This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
I am Simon M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from,Nottingham Trent University,UK
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Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
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In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
I am Simon M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from,Nottingham Trent University,UK
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistics.
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Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
I am Terry K. I am a Logistics Management Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Chicago, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Logistics Management. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Logistics Management Assignments.
In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.
Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.Keywords: Zeros of polynomial, Eneström - Kakeya theorem, Polar derivatives.
Title: On the Zeros of Polar Derivatives
Author: P. Ramulu, G.L. Reddy
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
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Abstract: We extend some existing results on the zeros of polar derivatives of polynomials by considering more general coefficient conditions. As special cases the extended results yield much simpler expressions for the upper bounds of zeros than those of the existing results.
Mathematics Subject Classification: 30C10, 30C15.Keywords: Zeros of polynomial, Eneström - Kakeya theorem, Polar derivatives.
Title: On the Zeros of Polar Derivatives
Author: P. Ramulu, G.L. Reddy
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
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International Journal of Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
A. JBILOU, Convexity of the Set of k-Admissible Functions on a Compact Kähler Manifold (2014), International Journal of Innovation and Scientific Research, vol. 4, no. 1, pp. 42–46.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
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GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
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Gopinath Rebala
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The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
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https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
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Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
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PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
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- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
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Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
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"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
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UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
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1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
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Orchestrator execution result
Defect reporting
SAP heatmap example with demo
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DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
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https://arxiv.org/abs/2306.08302
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Iterative procedure for uniform continuous mapping.
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.8, 2013
53
Iterative Procedure for Uniform Continuous Mapping.
Chika Moore1
Nnubia Agatha C1*
and. Mogbademu Adesanmi2
1. Department of Mathematics, Nnamdi Azikiwe University P.M.B 5025 Awka, Nigeria.
2. Department of Mathematics, University of Lagos, Lagos Nigeria.
* E-mail of the corresponding author: obijiakuagatha@ymail.com
Abstract
Let K be a closed convex nonempty subset of a normed linear space E and let
{ }N
iiT 1= be a finite family of self
maps on K such that T1 is a uniformly continuous uniformly hemicontractive map and T1(K) is a bounded set
with φ≠= = ))(( 1 i
N
i TFF I , sufficient conditions for the strong convergence of an N-step iteration process to a
fixed common point of the family are proved
Keywords: key words, uniformly continuous, uniformly hemicontractive, finite family, common fixed point,
Noor iteration, strong convergence.
1. Introduction
{ } ExfxfxEfxJ ∈∀==∈= ;,:)(
22*
Where E*
denotes the dual space of E and .,. denotes the
generalized duality pairing between E and E*
. The single-valued normalized duality mapping is denoted by j. A
mapping
*
: EET → is called strongly pseudocontractive if for all Eyx ∈, , there exist
)()( yxJyxj −∈− and a constant )1,0(∈K such that
2
)1()(, yxkyxjTyTx −−≤−− .
T is called strongly eractractivpseudocont−φ if for all Eyx ∈, , there exist )()( yxJyxj −∈− and
a strictly increasing function [ ) [ )1,01,0: →φ with 0)0( =φ such that
( ) .)(,
2
yxyxyxyxjTyTx −−−−≤−− φ
It is called generalized strongly eractractivpseudocont−ψ or uniformly pseudocontractive if for all
Eyx ∈, , there exist )()( yxJyxj −∈− and a strictly increasing function [ ) [ )1,01,0: →ψ with
0)0( =ψ such that
( )yxyxyxjTyTx −−−≤−− ψ
2
)(, .
Every strongly eractractivpseudocont−φ operator is a uniformly eractractivpseudocont−ψ
operator with [ ) [ )1,01,0: →ψ defined by sss )()( φψ = , but not conversely (see [13]).
These classes of operators have been studied by several authors (see, for example [3], [4], [7], [13], [19], [23],
[24] and references therein).
If I denotes the identity operator, then T is strongly pseudocontractive, strongly eractractivpseudocont−φ ,
generalized strongly eractractivpseudocont−ψ if and only if )( TI − is strongly accretive, strongly
accretive−φ , generalized strongly accretive−φ operators respectively. The interest in pseudocontractive
mappings is mainly due to their connection with the important class of nonlinear accretive operators. In recent
years, many authors have given much attention to approximate the fixed points of non-linear operators in Banach
space using the Ishikawa and Mann iterative schemes (see, for example [8], [10], [11] and references therein).
