Pure and Applied Mathematics Journal
Volume 4, Issue 3, June 2015, Pages: 70-74

An Extension of a Fixed Point Result in Cone Banach Space

Elvin Rada1, Agron Tato2

1Department of Mathematics, Facultyof Natural Sciences, University of Elbasan, Elbasan, Albania

2Department of Mathematics, Polytechnic University of Tirana, Tirana, Albania

Email address:

(E. Rada)
(A. Tato)

To cite this article:

Elvin Rada, Agron Tato. An Extension of a Fixed Point Result in Cone Banach Space. Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 70-74.doi: 10.11648/j.pamj.20150403.12


Abstract: In this paper we study a class of mappings in a ConeBanach Space which have at least one fixed point. More precisely for a closed and convex subset C of a cone Banach space with a generalized norm that satisfy a special condition. We are proposing some extensions of the results of Karapinar.

Keywords: Fixed Point, Contraction, Cone Banach Space


1. Introduction and Preliminaries

In recent times, fixed point theory has been developed considerably in partially ordered metric spaces; that is, metric spaces endowed with a partial ordering. The application of mathematical modelling is highly relevant for different fields of science, such as engineering, physics, economics etc. A central issue in this topic is the rate of convergence of iterative or approximation processes. In this study we indirectly address this problem by extending some convergence theorems.

Rzepecki[1] in 1980 was the first that introduced the concept of a generalized metric on a set in a way that , whereis Banach space andis a normal cone inwith partial order. In that paper, theauthor generalized the fixed point theorems of Maia type [2].

Seven years later Lin [3] considered the notion of K-metric space by replacing real numbers with cone K in metric function, that is .

In 2007, Huang and Zhang [4] announced the notion of cone metric spaces (CMS) by replacing real numbers with an ordering Banach space. In that paper, they also discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces.

Recently, many results on fixed point theorems have been extended to cone metric spaces.( see example [3-8]).Karapinar [5] proved some fixed point theorems for self mappings satisfying some contractive condition in these spaces. Notice also that in ordered abstract spaces, existence of some fix point theorems is presented and applied the resolution of matrix equations (see e.g. 12)

Throughout this paper  stands for real Banach space. Let  always be a closed nonempty subset of . is called cone if for all  and nonnegative numberswhereand

For a given cone , one can define a partial ordering (denoted by or )with respectto byif and only if . The notation indicates that and while will show where denotes the interior of . From now on, it isassumed that

The cone is called

normal if there is a number such that for all :

(1.1)

Regular if every increasing sequence which is bounded from above is convergent.

That is, if is a sequence such that for some , then there is such that

In the least positive integer , satisfying (1.1), is called the normal constant of .

Lemma 1.1. (see [6, 10])

(i) Every regular cone is normal.

(ii) For each , there is a normal cone with normal constant .

(iii) The cone is regular if every decreasing sequence which is bounded from below isconvergent.

Proofs of (i) and (ii) are given in [6]and the last one follows from definition.

Definition 1.2. (see [4])

Let be a nonempty set. Suppose the mapping satisfies

(M1)for all ,

(M2)if and only if

(M3)for all ,

(M4) for all ,

thenis called cone metric on , and the pair is called a cone metric space (CMS).

Example 1.3.

Let ,

,and . Define

 

by . Then is a CMS. Note that the cone is normal with the normal constant .

It is quite natural to consider Cone Normed Spaces (CNS).

Definition 1.4 (see[7], [11])

Let be a vector space over . Suppose the mappingsatisfies:

(N1) for all

(N2) if and only if

(N3)

(N4) for all

then ||.||pis called cone norm on X, and the pair (X, ||.||p) is called a cone normed space (CNS).

Note that each CNS is CMS. Indeed,

Definition 1.5.

Let be a CNS, and  a sequence in . Then

(i)converges to whenever for every withthere is a naturalnumberN, such that for all . It is denoted by or

(ii)is a Cauchy sequence whenever for every withthere is a natural number N, such that for all.

(iii) is a complete cone normed space if every Cauchy sequence is convergent.

