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)

Coincidence Points and Common Fixed Points for Four Self-Mappings via Weakly Compatible Mappings in Cone Metric Spaces

K. Prudhvi

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

Abstract: In this paper, we obtain coincidence points and common fixed point theorem for four self-mappings via weakly compatible mappings in cone metric spaces, where the cone is not necessarily normal. These results are improved and generalized several well- known comparable results existing in the references.

Keywords: cone metric space, common fixed point, coincide point

Citation: K. Prudhvi, Coincidence Points and Common Fixed Points for Four Self-Mappings via Weakly Compatible Mappings in Cone Metric Spaces. American Journal of Applied Mathematics and Statistics, 5 (3). 112-114. January 2017. doi:10.12691/ajams-5-3-5

1. Introduction

Huang and Zhang [8] introduced the concept of a cone metric space, they replaced set of real numbers by an ordered Banach space and proved some fixed point theorems for single mapping for contractive type conditions in cone metric spaces, using the normality condition. Later on many authors have [1,2,3,4,6,7,9,10,11] generalized and extended the results of Huang and Zhang [8] and obtained the common fixed points for two or more self-mappings for different types of contractive conditions in cone metric spaces with and without using the normality condition. In this paper, we obtained coincidence points and common fixed points for four self-mappings in cone metric spaces without the normality condition. Our results are extended and improved the results of Arshad, Azam and Vetro [5].

The following definitions are due to Huang and Zhang [8].

Definition 1.1. [8]

Let B be a real Banach space and P be a subset of B. The set P is called a cone if and only if:

(a). P is closed, non–empty and P ≠ {0};

(b). a,b ∈ℝ, a, b, x,y∈P ⇒ ax+by∈P;

(c). P ∩ (-P) = {0}.

Definition 1.2. [8]

Let P be a cone in a Banach space B, define partial ordering ‘≤’ with respect to P by x ≤ y if and only if y-x∈P. We shall write x<y to indicate x ≤ y but x ≠ y while x<<y will stand for y-x∈ int P, where, int P denotes the interior of the set P. This cone P is called an order cone.

Definition 1.3. [8] Let B be a Banach space and P ⊂ B be an order cone. The order cone P is called normal if there exists M > 0 such that for all x, y ∈ B,

The least positive number M satisfying the above inequality is called the normal constant of P.

Definition 1.4. [8] Let X be a nonempty set of B Suppose that the map d: X X→ B satisfies:

(d1). 0 < d(x, y) for all x, y ∈ X with x ≠ y and d(x, y) = 0 if and only if x = y;

(d2). d(x, y) = d(y, x) for all x, y ∈ X;

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

Then d is called a cone metric on X and (X, d) is called a cone metric space.

It is clear that the concept of a cone metric space is more general than that of a metric space.

Example 1.5. [8] Let B = ℝ², P = {(x, y)∈B such that: x, y ≥ 0} ⊆ ℝ², X = ℝ and d: X × X → B such that d(x , y) = (│x - y│, α│x - y│), where α ≥ 0 is a constant. Then (X, d) is a cone metric space.

Definition 1.6 [8] Let (X, d) be a cone metric space .We say that {xn} is said to be

(i) a convergent sequence if for any c>>0, there is a natural number N such that for all n>N, d(xn, x) <<c, for some fixed x in X. We denote this xn→x (as n→∞).

(ii) a Cauchy sequence if for every c in E with c>>0, there is a natural number N such that for all n, m>N, d(xn, xm) << c.

(iii) a cone metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent.

Definition 1.7. [5] A pair (S, T) of self-mappings on X is said to be weakly compatible if they commute at their coincidence point ( i. e., STx = TSx), whenever Sx = Tx).

2. Main Results

In this section, we prove a result on points of coincidence and common fixed points for self-mappings and then show that this result generalizes some of the recent results existing in the literature.

Lemma 2.1. Let X be a nonempty set and the mappings S, T, I and J: X→ X point of coincidence p in X. If {S, I} and {T, J} are weakly compatible if they commute at their coincidence p in X. If {S, I} and {T, J} are weakly compatibles, then S, T, I and J have a unique common fixed point.

Proof. Since, p is a coincidence point of S, T, I and J. Therefore p = Iq = Jq = Sq = Tq for some q∈ X. By weakly compatibility of {S, I} and {T, J} we have Sp = SIq = ISq = Ip, Tp = TJq = JTq = Jp.

It implies that Sp = Tp = Ip = Jp = r(say), then r is a point of coincidence of S,T, I and J. Therefore, p = r by uniqueness. Thus p is a unique common fixed point of S, T, I and J.

We adopt the technique which was used in [5].

Let (X, d) be a cone metric space and S, T, I and J be self-mappings on X such that S(X) ⊆ J(X), T(X) ⊆ I(X). Suppose x₀ ∈ X and x1, x₂ ∈ X is chosen such that S(x₀) = J(x₁), T(x₁) = I(x₂) continuing this process we can define {xn} by

Denote,

[picture] | (1)

The sequence {yn} is called an S-T-sequence with initial point x₀.

The following result is an extension of Proposition 3.2,. in [5].

Proposition 2.2. Let (X, d) be a cone metric space and P be an order cone. Let S, T, I, and J:X→X be such that S(X) ⊆ J(X), T(X) ⊆ I(X). Assume that the following conditions hold:

(i) d(Sx, Ty) ≤ α d(Ix, Sx) + β d(Jy, Ty) + γ d(Ix, Jy), for all x, y∈X with x ≠ y, where α, β, γ are non-negative real numbers with α + β + γ < 1;

(ii) d(Sx, Tx) < d(Ix, Sx) + d(Jx, Tx) , for all x∈X, whenever Sx ≠ Tx.

