- Research Article
- Open Access

# Fixed Point Theorems on Spaces Endowed with Vector-Valued Metrics

- Alexandru-Darius Filip
^{1}and - Adrian Petruşel
^{1}Email author

**2010**:281381

https://doi.org/10.1155/2010/281381

© A.-D. Filip and A. Petruşel. 2010

**Received:**2 July 2009**Accepted:**21 December 2009**Published:**17 January 2010

## Abstract

The purpose of this work is to present some (local and global) fixed point results for singlevalued and multivalued generalized contractions on spaces endowed with vector-valued metrics. The results are extensions of some theorems given by Perov (1964), Bucur et al. (2009), M. Berinde and V. Berinde (2007), O'Regan et al. (2007), and so forth.

## Keywords

- Banach Space
- Fixed Point Theorem
- Successive Approximation
- Contraction Principle
- Contractive Type

## 1. Introduction

The classical Banach contraction principle was extended for contraction mappings on spaces endowed with vector-valued metrics by Perov in 1964 (see [1]).

Let be a nonempty set. A mapping is called a vector-valued metric on if the following properties are satisfied:

for all ; if , then ;

for all ;

for all .

If , , , and , by (resp., ) we mean that (resp., ) for and by we mean that for .

A set equipped with a vector-valued metric is called a generalized metric space. We will denote such a space with . For the generalized metric spaces, the notions of convergent sequence, Cauchy sequence, completeness, open subset, and closed subset are similar to those for usual metric spaces.

the open ball centered in with radius , by the closure (in ) of the open ball, and by

the closed ball centered in with radius

If is a singlevalued operator, then we denote by the set of all fixed points of ; that is,

For the multivalued operators we use the following notations:

Now, if is a multivalued operator, then we denote by the fixed points set of , that is, .

The set is called the graph of the multivalued operator .

In the context of a metric space , if , then we will use the following notations:

It is well known that is a generalized metric, in the sense that if , then .

Throughout this paper we denote by the set of all matrices with positive elements, by the zero matrix, and by the identity matrix. If , then the symbol stands for the transpose matrix of . Notice also that, for the sake of simplicity, we will make an identification between row and column vectors in .

Recall that a matrix is said to be convergent to zero if and only if as (see Varga [2]).

Notice that, for the proof of the main results, we need the following theorem, part of which being a classical result in matrix analysis; see, for example, [3, Lemma , page 55], [4, page 37], and [2, page 12]. For the assertion (iv) see [5].

Theorem 1.1.

Let . The following are equivalents.

(i) is convergent towards zero.

(ii) as .

(iii)The eigenvalues of are in the open unit disc, that is, , for every with .

(v)The matrix is nonsingular and has nonnenegative elements.

(vi) and as , for each .

Remark 1.2.

Some examples of matrix convergent to zero are

(a)any matrix , where and ;

(b)any matrix , where and ;

(c)any matrix , where and .

For other examples and considerations on matrices which converge to zero, see Rus [4], Turinici [6], and so forth.

Main result for self contractions on generalized metric spaces is Perov's fixed point theorem; see [1].

Theorem 1.3 (Perov [3]).

Let be a complete generalized metric space and the mapping with the property that there exists a matrix such that for all .

If is a matrix convergent towards zero, then

(1) ;

(2)the sequence of successive approximations , is convergent and it has the limit , for all ;

On the other hand, notice that the evolution of macrosystems under uncertainty or lack of precision, from control theory, biology, economics, artificial intelligence, or other fields of knowledge, is often modeled by semilinear inclusion systems:

Hence, it is of great interest to give fixed point results for multivalued operators on a set endowed with vector-valued metrics or norms. However, some advantages of a vector-valued norm with respect to the usual scalar norms were already pointed out by Precup in [5]. The purpose of this work is to present some new fixed point results for generalized (singlevalued and multivalued) contractions on spaces endowed with vector-valued metrics. The results are extensions of the theorems given by Perov [1], O'Regan et al. [7], M. Berinde and V. Berinde [8], and by Bucur et al. [9].

