Journal of Inequalities and Applications Volume 2010, Article ID 437976,12pages doi:10.1155/2010/437976
Research Article
On the System of Nonlinear Mixed Implicit
Equilibrium Problems in Hilbert Spaces
Yeol Je Cho
1and Narin Petrot
21Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, South Korea
2Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Narin Petrot,[email protected]
Received 22 December 2009; Accepted 10 January 2010
Academic Editor: Jong Kyu Kim
Copyrightq2010 Y. J. Cho and N. Petrot. 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 use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problemsSMIEin Hilbert spaces. The algorithm for finding a solution of the problemSMIEis suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.
1. Introduction and Preliminaries
LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and · ,
respectively. LetF1, F2:H×H → Hbe given two bi-functions satisfyingFix, x 0 for all
x∈Handi1,2. LetT :H×H → Hbe a nonlinear mapping. LetCbe a nonempty closed
convex subset ofH. In this paper, we consider the following problem.
Findx∗, y∗∈Hsuch that
F1x∗, z
T1
x∗, y∗, z−x∗≥0, ∀z∈C,
F2
y∗, zT 2
x∗, y∗, z−y∗≥0, ∀z∈C. 1.1
The problem of type1.1is called the system of nonlinear mixed implicit equilibrium problems.
We denote by SMIEF1, F2, T1, T2, C the set of all solutions x∗, y∗ of the problem
Some examples of the problem1.1are as follows.
IIf Fix, z supξ∈M
ixξ, z−x, whereMi : C → 2
H is a maximal monotone
mapping fori1,2,then the problem1.1becomes the following problem.
Findx∗, y∗∈Hsuch that
0∈T1
x∗, y∗M 1x∗,
0∈T2
x∗, y∗M 2
y∗, 1.2
which is called the system of variational inclusion problems. In particular, whenT1 T2and
M1 M2,the problem1.2is reduced to the problem, so-called the generalized variational
inclusion problem, which was studied by Kazmi and Bhat1.
It is worth noting that the variational inclusions and related problems are being studied extensively by many authors and have important applications in operations research, optimization, mathematical finance, decision sciences, and other several branches of pure and applied sciences.
IIIfFix, z ψiz−ψixfor all x, z ∈ H, whereψi : H → Ris a real valued
function for eachi1,2.Then the problem1.1reduces to the following problem.
Findx∗, y∗∈Hsuch that
T1
x∗, y∗, z−x∗ψ
1z−ψ1x∗≥0, ∀z∈C,
T2
x∗, y∗, z−y∗ψ
2z−ψ2
y∗≥0, ∀z∈C. 1.3
Some corresponding results to the problem 1.3 were considered by Kassay and
Kolumb´an2whenψ1ψ20.
IIIFor eachi 1,2, letSi : H×H → H be a nonlinear mapping andλ, η fixed
positive real numbers. IfT1x, y λS1y, x x−yandT2x, y ηS2x, y y−xfor all
x, y∈H, then the problem1.3reduces to the following problem.
Findx∗, y∗∈Hsuch that
λS1
y∗, x∗x∗−y∗, z−x∗ψ
1z−ψ1x∗≥0, ∀z∈C,
ηS2
x∗, y∗y∗−x∗, z−y∗ψ
2z−ψ2
y∗≥0, ∀z∈C, 1.4
which is called the system of nonlinear mixed variational inequalities problems. A special case of
the problem1.4, whenS1S2andψ1ψ2, has been studied by He and Gu3.
IVIfψ1x ψ2 δCxfor allx∈C, whereδCis the indicator function ofCdefined
by
δK ⎧ ⎨ ⎩
0, if x∈K,
∞, otherwise,
1.5
Findx∗, y∗∈Csuch that
λS1
y∗, x∗x∗−y∗, z−x∗≥0, ∀z∈C,
ηS2
x∗, y∗y∗−x∗, z−y∗≥0, ∀z∈C, 1.6
which is called the system of nonlinear variational inequalities problems. Some corresponding
results to the problem1.6were studied by Agarwal et al. 4, Chang et al.5, Cho et al.
