𝑞 − 𝜇1𝐵1𝑞)2 ≤ ⟨𝑢𝑛− 𝜇1𝐵1𝑢𝑛− (𝑞 − 𝜇1𝐵1𝑞) , 𝐽 (V𝑛− 𝑝)⟩ = ⟨𝑢𝑛− 𝑞, 𝐽 (V𝑛− 𝑝)⟩ + 𝜇1⟨𝐵1𝑞 − 𝐵1𝑢𝑛, 𝐽 (V𝑛− 𝑝)⟩ ≤12[𝑢𝑛− 𝑞2+ V𝑛− 𝑝2 −𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞)) ] + 𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 , (57)
which implies that
V𝑛− 𝑝2≤ 𝑢𝑛− 𝑞2
− 𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞))
+ 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 .
(58)
Substituting (56) for (58), we get
V𝑛− 𝑝2≤ 𝑥𝑛− 𝑝2 − 𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) − 𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞)) + 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 + 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 . (59)
ByLemma 7, we have from (52) and (59)
𝑦𝑛− 𝑝2 ≤ 𝛽𝑛𝑆𝑛𝑥𝑛− 𝑝2+ (1 − 𝛽𝑛) V𝑛− 𝑝2 ≤ 𝛽𝑛𝑥𝑛− 𝑝2+ (1 − 𝛽𝑛) × [𝑥𝑛− 𝑝2 − 𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) − 𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞)) + 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 + 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 ] ≤ 𝑥𝑛− 𝑝2− (1 − 𝛽𝑛) × [𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) + 𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞))] + 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 + 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 , (60) which hence leads to
(1 − 𝛽𝑛) [𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) +𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞))] ≤ 𝑥𝑛− 𝑝2− 𝑦𝑛− 𝑝2 + 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 + 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 ≤ (𝑥𝑛− 𝑝 +𝑦𝑛− 𝑝)𝑥𝑛− 𝑦𝑛 + 2𝜇2𝐵2𝑝 − 𝐵2𝑥𝑛𝑢𝑛− 𝑞 + 2𝜇1𝐵1𝑞 − 𝐵1𝑢𝑛V𝑛− 𝑝 . (61)
From (46), (54), (61), condition (ii), and the boundedness of
{𝑥𝑛},{𝑦𝑛},{𝑢𝑛}, and{V𝑛}, we deduce that
lim
𝑛 → ∞𝜓1(𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞)) = 0,
lim
𝑛 → ∞𝜓2(𝑢𝑛−V𝑛+ (𝑝 − 𝑞)) = 0.
(62)
Utilizing the properties of𝜓1and𝜓2, we deduce that
lim 𝑛 → ∞𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞) = 0, lim 𝑛 → ∞𝑢𝑛−V𝑛+ (𝑝 − 𝑞) = 0. (63) From (63) we get 𝑥𝑛−V𝑛 ≤ 𝑥𝑛− 𝑢𝑛− (𝑝 − 𝑞) +𝑢𝑛−V𝑛+ (𝑝 − 𝑞) → 0 as 𝑛 → ∞, (64)
which together with (54), leads to lim 𝑛 → ∞𝑥𝑛− 𝐺𝑥𝑛 = 0, lim 𝑛 → ∞𝑥𝑛− 𝑆𝑛𝑥𝑛 = 0. (65) Since 𝑥𝑛− 𝑆𝑥𝑛 ≤ 𝑥𝑛− 𝑆𝑛𝑥𝑛 + 𝑆𝑛𝑥𝑛− 𝑆𝑥𝑛. (66)
Utilizing the assumption on{𝑆𝑛}andLemma 11, from (65) we
get
lim
Next, let us show that{𝑥𝑛}converges weakly to some𝑥 ∈ Δ.
Indeed, since 𝑋 is reflexive and{𝑥𝑛} is bounded, there
exists a subsequence{𝑥𝑛𝑖}of{𝑥𝑛}such that𝑥𝑛𝑖 ⇀ 𝑥 ∈ 𝐶. Then
byLemma 6, we obtain from (44), (65), and (67) that𝑥 ∈ Γ,
𝑥 ∈Fix(𝐺) = Ω, and𝑥 ∈Fix(𝑆) = ⋂∞𝑖=0Fix(𝑆𝑖). Thus,𝑥 ∈ Δ.
In addition, if{𝑥𝑚𝑗}is another subsequence of{𝑥𝑛}such that
𝑥𝑚𝑗 ⇀ ̂𝑥, then byLemma 6we also deduce from (44), (65),
and (67) that ̂𝑥 ∈ Δ. Thus, the limits lim𝑛 → ∞‖𝑥𝑛− 𝑥‖and
lim𝑛 → ∞‖𝑥𝑛 − ̂𝑥‖exist. Now we claim that𝑥 = ̂𝑥. Assume
that𝑥 ̸= ̂𝑥. Then in terms of Opial’s condition, we get
lim
𝑛 → ∞𝑥𝑛− 𝑥 =lim sup𝑖 → ∞ 𝑥𝑛𝑖− 𝑥 <lim sup
𝑖 → ∞ 𝑥𝑛𝑖− ̂𝑥 =lim sup 𝑛 → ∞ 𝑥𝑛− ̂𝑥 =lim sup𝑗 → ∞ 𝑥𝑚𝑗− ̂𝑥 <lim sup 𝑗 → ∞ 𝑥𝑚𝑗− 𝑥 =𝑛 → ∞lim 𝑥𝑛− 𝑥 , (68)
which leads to a contradiction. So, we must have 𝑥 = ̂𝑥.