## 2. Main Results

We start our considerations by a local fixed point theorem for a class of generalized singlevalued contractions.

Theorem 2.1.

for all . We suppose that

(1) is a matrix that converges toward zero;

(2)if is such that , then ;

(3)

Then

In addition, if the matrix converges to zero, then .

Proof.

Using , we have .

Thus, by we get that and hence . Similarly, .

Since , by we get

Thus and hence .

Inductively, we construct the sequence in satisfying, for all , the following conditions:

(i) ;

(ii) ;

(iii) .

Hence is a Cauchy sequence. Using the fact that is a complete metric space, we get that is convergent in the closed set . Thus, there exists such that

Next, we show that

Indeed, we have the following estimation:

We show now the uniqueness of the fixed point.

which implies Taking into account that is nonsingular and we deduce that and thus

Remark 2.2.

for some matrices with a matrix that converges toward zero, could be called an almost contraction of Perov type.

We have also a global version of Theorem 2.1, expressed by the following result.

Corollary 2.3.

If is a matrix that converges towards zero, then

(1) ;

(2)the sequence given by converges towards a fixed point of , for all ;

where

In addition, if the matrix converges to zero, then

Remark 2.4.

Any matrix , where and , satisfies the assumptions ( )-( ) in Theorem 2.1.

Remark 2.5.

Let us notice here that some advantages of a vector-valued norm with respect to the usual scalar norms were very nice pointed out, by several examples, in Precup in [5]. More precisely, one can show that, in general, the condition that is a matrix convergent to zero is weaker than the contraction conditions for operators given in terms of the scalar norms on of the following type:

or

.

As an application of the previous results we present an existence theorem for a system of operatorial equations.

Theorem 2.6.

Let be a Banach space and let be two operators. Suppose that there exist , such that, for each , one has:

(1)

(2)

In addition, assume that the matrix converges to .

Then, the system

has at least one solution . Moreover, if, in addition, the matrix converges to zero, then the above solution is unique.

Proof.

Hence, Corollary 2.3 applies in , with .

We present another result in the case of a generalized metric space but endowed with two metrics.

Theorem 2.7.

Let be a nonempty set and let be two generalized metrics on . Let be an operator. We assume that

(1)there exists such that

(2) is a complete generalized metric space;

(3) is continuous;

If the matrix converges towards zero, then

In addition, if the matrix converges to zero, then

Proof.

Letting we obtain that . Thus is a Cauchy sequence with respect to .

On the other hand, using the statement , we get

Hence, is a Cauchy sequence with respect to . Since is complete, one obtains the existence of an element such that with respect to .

We prove next that , that is, . Indeed, since , for all , letting and taking into account that is continuous with respect to , we get that .

The uniqueness of the fixed point is proved below.

In what follows, we will present some results for the case of multivalued operators.

Theorem 2.8.

Let be a complete generalized metric space and let , with for each . Consider a multivalued operator. One assumes that

(ii)there exists such that

(iii)if is such that , then .

If is a matrix convergent towards zero, then .

Proof.

By induction, we construct the sequence in such that, for all , we have

(1)

(2)

(3) .

By a similar approach as before (see the proof of Theorem 2.1), we get that is a Cauchy sequence in the complete space . Hence is convergent in . Thus, there exists such that

Next we show that .

Using and the fact that , for all , we get, for each , the existence of such that

Letting , we get Hence, we have and since and is closed set, we get that .

Remark 2.9.

where is a fixed point for the multivalued operator , and the pair is arbitrary.

We have also a global variant for the Theorem 2.8 as follows.

Corollary 2.10.

If is a matrix convergent towards zero, then .

Remark 2.11.

where are multivalued operators satisfying a contractive type condition (see also [9]).

The following results are obtained in the case of a set endowed with two metrics.