6, J. K. Kim and D. S. Kim,7and Verma8,9.
For the recent trends and developments in the problem1.6and its special cases, see
3,8–11and the references therein, for examples.
VIfS20, andS1:C → His a univariate mapping, then the problem1.6reduces
to the following problem.
Findx∗∈Hsuch that
S1x∗, z−x∗ ≥0, ∀z∈C, 1.7
which is known as the classical variational inequalityintroduced and studied by Stampacchia
12 in 1964. This shows that a number of classes of variational inequalities and related
optimization problems can be obtained as special cases of the system 1.1 of mixed
equilibrium problems.
Inspired and motivated by the recent research going on in this area, in this paper, we use the Wiener-Hopf equations and the Yosida approximation notion to suggest and
prove the existence and uniqueness of solutions for the problem1.1. We also discuss the
convergence criteria and stability of the iterative algorithm. The results presented in this paper improve and generalize many known results in the literature.
In the sequel, we need the following basic concepts and lemmas.
Definition 1.1Blum and Oettli13. A real valued bifunctionF:C×C → Ris said to be:
1monotoneif
Fx, yFy, x≤0, ∀x, y∈C; 1.8
2strictly monotoneif
Fx, yFy, x<0, ∀x, y∈C x /y; 1.9
3upper-hemicontinuousif
lim sup
t→0
Ftz 1−tx, y≤Fx, y, ∀x, y, z∈C. 1.10
Definition 1.2. 1A functionf:H → R∪ {∞}is said to be lower semicontinuousatx0if, for
allα < fx0,there exists a constantβ >0 such that
α≤fx, ∀x∈Bx0, β
whereBx0, βdenotes the ball with centerx0and radiusβ, that is,
Bx0, β
y:y−x0≤β
. 1.12
2The functionfis said to be lower semicontinuousonHif it is lower semicontinuous
at every point ofH.
Lemma 1.3Combettes and Hirstoaga14. LetCbe a nonempty closed convex subset ofHand Fbe a bifunction ofC×CintoRsatisfying the following conditions:
C1Fis monotone and upper hemicontinuous;
(C2)Fx,·is convex and lower semi-continuous for allx∈C. For allρ >0andx∈H, define a mappingTρF :H → Cas follows:
TF
ρx w∈C:ρFw, z w−x, z−w, ∀z∈C, ∀x∈H. 1.13
ThenTρF is a single-valued mapping.
Definition 1.4. Letρbe a positive number. For any bi-functionF :C×C → R,the associated
Yosida approximationFρoverCand the corresponding regularized operatorAFρare defined
as follows:
Fρx, z 1ρ
x−JF
ρx, z−x
, AF ρx ρ1
x−JF ρx
, 1.14
in whichJρFx∈Cis the unique solution of the following problem:
ρFJF ρx, z
JF
ρx−x, z−JρFx
≥0, ∀z∈C. 1.15
Remark 1.5. Definition 1.4 is an extension of the Yosida approximation notion introduced
in 15. The existence and uniqueness of the solution of the problem 1.15 follow from
Lemma 1.3.
Definition 1.6. LetM⊂H×Hbe a set-valued mapping.
1 Mis said to be monotoneif, for anyx1, y1,x2, y2∈M,
y1−y2, x1−x2
≥0. 1.16
2 A monotone operator M ⊂ H ×H is said to be maximal if M is not properly
contained in any other monotone operators.
Example 1.7Huang et al.16. LetFx, z supξ∈Mxξ, z−x,whereM : H → 2H is a maximal monotone mapping. Then it directly follows that
JF ρx
IρM−1x, AF
whereMρ : 1/ρI−IρM−1is the Yosida approximation ofM,and we recover the
classical concepts.
Using the idea as in Huang et al.16, we have the following result.