Therefore,𝑥𝑛 ⇀ 𝑥 ∈ Δ. This completes the proof.
Theorem 15. Let 𝑋be a uniformly convex and2-uniformly
smooth Banach space and let𝐶be a nonempty closed convex
subset of 𝑋 such that 𝐶 ± 𝐶 ⊂ 𝐶. Let Π𝐶 be a sunny
nonexpansive retraction from𝑋onto𝐶. Let𝑇and𝐹 : 𝑋 →
𝐶𝐵(𝑋)and let𝐴 : 𝐶 → 2𝐶be three multivalued mappings,
let𝑔 : 𝑋 → 𝐶be a single-valued mapping, and let𝑁(⋅, ⋅) :
𝑋 × 𝑋 → 𝐶be a single-valued continuous mapping satisfying
conditions (C1)–(C5) inTheorem 2. Let𝐵𝑖 : 𝐶 → 𝑋be𝛼𝑖
-inverse strongly accretive for𝑖 = 1, 2. Let{𝑆𝑖}∞𝑖=0be a countable
family of nonexpansive mappings of𝐶into itself such thatΔ :=
⋂∞𝑖=0Fix(𝑆𝑖) ∩ Ω ∩ Γ ̸= 0, whereΩis the fixed point set of the
mapping𝐺 = Π𝐶(𝐼 − 𝜇1𝐵1)Π𝐶(𝐼 − 𝜇2𝐵2)with0 < 𝜇𝑖< 𝛼𝑖/𝜅2
for𝑖 = 1, 2. Let{𝛼𝑛}, {𝛽𝑛} ⊂ [0, 1], and{𝜎𝑛} ⊂ (0, ∞)such
thatlim𝑛 → ∞𝛼𝑛𝜎𝑛 = 0 and∑∞𝑛=0𝛼𝑛𝜎𝑛 = ∞. For arbitrary
𝑥0 ∈ 𝐶define the sequence{𝑥𝑛}iteratively by(32), where{𝑢𝑛}
is defined by (33) for any 𝑤𝑛 ∈ 𝑇𝑥𝑛,𝑘𝑛 ∈ 𝐹𝑥𝑛 and some
𝜀 > 0. Then, there exists𝑑 > 0such that for0 < 𝛼𝑛𝜎𝑛 ≤ 𝑑,
for all𝑛 ≥ 0, {𝑥𝑛} converges strongly to 𝑝 ∈ Δ provided
lim𝑛 → ∞‖𝑥𝑛+1− 𝑥𝑛‖ = 0; in this case, for any𝑤 ∈ 𝑇𝑥,𝑘 ∈ 𝐹𝑥,
(𝑥, 𝑤, 𝑘)is a solution of the MVVI(15).
Proof. First of all, repeating the same arguments as those in
the proof ofTheorem 14, we can prove that for anyV ∈ 𝐶,
𝜆 > 0, there exists a point ̃𝑥 ∈ 𝐷(𝐴) = 𝐶such that(̃𝑥, 𝑤, 𝑘)is
a solution of the MVVI (14), for any𝑤 ∈ 𝑇̃𝑥and𝑘 ∈ 𝐹̃𝑥. In
addition, in terms ofProposition 5we know that𝑉 + 𝜆𝐴 ∘ 𝑔
is a single-valued mapping due to the fact that𝑉 + 𝜆𝐴 ∘ 𝑔is
𝜙-strongly accretive. Meantime, byLemma 13we know that
𝐺 : 𝐶 → 𝐶is nonexpansive.
Without loss of generality we may assume thatV= 0and
𝜆 = 1. Let𝑝 ∈ Δ and let𝑟 > 0be sufficiently large such
that𝑥0 ∈ 𝐵𝑟(𝑝) =: 𝐵. Let𝑀 := sup{‖𝑢‖ : 𝑢 ∈ 𝑁(𝑤, 𝑘) +
𝐴(𝑔(𝑥)), 𝑥 ∈ 𝐵, 𝑤 ∈ 𝑇𝑥, 𝑘 ∈ 𝐹𝑥}. Then as𝐴 ∘ 𝑔, 𝑇and𝐹
are𝐻-uniformly continuous on𝑋, for𝜀1 := 𝜙(𝑟)/8(1 + 𝜀)
and𝜀2 := 𝜙(𝑟)/8𝜇(1 + 𝜀),𝜀3 := 𝜙(𝑟)/8𝜉(1 + 𝜀), there exist
𝛿1, 𝛿2, 𝛿3 > 0such that for any𝑥, 𝑦 ∈ 𝑋, ‖𝑥 − 𝑦‖ < 𝛿1,
‖𝑥 − 𝑦‖ < 𝛿2, and ‖𝑥 − 𝑦‖ < 𝛿3 imply 𝐻(𝐴 ∘ 𝑔(𝑥), 𝐴 ∘
𝑔(𝑦)) < 𝜀1,𝐻(𝑇𝑥, 𝑇𝑦) < 𝜀2and𝐻(𝐹𝑥, 𝐹𝑦) < 𝜀3, respectively.