Noor [14] introduced the three-step iteration process for solving nonlinear operator equations in real Banach as
follows;
Let E be a real Banach space, K a nonempty convex subset of E and KKT →: , a mapping. For an arbitrary
Kx ∈0 , the sequence { } Kx nn ⊂
∞
=0
defined by
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.8, 2013
54
= 1 −∝ +∝ ,
= 1 − + ,
= 1 − + ,
(1)
Where {∝ },{ }and { }are three sequences in [0,1] is called the three-step iteration (or the Noor iteration).
When = 0, then the three-step iteration reduces to the Ishikawa iterative sequence if = = 0, then the
three-step iteration reduces to the Mann iteration.
Rafiq [21], recently introduced a new type of iteration-the modified three-step iteration process which is defined
as follows;
Let ∶ → be three mappings for any given ∈ , the modified three-step iteration is defined by
= 1 −∝ +∝ ,
= 1 − +
= 1 − +
(2)
Where {∝ },{ }and { } are three real sequences staisfying some conditions. It is clear that the iteration
scheme (2) includes Noor as special case.
Glowinski and Le Tallec [5] used the three-step iteration schemes to solve elastoviscoplasiticty, liquid crystal
and eigen-value problems. They have shown that the three-step approximation scheme performs better than the
two-step and one-step iteration methods. Haubruge et al. [6] have studied the convergence analysis of three-step
iteration schemes and applied these three-step iteration to obtain new splitting type algorithms for solving
variational inequalities, separable convex programming and minimization of a sum of convex functions. They
also proved that three-step iterations also lead to highly parallelized algorithms under certain conditions. Thus, it
is clear that three-step schemes play an important part in solving various problems, which arise in pure and
applied sciences.
Recently, Xue and Fan [23] used the iteration procedure define by (2) in their theorem as stated below.
Theorem 1.1 Let X be a real Banach space and K be a nonempty closed convex subset of X. Let T1, T2 and T3
be strongly pseudocontractive self maps of K with T1(K) bounded and T1, T2 and T3 uniformly continuous. Let
{ } be defined by (2),where { }, { }and { } are three real sequences in [0,1] such that (i) , →
0 → ∞. (ii) ∑∞
# = ∞, and $ ∩ $ ∩ $ ≠ 0 then the sequence{ } converges strongly to the
common fixed point of T1, T2 and T3
Olaleru and Mogbademu [17] established the strong convergence of a modified Noor iterative process when
applied to three generalized strongly eractractivpseudocont−φ , operators or generalized strongly
accretive−φ operators in Banach space. Thus, generalizing the recent results of Fan and Xue (2009). In fact
the stated and proved the following result.
Theorem 1.2 let E be real Banach space, K a nonempty closed convex subset of E,
eractractivpseudocont−φ mappings such that ' bounded. Let { } be a sequence defined by (2)
where {( }, { } and { } are three sequences in [0,1] satisfiying the following conditions:
lim →∞ ( = lim →∞ = lim →∞ = 0
∞=∑
∞
≥0n
nα if F(T1) F(T2) F(T3) ≠ ∅, then the sequence { } converges to the unique common fixed
points T1, T2 and T3 .
Remark 1.1 In theorem 1.2, it is required that;
All the 3 maps be generalized strongly eractractivpseudocont−φ with the same function ϕ (which is
rather strong function).
All the 3 maps are required to be uniformly continuous (and thus bounded).
Our purpose in this paper is to extend is to extend and generalized the result of Olaleru and Mogbademu (17) in
the following ways:
We introduce m-step iteration scheme
We extend the result to any finite family of m-maps.
The conditions of our theorems are less restrictive and more general than the one used in (17), (23). For instance,
the demand that the three maps must be uniformly continuous is weakened by allowing some of the maps to be
free.
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.8, 2013
55
1.1. The M-Step Iteration Process
Let K be a non-empty convex subset of a normed linear space E and let T: K K be a map. For any
given ∈ . The m-step iteration process is defined by
, =
,- = .1 − ( ,-/ + ( ,- ,-0 ; ' = 1, 2, … , 4
,5 = = , 6ℎ898 + 1 = ' 4:; 4; ≥ 0 (3)
For a finite family
m
iiT 1}{ = of m-maps, the m-step iterative process becomes
, =
,- = .1 − ( ,-/ + ( ,- 5 0- ,-0 ; ' = 1, 2, … , 4
,5 = = , (4)
6ℎ898 + 1 = ' 4:; 4 =:9 ' = >8 ?
