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The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
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Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
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The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

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Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

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A Result on “Common Fixed Points” for Pair of OWC- Mappings in C-MS

K. Prudhvi

Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad, Telangana State, India

Abstract: In this paper, we obtain a unique common fixed point theorem for pair of self-mappings of OWC(Occasionally Weakly Compatible) pair of mappings in C-MS(Cone -Metric Space) and also given the example for supporting this result. Our result is a generalization and improvement of the some results they are present in this references.

Keywords: fixed point, OWC(Occasionally Weakly Compatible), C-MS(Cone -Metric Space)

Received October 01, 2023; Revised November 02, 2023; Accepted November 10, 2023

Citation: K. Prudhvi, A Result on “Common Fixed Points” for Pair of OWC- Mappings in C-MS. American Journal of Applied Mathematics and Statistics, 11 (3). 98-99. January 2023. doi:10.12691/ajams-11-3-3

1. Introduction

Before 1968 fixed point theorems are used the Banach contraction principle, but in 1968, Kannan [9] proved a fixed point theorem for self- mapping satisfying the contractive condition which did not required the continuity condition at each point. Later on many mathematicians studied in this fixed point theory and established fixed point theorems. In 1997 Huang and Zhang [7] proved fixed point results in cone metric space they replaced the real numbers by an ordered Banach space. Later on many authors (for e.g., see, 1-4, 6, 8, &10-13) were proved in many ways of fixed point results and they extended, generalized and improved the results in different ways. In the like manner, Bhatt and Chandra [5] proved some fixed point theorems in C-MS using the OWC. In this paper, we obtained a unique common fixed point theorem for pair of OWC-mappings in C-MS and also given one example for supporting of the results.

2. Preliminaries

We need some of useful Definitions and Lemma’s for our main results, they are in [7].

Definition 2.1. Let a real Banach space S. And subset Q of S is called a cone iff

(d1) Q is non –empty and closed and Q ≠ {0};

(d2) αu + βv ϵ Q, for, α, β ϵ ℝ, α,β ≥ 0, u, v ϵ Q;

(d3) Q ∩ (-Q) = {0}.

A cone Q S, and define a partial ordering ‘≤ ‘ w. r. to Q by α ≤ β iff β – α ϵ Q. A cone Q is said to be a normal if there exists a number L > 0 such that for all α, β ϵ S,

“ 0 ≤ α ≤ β implies that ||α|| ≤ L ||β||”.

Then the least + ve number satisfying the above inequality is said to be a normal constant of Q, while

α<< β stands for β – α ϵ interior of Q.

Definition 2.2. Suppose that “X” be a nonempty set of “S”. And let the map ρ: X X → S satisfying

(i). 0 ≤ ρ(x, y) for all x, y ϵ Xand

ρ (x, y) = 0 iff x = y;

(ii). ρ(x, y) = ρ(y, x) for all x, y ϵ X;

(iii). ρ (x, y) ρ(x, z) + ρ (z, y), for all x, y, z ϵ X.

Then ρ said to be a cone metric on “X” and (X, ρ) is said to be a cone metric space.

Definition 2.3. Let (X, ρ) be a cone metric space.

(i) If {xn} is said to be a convergent sequence if for any

b>>0, there exists a natural number N such that

ρ (xn, x) << b, for all n > N and for some fixed

x in X. We denote this xnx, as n→∞.

(ii) If {xn} is said to be a Cauchy sequence if for every b

in S with b>>0, there exists a natural number N such

that ρ(xn, xm) << b, for all n, m > N.

Definition 2.4. A cone metric space (X, ρ) is said to be complete if every Cauchy sequence is convergent in it.

Definition 2.5 [8]. Let A and B be self-mappings of a set X. If u = Ax = Bx for some x in X, then x is said to be a coincidence point of A and B, and u is called a point of coincidence of A and B.

Proposition 2.1. Let A and B be OWC-mappings of a set X iff there is a point x in X which is coincidence point of A and B at which A and B are commute.

Lemma 2.1. Let X be a set, A, B are OWC-mappings of X. If A and B have a unique point of coincidence u = Ax = Bx, then u is the unique common fixed point of A and B.

3. Main Results

Now we are proving our main result theorem.

Note: Suppose that Ф: ℝ⁺ → ℝ⁺ is a function satisfying the condition Ф(t) < t, for each t > 0.

Theorem3.1. Let (X, ρ) be a C-MS and “S” be a normal cone. And suppose p and q are two self- mappings of X and satisfying the following conditions”

(i) (px, py) ≤ Φ(max{(qx, qy), (qx, py)

+ (qy, px) / 2, (qy, py)}),

for all x, y ϵ X.