Complete cone normed spaces will be called cone Banach spaces.

Lemma 1.6.

Let be a CNS, P a normal cone with normal constant K, and a sequence in X. Then,

(i)the sequence converges to if and only if , as;

(ii)the sequence is Cauchy if and only if as

(iii)the sequence converges to and the sequence converges to then

The proof is direct by applying [4]to the cone metric space

where , for all .

Lemma 1.7. [7].

Let (X, ||.||p) be a CNS over a cone P in E. Then:

(i)and .

(ii)If then there exists such that implies

(ii)For any given and , there exists such that.

(iv)If are sequences in E such that , and , for all then .

The proofs of the first two parts followed from the definition of . The third partis obtained by the second part. Namely, if is given then find such that , implies . Then findsuch that and hence Sinceis closed, the proof of fourth part is achieved.

2. Main Results

Theorem 2.1.

Let C be a closed and convex subset of a cone Banach space X with norm and a mapping which satisfies the condition

(2.1)

for all  and, where,Then  has at least one fixed point.

Proof. Let  be an arbitrary point, Define a sequence  as follows:

(2.2)

Notice that

(2.3)

which can write

(2.4)

for If we substitute  then from (2.1) and (2.4) we obtain

(2.5)

Thus,  where or . Hence, the sequence is a Cauchy sequence in C then it converges to the point . Regarding the inequalityand the Lemma (1.6) we have

.

If we substitute  and  then inequation(2.1) and (2.3) imply

When , one can get , that is .

If the coefficient take the value , one obtains the follow theorem

Theorem 2.2. (Karapinar (2009))

Let  be a closed and convex subset of a cone Banach space X with norm and a mapping which satisfies the condition

for all  and , where . Then, T has at least one fixed point.

Theorem 2.3.

Let  be a closed and convex subset of a cone Banach space X with norm and  a mapping which satisfies the condition.

(2.6)

for all  and , where. Then,  has at least one fixed point.

Proof. Construct the sequence in same way as in proof of the theorem (2.1)and equalities (2.2) and (2.3) hold. Let we see also

(2.7)

from that

(2.8)

Triangle inequality for the points and  implies

(2.9)

then by (2.4), (2.7) and (2.8) we obtain

(2.10)

Replacing  and  in (2.6) we have

(2.11)

regarding (2.9), (2.10) and (2.8) one can obtain

(2.12)

by ( 2.8)

then or  . For these values, the sequence is a Cauchy sequence that converges to any . Since the sequence  also converges to  as in proof of the theorem (2.1)under the assumption that and . By means of lemma (1.6) we deduce which is equivalent with .

Corrollary 2.4. [5]

Let  be a closed and convex subset of a cone Banach space  with norm  and a mapping which satisfies the condition

for all  and , where Then,  has at least one fixed point.

Theorem 2.5.

Let be a closed and convex subset of a cone Banach space  with norm and a mapping which satisfies the condition

(2.13)

where. Then  has one fixed point.

Proof.Let we regard again three points . The triangle property one can write

replacing two terms of the right side with (2.3) and (2.7) we haveIf we substitute the right side to theinequation (2.13) on can obtain

(2.14)

The sequence is a Cauchy sequence  and also convergent if is satisfied the condition  or .

Notice that we get another proof for well knownBanach Theorem in cone Banach space.

Theorem 2.6.

Let be a closed and convex subset of a cone Banach space  with norm and a mapping which satisfies the condition

(2.14)

for all , , where

 . Then, has at least one fixed point.

Proof.Considering that the sequence is the same as one as is constructed in previoustheorems. Replace in inequality  and  one can obtain

(2.15)

regarding inequalities (2.3) and (2.7) we have

or

The inequality .

implies that if  above sequence is a Cauchy sequence. Further, proof is same as above theorems.

Corrollary 2.7. [5]

Let be a closed and convex subset of a cone Banach space  with norm and a mapping which satisfies the condition

for all . Then, has at least one fixed point.

This proposition derives from above theorem for.


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