Lemma 1.8. If F : C×C → Ris a monotone function, then the operatorJρF is a nonexpansive mapping, that is,
JF
ρx−JρFy≤x−y, ∀x, y∈H. 1.18
Proof. From1.15, for allx, y∈H, we can obtain
ρFJF ρx, JρF
yJF
ρx−x, JρF
y−JF ρx
≥0, 1.19
ρFJF ρ
y, JF ρx
JF
ρ
y−y, JF
ρx−JρF
y≥0. 1.20
By adding1.19with1.20and using the monotonicity ofF, we have
x−y−JF
ρx−JρF
y, JF
ρx−JρF
y≥0, 1.21
and so
JF
ρx−JρFy
2
≤JF
ρx−JρFy, JρFx−JρFy
≤x−y, JF
ρx−JρF
y
≤x−yJF
ρx−JρF
y.
1.22
This implies thatJρF is a nonexpansive mapping. This completes the proof.
Now, for solving the problem1.1, we consider the following equation: letx, y ∈
H×Handρ1, ρ2be fixed positive real numbers. Findw1, w2∈H×Hsuch that
T1
x, yAF1
ρ1w1 0, xJ
F1
ρ1w1,
T2
x, yAF2
ρ2w2 0, yJ
F2
ρ2w2.
1.23
Lemma 1.9. x, y∈H×His a solution of the problem1.1if and only if the problem1.23has a solutionw1, w2,where
xJF1
ρ1w1, w1x−ρ1T1
x, y,
yJF2
ρ2w2, w2y−ρ2T2
that is,
xJF1
ρ1
x−ρ1T1
x, y,
yJF2
ρ2
y−ρ2T2
x, y. 1.25
Proof. The proof directly follows from the definitions ofJF1
ρ1 andJ
F2
ρ2.
In this paper, we are interested in the following class of nonlinear mappings.
Definition 1.10. 1A mappingT :H → His said to be ν-strongly monotoneif there exists a constantν >0 such that
Tx−Ty, x−y≥νx−y2, ∀x, y∈H
; 1.26
2A mappingT : H×H → His said to beτ, σ-Lipschitz if there exist constants
τ, σ >0 such that
T
x1, y1
−Tx2, y2≤τx1−x2σy1−y2, ∀x1, x2, y1, y2∈H. 1.27
2. Existence of Solutions of the Problem
1.1
In this section, we give an existence theorem of solutions for the problem1.1. Firstly, in view
ofLemma 1.9, we can obtain the following, which is an important tool, immediately.
Lemma 2.1. Letx, y ∈ H×H. Thenx, y ∈SMIEF1, F2, T1, T2, Cif and only if there exist
positive real numbersρ1, ρ2such thatx, yis a fixed point of the mappingGρ1,ρ2 :H×H → H×H
defined by
Gρ1,ρ2
x, yAρ1
x, y, Bρ2
x, y, ∀x, y∈H×H, 2.1
whereAρ1, Bρ2:H×H → Hare defined, respectively, by
Aρ1
x, yJF1
ρ1
x−ρ1T1
x, y,
Bρ2
x, yJF2
ρ2
y−ρ2T2
x, y. 2.2
Now, we are in position to prove the existence theorem of solutions for the problem
1.1.
Theorem 2.2. For eachi1,2, letFi:H×H → Rbe a monotone bi-function. LetT1:H×H → H
T2 : H×H → H be aν2-strongly monotone with respect to the second argument andτ2, σ2
-Lipschitz mapping. Suppose that there are positive real numbersρ1, ρ2such that
1−2ρ1ν1ρ21τ12
1/2
ρ2τ2<1,
1−2ρ2ν2ρ22σ22
1/2
ρ1σ1<1.
2.3
ThenSMIE F1, F2, T1, T2, Cis a singleton.