+ 1
4
@ = 4 A
+ 1
4
B − A
+ 1
4
BC , ≥ 0
In the case where at least one of the maps in the finite family has some asymptotic behaviour (satisfies an
asymptotic condition) then the iterative process becomes:
, =
,- = .1 − ( ,-/ + ( ,-
r
imT −+1 ,-0 ; ' = 1, 2, … , 4
,5 = = , (5)
With n and m as in equation (4) and 9 = 1 − D
5
E
We need the following lemma in this work:
Lemma 1.1 [13] Let {F }, { }, { } be sequence of nonnegative numbers satisfying the conditions:
,
0
∞=∑
∞
≥n
nβ → 0 as n→ ∞ and = 0{ }. Suppose that
;)( 1
22
1 nnnnn γµψβµµ +−≤ ++ = 1, 2, … …
Where H:[0,1) [0,1) is a strictly increasing function with H(0) = 0. Then F → 0 as → ∞.
2. Main Result
2.1 Theorem 2.1 Let E be a normed linear space and K a nonempty closed convex subset of E, let {Ti}
m
i 1= be
a finite family of self maps on the K such that :
• T1(K), T2(K) are bounded
• T1 is a uniformly continuous uniformly hemicontractive map on K
•
m
iF 1== I F(Ti) ∅ where F(Ti) is the set of fixed points of Ti in K
Starting with an arbitrary ∈ , let { } be the iterative sequence defined by
, =
,- = .1 − ( ,-/ + ( ,- imT −+1 ,-0 ; ' = 1, 2, … , 4 − 1
,5 = (6)
Where {( ,-} ⊂ 0,1 is a finite family of real sequence such that },...2,1{;0lim , miin
n
∈∀=
∞→
α and
.
0
, ∞=∑
∞
≥n
mnα Then, { nx } converges strongly to a common fixed point of the finite family.
Proof Let ∗
J $. It suffices to prove that:
• { } is bounded.
• Let K = L ,50 − L Then K → 0 as n→ ∞
• converges to ∗
.
Now, since T1(K) is bounded, let D1 = M − ∗M + NO # L ,50 − ∗
L < ∞. We establish by
induction that M − ∗M ≤ R ∀ ≥ 0. The case n = 0 is trivial, so assume it is true for n = v +1
M T − ∗M ≤ .1 − (T,5/M T − ∗M + (T,5L T,50 − ∗
L ≤ D1
Thus, M − ∗M ≤ R ∀ ≥ 0 which gives { } are bounded.
More so, since T1(K), T2(K) and { } are bounded sets, let
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56
D2 = NO # {M − ∗M + L ,50 − ∗
L + L ,50 − ∗
L < ∞
Then,
L ,50 − L ≤ ( ,50 L − ,50 L + ( ,5L − ,50 L
≤ ( ,50 M − ∗M + L ,50 − ∗
L + ( ,5M − ∗M + L ,50 − ∗
L)
≤ ( ,50 + ( ,5 M − ∗M + max {L ,50 − ∗
L, L ,50 − ∗
L})
≤ ( ,50 + ( ,5 D2
Thus, 011, →− +− nmn xy as ∞→n .
Then by uniform continuity of T1, 0→nδ as ∞→n . Thus proving (ii).
Also,
))(,( *
1
*
1
2*
1 xxjxxxx nnn −−=− +++
))(,())(,)(1( *
1
*
1,1,
*
1
*
1, xxjxyTxxjxx nmnmnnnmn −−+−−−= +−++ αα
))(,()1( *
1
*
11,
*
1111,1,
*
1
*
, xxjxxTxxxTyTxxxx nnmnnnmnmnnnmn −−+−−+−−−≤ ++++−+ ααα
)()1( *
11
2*
1,
*
1111,1,
*
1
*
, xxxxxxxTyTxxxx nnmnnnmnmnnnmn −−−+−−+−−−≤ ++++−+ ψααα
)()1(
2
1
])1[(
2
1 *
11
2*
1,
2*
1,
2*
1
2*2
, xxxxxxxxxx nnmnnnmnnnmn −−−+−++−+−−≤ ++++ ψαδαα
So that
2*
1,
2*
1,,
2*
1
2*2
,
2*
1 2
2
1
)1(2 xxxxxxxxxx nmnnnmnnmnnnmnn −+−++−+−−≤− ++++ αδαδαα
)( *
11, xxnmn −− +ψα
Hence,
2*
11,,
2*
1
2
,
2*
,, 2)1()21( xxxxxx nmnnmnnmnnnmnmn −−+−−≤−−− ++ ψαδααδαα
(7)
Since
0, →inalpha as ,in ∞∀→ there exist Nn ∈0 such that 1)2(1
2
1 ,0 <+−<≥∀ nmnn n δα
)(
)2(1
2
)2(1)2(1
)1( 2*
11
,
,
,
,2*
,
2
,2*
1 xxxxxx n
nmn
mn
nmn
nmn
n
nmn
mn
n −
+−
−
+−
+−
+−
−
≤− ++ ψ
δα
α
δα
δα
δα
α
)(22)(2
2*
11,,,,
2*
1 xxDxx nmnnmnnmnmnn −−+++−≤ ++ ψαδαδαα
Let
*
xxnn −=µ ; mnn ,2αβ = ,
2
, )( Dnmnn δαδυ ++= . Then, we have that
)( 11
22
1 ++ −+≤ nnnnnn µψβυβµµ (8)
By Lemma, 0→nµ as ∞→n . Hence, the theorem.