Theorem 2.12.

Let be a complete generalized metric space and another generalized metric on . Let be a multivalued operator. One assumes that

(i)there exists a matrix such that , for all ;

(ii) has closed graph;

If is a matrix convergent towards zero, then .

Proof.

Let such that .

Consequently, we construct by induction the sequence in which satisfies the following properties:

(1) , for all ;

(2) , for all .

We show that is a Cauchy sequence in with respect to . In order to do that, let . One has the estimation

Since the matrix converges towards zero, one has as . Letting one get which implies that is a Cauchy sequence with respect to .

Using , we obtain that as . Thus, is a Cauchy sequence with respect to too.

Since is complete, the sequence is convergent in . Thus there exists such that with respect to .

Finally, we show that .

Since , for all and has closed graph, by using the limit presented above, we get that , that is, .

- (1)
Theorem 2.12 holds even if the assumption is replaced by

- (2)

Theorem 2.14.

Let be a complete generalized metric space and another generalized metric on . Let , with for each and let be a multivalued operator. Suppose that

(i)there exists such that , for all ;

(ii) has closed graph;

(iv)if is such that , then ;

(v)

Then .

Proof.

which implies .

Since , there exists such that

which implies that , that is, .

For , there exists such that

and thus , that is, .

Inductively, we can construct the sequence which has its elements in the closed ball and satisfies the following conditions:

(1) , for all ;

(2) , for all .

By a similar approach as in the proof of Theorem 2.12, the conclusion follows.

A homotopy result for multivalued operators on a set endowed with a vector-valued metric is the following.

Theorem 2.15.

Let be a generalized complete metric space in Perov sense, let be an open subset of , and let be a closed subset of , with . Let be a multivalued operator with closed (with respect to ) graph, such that the following conditions are satisfied:

(a) , for each and each ;

(b)there exist such that the matrix is convergent to zero such that for each , for each and all , there exists with .

(c)there exists a continuous increasing function such that for all , each and each there exists such that ;

(d)if are such that , then ;

Then has a fixed point if and only if has a fixed point.

Proof.

When , we obtain and, thus, is -Cauchy. Thus is convergent in . Denote by its limit. Since and since is -closed, we have that . Thus, from (a), we have . Hence . Since is totally ordered we get that , for each . Thus is an upper bound of . By Zorn's Lemma, admits a maximal element . We claim that . This will finish the proof.

Suppose . Choose with for each and such that , where . Since , by (c), there exists such that . Thus, .

Since , the multivalued operator satisfies, for all , the assumptions of Theorem 2.1 Hence, for all , there exists such that . Thus . Since , we immediately get that . This is a contradiction with the maximality of .

Conversely, if has a fixed point, then putting and using first part of the proof we get the conclusion.

Remark 2.16.

Usually in the above result, we take . Notice that in this case, condition (a) becomes

, for each and each .

Remark 2.17.

If in the above results we consider , then we obtain, as consequences, several known results in the literature, as those given by M. Berinde and V. Berinde [8], Precup [5], Petruşel and Rus [11], and Feng and Liu [12]. Notice also that the theorems presented here represent extensions of some results given Bucur et al. [9], O'Regan and Precup [13], O'Regan et al. [7], Perov [1], and so forth.

Remark 2.18.

Notice also that since is a particular type of cone in a Banach space, it is a nice direction of research to obtain extensions of these results for the case of operators on -metric (or -normed) spaces (see Zabrejko [14]). For other similar results, open questions, and research directions see [7, 11–13, 15–18].

## Declarations

### Acknowledgments

The authors are thankful to anonymous reviewer(s) for remarks and suggestions that improved the quality of the paper. The first author wishes to thank for the financial support provided from programs co-financed by The Sectoral Operational Programme Human Resources Development, Contract POS DRU 6/1.5/S/3-"Doctoral studies: through science towards society".

## Authors’ Affiliations

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