Proof. Notice that, in view of Lemma 2.1, it is sufficient to show that the mapping Gρ1,ρ2
defined in Lemma 2.1has the unique fixed point. SinceJF1
ρ1 is nonexpansive, we have the
following estimate:
Aρ
1
x1, y1
−Aρ1
x2, y2
JF1
ρ1
x1−ρ1T1
x1, y1
−JF1
ρ1
x2−ρ1T1
x2, y2
≤x1−ρ1T1
x1, y1
−x2−ρ1T1
x2, y2
≤x1−x2−ρ1
T1
x1, y1
−T1
x2, y1ρ1T1
x2, y1
−T1
x2, y2.
2.4
SinceT1 : H×H → H is aτ1, σ1-Lipschitz mapping and, for all w ∈ H, the mapping
T1·, w:H → His aν1-strongly monotone, we obtain
x1−x2−ρ1T1x1, y1−T1x2, y12
x1−x22−2ρ1
x1−x2, T1
x1, y1
−T1
x2, y1
ρ2
1T1x1, y1−T1x2, y12
≤ x1−x22−2ρ1ν1x1−x22ρ21τ12x1−x22
1−2ρ1ν1ρ12τ12
x1−x22,
T1
x2, y1
−T1
x2, y2≤σ1y1−y2.
2.5
Consequently, from2.4-2.5, it follows that
Aρ
1
x1, y1
−Aρ1
x2, y2≤
1−2ρ1ν1ρ21τ12
1/2
Next, we have the following estimate:
Bρ
2
x1, y1
−Bρ2
x2, y2
JF2
ρ2
y1−ρ2T2
x1, y1
−JF2
ρ2
y2−ρ2T2
x2, y2
≤y1−ρ2T2
x1, y1
−y2−ρ2T2
x2, y2
≤y1−y2−ρ2
T2
x1, y1
−T2
x1, y2ρ2T2
x1, y2
−T2
x2, y2
≤1−2ρ2ν2ρ22σ22
1/2
y1−y2ρ2τ2x1−x2.
2.7
From2.6and2.7, we have
Aρ1
x1, y1
−Aρ1
x2, y2Bρ2
x1, y1
−Bρ2
x2, y2
≤max{κ1, κ2}
x1−x2y1−y2,
2.8
where
κ1l1ρ2τ2, κ2l2ρ1σ1,
l1 :
1−2ρ1ν1ρ21τ12
1/2
, l2:
1−2ρ2ν2ρ22σ22
1/2
. 2.9
Now, define the norm · onH×Hby
x, yxy, ∀x, y∈H×H. 2.10
Notice thatH×H, · is a Banach space and
Gρ1,ρ2x1, y1−Gρ1,ρ2x2, y2≤max{κ
1, κ2}x1, y1−x2, y2. 2.11
By the condition2.3, we have max{κ1, κ2} < 1, which implies that Gρ1,ρ2 is a contraction
mapping. Hence, by Banach contraction principle, there exists a uniquex, y∈H×Hsuch
thatGρ1,ρ2x, y x, y.This completes the proof.
3. Convergence and Stability Analysis
In view ofLemma 2.1, for the fixed point formulation of the problem2.1, we suggest the
3.1. Mann Type Perturbed Iterative Algorithm (MTA)
For any x0, y0 ∈ H×H, compute approximate solutionxn, yn ∈ H ×H given by the
iterative schemes:
x0∈H,
xn1 1−αnxnαnJρF11
xn−ρ1T1
xn, yn,
yn1 1−αnynαnJF2
ρ2
yn−ρ2T2
xn, yn, ∀n≥0,
3.1
where{αn}is a sequence of real numbers such thatαn∈0,1and∞n0αn∞.
In order to consider the convergence theorem of the sequences generated by the
algorithmMTA, we need the following lemma.
Lemma 3.1. Let{an}and{bn}be two nonnegative real sequences satisfying the following conditions. There exists a positive integern0such that
an1≤1−λnanbn, ∀n≥n0, 3.2
where{λn} ⊂0,1with∞n0λn∞andbnoλn. Thenlimn→ ∞an0.
Now, we prove the convergence theorem for a solution for the problem1.1.