2.2 Theorem 2.2
Let E be a normal linear space, K a non-empty closed convex subset of E, and let
m
iiT 1}{ = be a finite family
of self maps on K such that:
• T1(K) is bounded
• T1is a uniformly continuous uniformly hemicontractive map on K
• T2, …,Tm are bounded maps
• φ≠= = )(1 i
m
i TFF I where F(Ti) is the set of fixed points of Ti in K
Starting with an arbitrary Kx ∈0 , let { nx } be the iterative sequence defined by Equation 6. Then, { nx }
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57
converges strongly to a common fixed point of the finite family.
Proof Let Fx ∈*
. Then
))(,( *
1
*
1
2*
1 xxjxxxx nnn −−=− +++
))(,())(,)(1( *
1
*
1,1,
*
1
*
1, xxjxyTxxjxx nmnmnnnmn −−+−−−= +−++ αα
))(,()1( *
1
*
11,
*
1111,1,
*
1
*
, xxjxxTxxxTyTxxxx nnmnnnmnmnnnmn −−+−−+−−−≤ ++++−+ ααα
)()1( *
11
2*
1,
*
1111,1,
*
1
*
, xxxxxxxTyTxxxx nnmnnnmnmnnnmn −−−+−−+−−−≤ ++++−+ ψααα
)()1(
2
1
])1[(
2
1 *
11
2*
1,
2*
1,
2*
1
2*2
, xxxxxxxxxx nnmnnnmnnnmn −−−+−++−+−−≤ ++++ ψαδαα
Where 111,1 +− −= nmnn xTyTδ ; So that
2*
1,
2*
1,,
2*
1
2*2
,
2*
1 2
2
1
)1(2 xxxxxxxxxx nmnnnmnnmnnnmnn −+−++−+−−≤− ++++ αδαδαα
)( *
11, xxnmn −− +ψα
So that
2*
11,,
2*
1
2
,
2*
,, 2)1()21( xxxxxx nmnnmnnmnnnmnmn −−+−−≤−−− ++ ψαδααδαα (9)
Now
L ,50 − L ≤ ( ,50 L − ,50 L + ( ,5L − ,50 L
≤ ( ,50 M − ∗M + L ,50 − ∗
L + ( ,5M − ∗M + L ,50 − ∗
L)
≤ ( ,50 + ( ,5 M − ∗M + max {L ,50 − ∗
L, L ,50 − ∗
L})
Since T1(K) is bounded by the same argument in the prove of part (i) of theorem 2.1 we establish that { nx }
is bounded. Since T2,…,Tn are bounded maps, we have that { nT nx } is bounded. Let D2 =
max {R , NO # M 5 − ∗M < ∞
L , − ∗
L ≤ .1 − ( , /M − ∗M + ( , M 5 − ∗M ≤ D ∀ ≥ 0 (10)
Thus, { 1−mT 1,ny }is bounded. Let D3 = = max {R , NO # L 50 , − ∗
L < ∞
L , − ∗
L ≤ .1 − ( , /M − ∗M + ( , L 50 , − ∗
L ≤ D ∀ ≥ 0 (11)
So, { 2−mT 2,ny }is bounded. Let D4 = max {R , NO # L 50 , − ∗
L < ∞. Proceeding thus, we obtain
that if { iny , }is bounded then { imT − iny , }is bounded, and Di+2 = max {R , NO # L 50- ,- − ∗
L < ∞
so that
L ,- − ∗
L ≤ .1 − ( ,- /M − ∗M + ( ,- L 50- ,- − ∗
L ≤ DX ∀ ≥ 0 (12)
Hence, {yn,i+1} and {Tm-1-iyn,i+1} are bounded. We have thus established that there exists a constant Do>0
such that
}.,...,2,1{},,max{ *
,
*
,
*
0 mxyTxyxxD iiniminn ∈∀−−−≥ − Thus, let D=2D0
0)( ,1,1, →+≤− −+− Dxy mnmnnimn αα as ∞→n (13)
So that 0→nδ as 0→n . There exists Nn ∈0 such that 0nn ≥∀
)(
)2(1
2
)2(1)2(1
)1( 2*
1
,
,
,
,2*
,
2
,2*
1 xxxxxx n
nmn
mn
nmn
nmn
n
nmn
mn
n −
+−
−
+−
+−
+−
−
≤− ++ ψ
δα
α
δα
δα
δα
α
)(22)(2
2*
11,,
2
,,
2*
1 xxDxx nmnnmnnmnmnn −−+++−≤ ++ ψαδαδαα
Let ;*
xxnn −=µ mnn ,2αβ = ,
2
, )( Dnmnn δαδυ ++= . Then, we have that
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58
)( 11
22
1 ++ −+≤ nnnnnn µψβυβµµ (14)
By Lemma 1.1, 0→nµ as ∞→n . Hence, the theorem.
