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

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Unique Fixed Point Theorem on a Generalized d – Cyclic Contraction Mapping in d-Metric Spaces

K. Prudhvi

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

Abstract: In this paper, we prove a unique fixed point theorem for generalized d-cyclic contraction in dislocated metric spaces (d-metric spaces). Our result generalizes, extends and improves some known results existing in the references.

Keywords: dislocated metric space, fixed point, cyclic mapping, d-cyclic contraction

Citation: K. Prudhvi, A Unique Fixed Point Theorem on a Generalized d – Cyclic Contraction Mapping in d-Metric Spaces. American Journal of Applied Mathematics and Statistics, 6 (3). 107-108. January 2018. doi:10.12691/ajams-6-3-3

1. Introduction

In 2000, Hitzler and Seda [2] have introduced the notion of dislocated metric space(also called d-metric space) and established some fixed point theorems in complete dislocated metric spaces, dis-located metric space plays an important role in Topology, Logical programming and in electronics engineering. In 2003, Kirk et al. [5] have introduced the notion of cyclic contraction and they obtained some fixed point theorems for cyclic contractions in dislocated metric spaces. In 2013, George et al. [1] have obtained some fixed point results on d-cyclic contractions in dislocated metric spaces. In this paper, we obtain a unique fixed point theorem for a generalized d-cyclic contraction in dislocated metric spaces.

Definition 1.1 [2]. Let X be a non-empty set and let d:X × X→[0,∞) be a function satisfying the following conditions

(d1)d(x, y) = d(y, x).

(d2)d(x, y) = d(y, x) = 0 ⇒ x = y.

(d3)d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.

Then d is called dislocated metric or d-metric on X.

Definition 1.2 [2]. A sequence {xn} in a d-metric space (X, d) is called a Cauchy sequence if for given ∊ > 0, there exists n₀ ∊ℕ such that for all m, n ≥ 0, we have d(xm, xn) < ϵ.

Definition 1.3 [2]. A sequence {xn} in a d-metric space (X, d) d-converges with respect to d if there exists x∊X such that d(xn, x) → 0 as n→∞. In this case x is called limit of {xn}( in d) and we write xn → x.

Definition 1.4 [2]. A d-metric space (X, d) is called d-complete if every Cauchy sequence in it is d-convergent.

Definition 1.5 [5]. Let A and B be non-empty subsets of a metric space (X, d). A cyclic map

T: A ∪ B →A ∪ B is said to be cyclic map if T(A) ⊂ B and T(B) ⊂ A.

Definition 1.6 [5]. Let A and B be non-empty subsets of a metric space (X, d). A cyclic map T:A ∪ B →A ∪ B is said to be a cyclic contraction if there exists k∈(0,1) such that d(Tx, Ty) ≤ kd(x, y) for all x∈A and y ∈ B.

We define a generalized d-cyclic contraction mapping in the following way.

Definition 1.7. Let A and B be non-empty subsets of a d-metric space (X, d). A cyclic map T:A ∪ B →A ∪ B is said to be a generalized d-cyclic contraction if there exists α, β, γ > 0 satisfying α + 2β + 4γ < 1 such that

for all x ∈A and y ∈ B.

2. Fixed Point Theorem

Theorem 2.1. Let (X, d) be a complete d-metric space, A and B be non-empty subsets of X and T:A ∪ B →A ∪ B be a generalized d- cyclic contraction in X. Then T has a unique fixed point in A ∩ B.

Proof. Fix x ∈ A. By the definition 1.7 there exists α, β, γ > 0, α + 2β + 4γ < 1 such that

Where, b = (α + β + 3 γ) /1- (β+ γ) < 1.

By induction, we have d(Tn+1x, Tnx) ≤ bn d(Tx, x), more generally, for m>n, we have

Since, b < 1, so as m, n → ∞ we have

Hence, d(Tmx, Tnx) → 0 ,as m, n → ∞.

Therefore, {Tnx} is a Cauchy sequence. Since (X, d) is complete so {Tnx} converge to some point z ∈ X. Since {T2nx} ⊆ A and {T2n-1x} ⊆ B and so z ∈ A ∩ B.

We claim that Tz = z.

Taking limit n→∞, we obtain

Thus, z is a fixed point.

To show the uniqueness, let us assume that there exists two fixed points say z₁ and z₂ such that Tz₁ = z₁ and Tz₂ = z₂.

Now,

Since, α + 2β + 2γ < 1.

Therefore, T has a unique fixed point in A ∩ B.

This completes the proof of the theorem.

Remark 2.2. If we choose β = γ = 0 in the above theorem then we get the d-cyclic contraction theorem3.3 in [1].

Remark 2.3. If we choose α = γ = 0 in the above theorem then we get the Kannan type d-cyclic contraction theorem3.6 in [1].

Remark 2.4. If we choose α = β = 0 in the above theorem then we get the Chatterjee type d-cyclic contraction theorem3.8 in [1].

3. Conclusion

The above Theorem 2.1 is a generalization of Theorems 3.3., 3.6., and 3.8., in [1].

References

[1]   | R. George, R. Rajagopalam and S.Vinayagam, Cyclic contractions and fixed points in dislocated metric spaces, Int. Journal of Math. Analysis, Vol.7, 2013, no.9, 403-411.
[2]   | P.Hitzler and A.K. Seda, Dislocated topologies, Proc. Of the Slovakian conference in applied mathematics, (2000), Bratislava.
[3]   | Erdal Karapinar, Best proximity on different types contractions, Applied Mathematics and information Science, 5(3), (2011), 558-569.
[4]   | S. Karpagam and Sushma Agarwal, Best proximity points theorems for cyclic Meir Keeler contraction maps, Nonlinear Analysis, 74(2011), 1040-1046.
[5]   | W.A.Kirik and P.S. Srinivasan and P.Veeramani, Fixed points for mapping satisfying cyclic contractive conditions, Fixed point theory,4,((2003), 79-89.
[6]   | G. Petrushel, Cyclic representations and periodic points, Studia Univ.Babes - Bloyai Math, 50, 107-112.
[7]   | V. Sankar Raj and P. Veeramani, Best proximity pair theorems for relatively non expansive mappings, Applied General Topology, Vol.10, No.1, (2009), 21-28.