Theorem 3.2. If all the conditions of theTheorem 2.2hold, then the sequence{xn, yn}inH×H generated by the algorithm3.1converges strongly to the unique solution for the problem1.1.
Proof. It follows fromTheorem 2.2that there exists x∗, y∗ ∈ H×H which is the unique
solution for the problem1.1. Moreover, in view ofLemma 2.1, we have
x∗JF1
ρ1
x∗−ρ 1T1
x∗, y∗,
y∗JF2
ρ2
y∗−ρ 2T2
x∗, y∗. 3.3
SinceJF1
ρ1 is nonexpansive, from the iterative sequences3.1and3.3, it follows that
xn1−x∗
1−αnxnαnJF1
ρ1
xn−ρ1T1
xn, yn−x∗
≤1−αnxn−x∗αnJF1
ρ1
xn−ρ1T1
xn, yn−JF1
ρ1
x∗−ρ 1T1
x∗, y∗
≤1−αnxn−x∗αnxn−x∗−ρ1
T1
xn, yn−T1
x∗, y∗
≤1−αnxn−x∗αnxn−x∗−ρ1
T1
xn, yn−T1
x∗, y
n
αnρ1T1
x∗, y
n−T1
x∗, y∗.
Next, we have the following estimate:
xn−x∗−ρ 1
T1
xn, yn−T1
x∗, y
nρ1T1
x∗, y
n−T1
x∗, y∗
≤1−2ρ1ν1ρ21τ12
1/2
xn−x∗ρ1σ1yn−y∗.
3.5
Substituting3.5into3.4yields that
xn1−x∗ ≤1−αnxn−x∗αn
1−2ρ1ν1ρ21τ12
1/2
xn−x∗αnρ1σ1yn−y∗. 3.6
Similarly, we have
yn1−y∗≤1−αny
n−y∗αn
1−2ρ2ν2ρ22σ22
1/2
yn−y∗αnρ2τ2xn−x∗.
3.7
Thus, from3.6and3.7, we have
xn1, yn1
−x∗, y∗
≤1−αnxn, yn−x∗, y∗αnmax{κ1, κ2}xn, yn−x∗, y∗
1−αn1−max{κ1, κ2}xn, yn−x∗, y∗,
3.8
whereκ1andκ2are given in2.9. Setting
anxn, yn−x∗, y∗,
λnαn1−max{κ1, κ2}, bn0, ∀n≥1.
3.9
From the condition2.3, it follows that max{κ1, κ2}<1 and so{λn} ⊂0,1. Moreover, since
∞
n0αn ∞, we have
∞
n0λn ∞. Hence all the conditions ofLemma 3.1are satisfied and
soan → 0 asn → ∞,that is,
xn, yn−x∗, y∗−→0 n−→ ∞. 3.10
Thus the sequence{xn, yn}inH×Hconverges strongly to a solutionx∗, y∗for the problem
1.1. This completes the proof.
3.2. Stability of the Algorithm (MTA)
Consider the following definition as an extension of the concept of stability of the iterative
Definition 3.3Kazmi and Khan18. LetHbe a Hilbert space andA, B :H×H → Hbe
nonlinear mappings. LetG : H×H → H×H be defined asGx, y Ax, y, Bx, y
for allx, y ∈H×Handx0, y0∈H×H. Assume thatxn1, yn1 fG, xn, yndefines
an iterative procedure which yields a sequence {xn, yn} inH×H. Suppose thatFG
{x, y∈H×H:Gx, y x, y}/∅and the sequence{xn, yn}converges to somex∗, y∗∈
FG. Let{un, vn}be an arbitrary sequence inH×Hand
δnun, vn−fG, xn, yn, ∀n≥0. 3.11
If limn→ ∞δn 0 implies that limn→ ∞un, vn x∗, y∗, then the iterative procedure {xn, yn}is said to beG-stableor stablewith respect toG.
