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Anne Lanham, Lillian Mwanri

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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.

Keywords: fixed point, common fixed point, ordered cone metric space, normal cone, nonnormal cone

Citation: K. Prudhvi, A Common Fixed Point Result in Ordered Complete Cone Metric Spaces. American Journal of Applied Mathematics and Statistics, 4 (2). 43-45. January 2016. doi:10.12691/ajams-4-2-3

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

[picture] | (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, x₀x₁…xnxn+1…. In particular, x2n and x2n+1 are comparable, by (1) we have

It follows that

That is,

[picture] | (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→+ ∞

[picture] | (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.
[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).
[4]   | I. Altun, B. Durmaz, Some fixed point theorems on ordered cone matric spaces, Rend. Circ. Mat. Palermo 58(2009) 319-325.
[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.
[6]   | D. IIic, V. Rakocevic, Quasi-contraction on a cone metric space, Appl. Math. Lett.22(2009)728-731.
[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.
[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.
[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.