Let 𝑑 := (1/2)min{𝛿2/𝑀, 𝛿3/𝑀, 𝛿1/𝑀, 𝑟/𝑀}. Taking into
account lim𝑛 → ∞𝛼𝑛𝜎𝑛= 0we may assume that0 < 𝛼𝑛𝜎𝑛≤ 𝑑,
for all𝑛 ≥ 0.
Let us show that𝑥𝑛 ∈ 𝐵for all𝑛 ≥ 0. We show this by
induction. First,𝑥0∈ 𝐵by construction. Assume that𝑥𝑛 ∈ 𝐵.
We show that𝑥𝑛+1 ∈ 𝐵. If possible we assume that𝑥𝑛+1 ∉ 𝐵,
then‖𝑥𝑛+1− 𝑝‖ > 𝑟. Further from (32) it follows that
𝑦𝑛− 𝑝 ≤ 𝛽𝑛𝑆𝑛𝑥𝑛− 𝑝 + (1 − 𝛽𝑛) 𝐺𝑥𝑛− 𝑝
≤ 𝛽𝑛𝑥𝑛− 𝑝 + (1 − 𝛽𝑛) 𝑥𝑛− 𝑝
= 𝑥𝑛− 𝑝 .
(69)
Repeating the same arguments as those of (37) in the proof
ofTheorem 14, we can get
𝑥𝑛+1− 𝑝2 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑥𝑛+1− 𝑝)𝑥𝑛+1− 𝑝 + 2𝛼𝑛𝜎𝑛[𝑁 (𝑤𝑛+1, 𝑘𝑛+1) − 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛+1− 𝑢𝑛]𝑥𝑛+1− 𝑝 ≤ 𝑥𝑛− 𝑝2− 2𝛼𝑛𝜎𝑛𝜙 (𝑥𝑛+1− 𝑝)𝑥𝑛+1− 𝑝 + 2𝛼𝑛𝜎𝑛[𝑁 (𝑤𝑛+1, 𝑘𝑛+1) − 𝑁 (𝑤𝑛+1, 𝑘𝑛) + 𝑁 (𝑤𝑛+1, 𝑘𝑛) − 𝑁 (𝑤𝑛, 𝑘𝑛) + 𝑢𝑛+1− 𝑢𝑛]𝑥𝑛+1− 𝑝 . (70)
Utilizing (33) and (70) and repeating the same arguments
as those of (40) in the proof ofTheorem 14, we can derive
‖𝑥𝑛+1− 𝑝‖2 ≤ ‖𝑥𝑛− 𝑝‖2. So, we have‖𝑥𝑛+1 − 𝑝‖ ≤ 𝑟, a
contradiction. Therefore,{𝑥𝑛}is bounded.
Next let us show that𝑥𝑛 → 𝑥as𝑛 → ∞.
Indeed, since ‖𝑥𝑛+1 − 𝑥𝑛‖ → 0as𝑛 → ∞, we have
that 𝐻(𝐴(𝑔(𝑥𝑛+1)), 𝐴(𝑔(𝑥𝑛))) → 0 and ‖𝑁(𝑤𝑛+1, 𝑘𝑛+1) −
𝑁(𝑤𝑛, 𝑘𝑛)‖ → 0as𝑛 → ∞. The conclusion now follows
from inequality (70) with the use ofProposition 3and hence
(𝑥, 𝑤, 𝑘)is a solution of the MVVI (15) for any 𝑤 ∈ 𝑇𝑥,
𝑘 ∈ 𝐹𝑥. This completes the proof.
Remark 16. Theorems14and15improve, extend, supplement,
and develop [4, Theorem 3.2] and [2, Theorem 3.1] in the following aspects.
(i) The problem of finding a point of⋂∞𝑖=0Fix(𝑆𝑖) ∩ Ω ∩ Γ
in Theorems14and15is more general and more subtle
than every one of the problem of finding a point ofΓ
in [4, Theorem 3.2] and the problem of finding a point
of⋂∞𝑖=1Fix(𝑆𝑖) ∩ Ωin [2, Theorem 3.1].
(ii) The iterative scheme in [2, Theorem 3.1] is extended to
develop the iterative scheme (32) of Theorems14and
15by virtue of the iterative schemes of [4, Theorems
3.2]. The iterative scheme (32) of Theorems 14 and
