A Common Fixed Point Result in Ordered Complete Cone Metric Spaces

K. Prudhvi

American Journal of Applied Mathematics and Statistics

A Common Fixed Point Result in Ordered Complete Cone Metric Spaces

K. Prudhvi

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

Abstract

In this paper, we prove a common fixed point theorem for ordered contractions in ordered cone metric spaces without using the continuity. Our result generalizes some recent results existing in the references.

Cite this article:

  • K. Prudhvi. A Common Fixed Point Result in Ordered Complete Cone Metric Spaces. American Journal of Applied Mathematics and Statistics. Vol. 4, No. 2, 2016, pp 43-45. http://pubs.sciepub.com/ajams/4/2/3
  • Prudhvi, K.. "A Common Fixed Point Result in Ordered Complete Cone Metric Spaces." American Journal of Applied Mathematics and Statistics 4.2 (2016): 43-45.
  • Prudhvi, K. (2016). A Common Fixed Point Result in Ordered Complete Cone Metric Spaces. American Journal of Applied Mathematics and Statistics, 4(2), 43-45.
  • Prudhvi, K.. "A Common Fixed Point Result in Ordered Complete Cone Metric Spaces." American Journal of Applied Mathematics and Statistics 4, no. 2 (2016): 43-45.

Import into BibTeX Import into EndNote Import into RefMan Import into RefWorks

1. Introduction

In 2007, Huang and Zhang [5] introduced the concept of a cone metric space and proved some fixed point theorems in cone metric space. Later on, many authors have generalized and extended the fixed point theorems of Huang and Zhang [5]. Fixed point theorems in partially ordered set was studied by Ran and Reurings [9], Nieto and Lopez [8]. Subsequently, many authors (see, e. g., [1, 2, 6]) were investigated the fixed point results on ordered metric spaces. Altun and Durmaz [4], Altun , Damnjanovic and Djoric [3] obtained fixed point theorems in ordered cone metric spaces. Recently, Kadelburg, Pavlovic and Radenovic [7] proved some common fixed point theorems in ordered contractions and quasicontractions in ordered cone metric spaces. In this paper, we proved a common fixed point theorem in ordered cone metric spaces without using the continuity. Our result, generalizes the results of [7].

The following definitions are in [5].

Definition 1.1. [5] Let E be a real Banach space and P be a subset of E. The set P is called a cone if and only if:

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

(b). a, b , a,b , x,y P imply ax+by P;

(c). x P and –x P implies x = 0.

Definition 1.2.[5] Let P be a cone in a Banach space E, define partial ordering with respect to P by if and only if y-xP. We shall write x y to indicate 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.[5] Let E be a Banach space and PE be an order cone. The order cone P is called normal if there exists L>0 such that for all x, yE,

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

Most of ordered Banach spaces used in applications posses a cone with the normal constant K = 1.

Definition 1.4. [5] Let X be a nonempty set of E. Suppose that the map d: X X→ E satisfies:

(d1). 0 d(x, y) for all x, y X 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.

Remark 1.5. [7] (1) If u v and v w , then u w.

(2) If 0u c for each c int P, then u = 0.

(3) If a b + c for each c int P, then a b.

(4) If 0 x y and 0 ≤ a , then 0 ax ay .

(5) If 0 xn yn , for each n, and , then 0 x y.

(6) If 0 d(xn, yn) bn and bn → 0, then, d(xn,x)c where xn , x are respectively, a sequence and a given point in X.

(7) If E is a real Banach space with a cone P and if a λa where a P and 0 < λ < 1, then a = 0.

(8) If c int P, 0 an and an → 0, then there exists n0 such that for all n > n0 we have an c.

2. Main Result

In this section, we prove a common fixed point theorem in an ordered complete cone metric spaces.