Then every S-T-sequence with initial point x₀∈X is a Cauchy sequence.

Proof. Let x0 be an arbitrary point in X and {yn} be an S-T-sequence with initial point x0 . First we assume that y2n ≠ y2n+1 for all n. Then

[picture] | (2)

Similarly, we obtain that

[picture] | (3)

Now from (2) and (3), we deduce that

[picture] | (4)

And

[picture] | (5)

Let M = (α + γ) /(1-β), N = (β + γ) /(1-α).

Then MN < 1. Now for any n > m, we have

In analogous way, we again

And d(y2n+1, y2m) ≤ (N+1) (MN)m+1 / 1-MN d(y₁, y₀).

Thus, for n > m > 0,

For each c >> 0, choose δ > 0 such that c – x∈int P, where ||x||<< δ, that is, x << c, for this δ, we can choose a natural number N₁ such that ||bm d(y₁, y₀)||< δ, for m > N₁ .

Thus we get that d(yn, ym) ≤ bm d(y₁, y₀) << c, for all n > m > N₁. Therefore {yn} is a Cauchy sequence in (X, d). Now we suppose that y2n = y2n+1 for some n∈N₁. If x2n = x2n+1 , by (ii) we have

which implies that y2n = y2n+1.

If x2n ≠ x2n+1 , we use (i) to obtain y2n = y2n+1.

Similarly, we deduce that y2n+1 = y2n+2 and so yn = ym for every n ≥ m. Hence, {yn} is a Cauchy sequence.

Theorem 2. 3. Let (X, d) be a cone metric space and P be an order cone, let S, T, I and J: X → X be such that S(X) ⊆ J(X), T(X) ⊆ I(X). Assume that the following conditions are hold.

(i) (Sx, Ty) ≤ α d(Ix, Sx) + β d(Jy, Ty) + γ d(Ix, Jy), for all x, y∈X with x ≠ y, where α, β, γ are non-negative real numbers with α + β + γ < 1;

(ii) d(Sx, Tx) < d(Ix, Sx) + d(Jx, Tx) , for all x∈X, whenever Sx ≠ Tx.

If one of S(X), T(X), I(X) and J(X) is a complete subspace of X, then S, T and f have a unique point of coincidence. Moreover, if {S, I} and {T, J} are weakly compatible, then S, T, I and J have a unique common fixed point.

Proof. Let x0 be an arbitrary point in X. By Proposition 2.2 every S-T-sequence {yn} with initial point x₀ is a Cauchy sequence. Suppose J(X) is complete there exists p∈ J(X) such that y2n = Sx2n = Jx2n+1 → p as n→∞. We can find a u∈X such that Ju = p (If S(X) is complete, there exists p∈S(X)⊆J(X) , then the conclusions remain the same). Now we show that Tu = p. By (i), we have

Fix 0<<c and choose n₁ ∈N be such that d(y2n-1, y2n) << (1- α)/(β + γ) c/2, and d(y2n, p) << (1- α)/( γ+1) c/2 for all n ≥ n₁.

Consequently, d(p, Sv) << c/2 + c/2 = c.

⇒ d(p, Sv) << c and hence d(p, Sv) << c/m for every m∈N. From c/m – d(p, Sv) ∈int P, being P is closed as m→∞, we deduce –d(p, Sv)∈P and so d(v, Sp) = 0.

⇒ p = Sv.

Therefore, Sv = p = Iv.

Therefore, Tu = Ju = Sv = Iv (= p).

⇒ p is a point of coincidence of S, T, I and J have a unique point of coincidence. For his we assume that there exists another point p* in X such that p* = Tu* = Tu* = Sv* = Iv*, then

We deduce p = p*. Since, {S, I} and {T, J} are weakly compatible, then p is the unique point of coincidence of S, T, I and J then by the above Proposition we get that p is the unique common fixed point of S, T, I and J.

Remark 2.4.

(i) If we choose I = J = f in the above Theorem 2.3., we can get the Theorem 3.3., of [5].

The above Theorem 2.3., generalizes the Theorem 3.3., of [5].

(ii) If we choose S = T and I = J = f in the above Theorem 2.3., we can get the Theorem 3.4., of [5].

The above Theorem 2.3., generalizes the Theorem 3.4., of [5].

References

[1]   | M. Aamri and D. E. Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270, 181-188, 2002.
[2]   | M. Abbas and G. Jungck, Common fixed point results for non commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341, 416-420, 2008.
[3]   | M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. Vol.22, no.4, 511-515, 2009.
[4]   | I. Altun, and G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Ciric. Mat. Palermo 58, 319-325, 2009.
[5]   | M. Arshad, A. Azam and P. Vetro, Some common fixed point results in cone metric spaces, Fixed Point Theory and Applications, Volume, Article ID493965, 11, 2009.
[6]   | P. N. Dutta and B. S. Choudary, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008, Article ID 406386, 8, 2008.
[7]   | O. Ege and I. Karaca, Nielsen fixed point theory for digital images, Journal of Computational Analysis and Applications, 22(5), 874-880, 2017.
[8]   | L. G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332(2), 1468-1476, 2007.
[9]   | D. IIic, V. Rakocevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341, 876-882, 2008.
[10]   | Sh. Rezapour and R. Hamlbarani, Some notes on the paper cone metric spaces and fixed point theorem of contractive mappings, J. Math. Anal. Appl. 345, 719-724, 2008.
[11]   | P. Vetro, Common fixed points in cone metric spaces, Rendiconti del Circolo Matematico di Palermo, 56(3), 464-468, 2007.