A map EEA →: is said to be accretive if Eyx ∈∀ , and 0>∀α
yxAyAxyx −≥−+− )(α . (15)
If the above holds, Eyx ∈∀ , and }0|{:)( =∈=∈∀ AwEwAZy (the zero set of A), then A is said to
be quasi-accretive. It is easy to see that T is hemicontractive if and if A=I-T is quasi-accretive. We have the
following theorem as an easy corollary to Theorem 2.2 and Theorem 2.1
2.3 Theorem 2.3
Let E be a normed liner space and let },...,2,1{;: miEEAi ∈→ be a finite family of maps such that:
• The simultaneous nonlinear equations },...,2,1{;0 mixAi ∈= have a common solution Ex ∈*
• R(I-Ai) is bounded.
• (I-A2), …, (I-Am) are bounded maps.
Starting with an arbitrary Ex ∈0 ,define the iterative sequence { nx }by
nn xy =0,
miyAxy iniminninin ,...,1;)1()1( 1,1,,, =−+−= −−+αα
miyTx iniminnin ,...,1;)1( 1,1,, =+−= −++αα
1, += nmn xy where in ≡+1 mod m; (16)
Where { in,α } )1,0(⊂ is a family of real sequences such that },...,2,1{;0lim , miin
n
∈∀=
∞→
α and
∑
∞
≥
∞=
0
,
n
mnα . Then, {xn} converges strongly to a solution of the simultaneous nonlinear equations.
Proof: Let Ti = I-Ai. Then Ti is a uniform continuous uniformly hemicontraction. Further,
nn xy =0,
miyAxy iniminninin ,...,1;)1()1( 1,1,,, =−+−= −−+αα
miyTx iniminnin ,...,1;)1( 1,1,, =+−= −++αα
1, += nmn xy where in ≡+1 mod m;
Thus, Theorem 2.2 applies and we have the stated results. Similarly
Theorem 2.4 Let E be a normed linear space and let },...,2,1{;: miEEAi ∈→ be a finite family of
maps such that:
• The simultaneous nonlinear equations Aix= 0 {1,2,…,m} have a common solution Ex ∈*
that is
φ≠= )(1 i
m
i AZI .
• R(I-A1), R(I-A2) are bounded
• Ai is a uniformly continuous quasi-accretive map
Starting with an arbitrary ,0 Ex ∈ define the iterative sequence {xn} by equation (16). Then {xn} converges
strongly to a solution of the simultaneous nonlinear equations.
3. ConclusionRemark
A Theorem 2.1 extends theorem 1.2 in the following ways; in Equation (6) of Theorem 2.1, let m = 3, ( , =
( , ( , ≡ , ( , ≡ , , ≡ , 0 ≡ , then we have Equation (2) of Theorem 1.2. Theorem 2.1 is
proved for any finite family of maps, so if they are just three in the family, we have Theorem 1.2. More so, the
conditions of Theorem 2.1. For instance, whereas the three maps in Theorem 1.2 are required to be uniformly
continuous and uniformly pseudocontractive with the same function H, Theorem 2.1 requires that only one map
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in the family satisfies such conditions. Also, Theorem 2.2 extends Theorem 1.2 in a similar manner. Hence,
Theorem 2.1, Theorem 2.2 and their corollaries are rather interesting.
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