Theorem 3.4. Assume that all the conditions of Theorem 2.2 hold. Let {un, vn} be an arbitrary sequence inH×Hand define{δn} ⊂0,∞by
δnun1, vn1−Cn, Dn, 3.12
where
Cn 1−αnxnαnJF1
ρ1
xn−ρ1T1
xn, yn,
Dn 1−αnynαnJF2
ρ2
yn−ρ2T2
xn, yn,
3.13
where {xn, yn} is a sequence defined in 3.1. If Gρ1,ρ2 is defined as in 2.1, then the iterative procedure generated by3.1isGρ1,ρ2-stable.
Proof. Assume that limn→ ∞δn0. Letx∗, y∗be the unique fixed point of the mappingGρ1,ρ2. This means that
x∗JF1
ρ1
x∗−ρ 1T1
x∗, y∗,
y∗JF2
ρ2
y∗−ρ 2T2
x∗, y∗. 3.14
Now, from3.12and3.13, it follows that
un1, vn1−x∗, y∗ ≤δ
nCn−x∗Dn−y∗. 3.15
Notice thatCn, Dn {xn1, yn1}for eachn≥1, which implies that
lim
n→ ∞Cnx
∗, lim
n→ ∞Dny
∗. 3.16
Using3.16and the assumption limn→ ∞δn0, it follows from3.15that
lim
n→ ∞un1, vn1
x∗, y∗. 3.17
Acknowledgments
The first author was supported by the Korea Research Foundation Grant funded by
the Korean Government KRF-2008-313-C00050. The second author was supported by
the Commission on Higher Education and the Thailand Research Fund project no.
MRG5180178.
References
1 K. R. Kazmi and M. I. Bhat, “Convergence and stability of iterative algorithms for some classes of general variational inclusions in Banach spaces,”Southeast Asian Bulletin of Mathematics, vol. 32, no. 1, pp. 99–116, 2008.
2 G. Kassay and J. Kolumb´an, “System of multi-valued variational inequalities,” Publicationes Mathematicae Debrecen, vol. 56, no. 1-2, pp. 185–195, 2000.
3 Z. He and F. Gu, “Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces,”Applied Mathematics and Computation, vol. 214, no. 1, pp. 26–30, 2009.
4 R. P. Agarwal, Y. J. Cho, J. Li, and N. J. Huang, “Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 435–447, 2002.
5 S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces,”Applied Mathematics Letters, vol. 20, no. 3, pp. 329–334, 2007.
6 Y. J. Cho, Y. P. Fang, N. J. Huang, and H. J. Hwang, “Algorithms for systems of nonlinear variational inequalities,”Journal of the Korean Mathematical Society, vol. 41, no. 3, pp. 489–499, 2004.
7 J. K. Kim and D. S. Kim, “A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces,”Journal of Convex Analysis, vol. 11, no. 1, pp. 235–243, 2004.
8 R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,”Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025–1031, 2001.
9 R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,”Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203–210, 2004.
10 H. Nie, Z. Liu, K. H. Kim, and S. M. Kang, “A system of nonlinear variational inequalities involving strongly monotone and pseudocontractive mappings,”Advances in Nonlinear Variational Inequalities, vol. 6, no. 2, pp. 91–99, 2003.
11 R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,”Applied Mathematics Letters, vol. 18, no. 11, pp. 1286–1292, 2005.
12 G. Stampacchia, “Formes bilin´eaires coercitives sur les ensembles convexes,”Comptes Rendus de l’Acad´emie des Sciences: Paris, vol. 258, pp. 4413–4416, 1964.
13 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
14 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
15 A. Moudafi and M. Th´era, “Proximal and dynamical approaches to equilibrium problems,” in
Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), vol. 477 ofLecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999.
16 N.-J. Huang, H. Lan, and Y. J. Cho, “Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings,”Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 608–618, 2006.
17 A. M. Harder and T. L. Hicks, “Stability results for fixed point iteration procedures,”Mathematica Japonica, vol. 33, no. 5, pp. 693–706, 1988.