Theorem 2.1. Let (X, , d) be an ordered complete cone metric cone space. Let (f, g) be weakly increasing pair of self-maps on X w. r. t. . Suppose that the following conditions hold:

(i) there exists p, q, r, s, t ≥ 0 satisfying p + q + r + s + t < 1 and q = r or s = t, such that

(1)

for all comparable x, yX;

(ii) if a nondecreasing sequence {xn} converges to xX, then xn x for all n. Then, f and g have a common fixed point in X.

Proof. Let x0 X be arbitrary and define a sequence {xn} by x2n+1 = fx2n and x2n+2 = gx2n+1 for all nN. Since, (f, g) is weakly increasing , it can be easily shown that the sequence {xn} is nondecreasing w. r. t. , that is, x0x1xnxn+1…. In particular, x2n and x2n+1 are comparable, by (1) we have

It follows that

That is,

(2)

Similarly, we obtain

From (1) and (2), by induction, we obtain that

and

Let

In the case q = r,

Now, for n < m we have

Similarly, we obtain

and

Hence, for n < m

where bn → 0, as n→∞.

By using (8) and (1) of Remark 1.5 and only the assumption that the underlying cone is solid, we conclude that {xn} is a Cauchy sequence.

Since (X, d) is complete, there exists uX such that xn →u (as n→∞).

Letting n→+ ∞

(3)

Let c 0 be given. Choose a natural number N1 such that d(u, gu) c. Then from (3) we get that d(fu, u) c.

Since c is arbitrary, we get that

Noting that → 0 as m→∞, we conclude that as m→∞.

Hence, P is closed, then - d(fu, u) P.

Thus d(fu, u) P (-P). Hence d(fu, u) = 0.

Therefore, fu = u.

And

Letting n→+ ∞

That is, fu = gu.

Now we show that fu = gu = u. By (1), we have

Letting n→+ ∞

Therefore, fu = gu = u and u is a common fixed point of f and g.

Now, we consider the case when condition (ii) is satisfied. For the sequence {xn} we have xn → u X(as n→∞) and xn u(n). By the construction, fxn → u and gxn → u(as n→∞).

Let us prove that u is a common fixed point of f and g. Putting x = u and y = xn in (1)(since they are comparable) we get that

For the first and fourth term of the right hand side we have d(xn,u)c and d(u, gxn )c( for cint P arbitrary and n ≥ n0). For the second term d(u ,f u) ≼ d(u ,xn) + d(xn ,gxn) + d( gxn , fu)(again the first term n the right can be neglected) and for the fifth term d(xn ,f u) ≼ d( xn , gxn) + d(gxn , fu). It follows that

But xn → u and gxn → u ⇒ d(xn, gxn) c, which means that d(fu, gxn)<<c, that is, gxn →fu. It follows that, fu = u and in a symmetric way ( by using that u⊑u), gu =u.

Remark 2.2. If we choose f and g are continuous mappings in the above Theorem 2.1, then we get the Theorem 2.1 of [7].

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.
In article      View Article
 
[2]  M. Abbas , B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 21(2008)511-515.
In article      
 
[3]  I. Altun, B. Damnjanovic, D. Djoric, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. (2009).
In article      
 
[4]  I. Altun, B. Durmaz, Some fixed point theorems on ordered cone matric spaces, Rend. Circ. Mat. Palermo 58(2009) 319-325.
In article      View Article
 
[5]  L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332(2)(2007) 1468-1476.
In article      View Article
 
[6]  D. IIic, V. Rakocevic, Quasi-contraction on a cone metric space, Appl. Math. Lett.22(2009)728-731.
In article      View Article
 
[7]  Z. Kadelburg , M. Pavlovic and S. Radenovic, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comp. and Math. with Appl. 59(2010) 3148-3159.
In article      View Article
 
[8]  J.J. Nietro, R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(2005)223-239.
In article      View Article
 
[9]  A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc. 132.
In article      
 
  • CiteULikeCiteULike
  • MendeleyMendeley
  • StumbleUponStumbleUpon
  • Add to DeliciousDelicious
  • FacebookFacebook
  • TwitterTwitter
  • LinkedInLinkedIn