(ii) p and q are OWC.

Then, p and q are having a unique common fixed point in X.

Proof. By the condition (ii) p and q are OWC, then there exists a point α ϵ X such that pα = qα, pqα = qpα.

Claim: “pα” is a unique common fixed point of p and q. Now we assert that “pα” is a fixed point of p. For if

ppα ≠ pα, then form (i) we get that

(pα,ppα) ≤ Φ(max{(qα, qpα), (qα, ppα)

+ (qpα, pα) / 2, (qpα, ppα)})

≤ Φ(max{(pα, pqα), (pα, ppα)

+ (pqα, pα) / 2, (pqα, ppα)})

≤ Φ(max{(pα, ppα), (pα, ppα)

+ (ppα, pα) / 2, (ppα, ppα)})

≤ Φ(max{(pα, ppα), 2(pα, ppα) /2, 0})

≤ Φ(max{(pα, ppα), (pα, ppα) })

< (pα, ppα), which is a contradiction.

Hence ppα = pα and ppα = pqα = qpα = pα. Thus pα is a common fixed point of p and q.

Uniqueness: Suppose that α, β ϵ X such that pα = qα = α and pβ = qβ = β and α ≠ β, then by (i) we get that

(α, β) = (pα, pβ )

≤ Φ(max{(qα, qβ), (qα, pβ) +

(qβ, pα) / 2, (qβ, pβ)})

≤ Φ(max{(α, β), (α, β) +

(β, α) / 2, (β, β)})

≤ Φ(max{(α, β), 2(α, β) / 2, 0)})

≤ Φ(max{(α, β), (α, β) }) <(α, β),

which is a contradiction.

Therefore α = β. Therefore p and q having a unique common fixe point. Hence proved.

Example 3.2. Let B = ℝ², S = {( α, β)ϵB/ α, β ≥ 0⸦ℝ2 }

and define ρ: ℝ× ℝ → B by

ρ(α, β) = (⎹ α – β⎹, λ⎹ α – β⎹ ), where λ > 0is a constant.

Define p, q: X → X by, p(α) = and q(α) = ,

α ϵ X. And clearly (X, ρ) is a C- MS, p and q are OWC and p, q satisfy the condition (i) and also 1 is the unique common fixed point of p and q.

4. Conclusion

In this research article we obtained some results, these results are more general and improved than the results of [5].

ACKNOWLEDGEMENTS

The author is grateful to the reviewers to review this research article and also given great suggestions for improvement this research article.

References

[1]   | M. Abbas and G. Jungck, Common fixed point results for non commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341(2008) 416-420.
[2]   | M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 21(2008)511-515.
[3]   | I. Altun, B. Damnjanovic, D. Djoric, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. (2009)doi:10.1016/j.aml.2009.09.016.
[4]   | I. Altun, B. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo 58(2009) 319-325.
[5]   | Arvind Bhatt and Harish Chandra, Occasionally weakly compatible mappings in cone metric space, Applied Mathematical Sciences, Vol. 6, 2012, no. 55, 2711 – 2717.
[6]   | Guangxing Song, Xiaoyan Sun, Yian Zhao, Guotao Wang, New common fixed point theorems for maps on cone metric spaces, Appl. Math. Lett. 32(2010)1033-1037.
[7]   | L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings J. Math. Anal. Appl. 332(2)(2007) 1468- 1476.
[8]   | G.Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly compatible Mappings, Fixed Point Theory, 7(2006), 286-296.
[9]   | R. Kannan, Some results on fixed points, Bull.Calcutta Math.Soc. vol. 60, (1968) pp. 251-258.
[10]   | M.Kır, Sayed K. Elagan, Ö.Özer, 2019, Fixed point theorem for contraction of Almost Jaggi type contractive mappings, Journal of Applied & Pure Mathematics, 1(2019), No. 5 - 6, pp. 329 – 339.
[11]   | K. Prudhvi, A Study on Fixed Point Results for OWC Self-Maps on C-Metric Space, Asian Research Journal of Mathematics,18(9),(2022),54-57.
[12]   | K. Prudhvi, A Unique Common Fixed Point Theorem for a Metric Space with the Property (E.A), American Journal of Applied Mathematics and Statistics, Vol.11., No.1, (2023), 11-12.
[13]   | K. Prudhvi, Generalized Fixed Points for Four Self - Mappings with the property OWC in CMS, Asian Research Journal of Mathematics, Vol.9, Issue. 5.,(2023), 37- 40.