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

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## 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 Open image in new window be a nonempty set. A mapping Open image in new window is called a vector-valued metric on Open image in new window if the following properties are satisfied:

Open image in new window for all Open image in new window ; if Open image in new window , then Open image in new window ;

Open image in new window for all Open image in new window ;

Open image in new window for all Open image in new window .

If Open image in new window , Open image in new window , Open image in new window , and Open image in new window , by Open image in new window (resp., Open image in new window ) we mean that Open image in new window (resp., Open image in new window ) for Open image in new window and by Open image in new window we mean that Open image in new window for Open image in new window .

A set Open image in new window equipped with a vector-valued metric Open image in new window is called a generalized metric space. We will denote such a space with Open image in new window . 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 Open image in new window with radius Open image in new window , by Open image in new window the closure (in Open image in new window ) of the open ball, and by

the closed ball centered in Open image in new window with radius Open image in new window

If Open image in new window is a singlevalued operator, then we denote by Open image in new window the set of all fixed points of Open image in new window ; that is, Open image in new window

For the multivalued operators we use the following notations:

Now, if Open image in new window is a multivalued operator, then we denote by Open image in new window the fixed points set of Open image in new window , that is, Open image in new window .

The set Open image in new window is called the graph of the multivalued operator Open image in new window .

In the context of a metric space Open image in new window , if Open image in new window , then we will use the following notations:

It is well known that Open image in new window is a generalized metric, in the sense that if Open image in new window , then Open image in new window .

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

Recall that a matrix Open image in new window is said to be convergent to zero if and only if Open image in new window as Open image in new window (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 Open image in new window , page 55], [4, page 37], and [2, page 12]. For the assertion (iv) see [5].

Theorem 1.1.

Let Open image in new window . The following are equivalents.

(i) Open image in new window is convergent towards zero.

(ii) Open image in new window as Open image in new window .

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

(v)The matrix Open image in new window is nonsingular and Open image in new window has nonnenegative elements.

(vi) Open image in new window and Open image in new window as Open image in new window , for each Open image in new window .

Remark 1.2.

Some examples of matrix convergent to zero are

(a)any matrix Open image in new window , where Open image in new window and Open image in new window ;

(b)any matrix Open image in new window , where Open image in new window and Open image in new window ;

(c)any matrix Open image in new window , where Open image in new window and Open image in new window .

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 Open image in new window be a complete generalized metric space and the mapping Open image in new window with the property that there exists a matrix Open image in new window such that Open image in new window for all Open image in new window .

If Open image in new window is a matrix convergent towards zero, then

(1) Open image in new window ;

(2)the sequence of successive approximations Open image in new window , Open image in new window is convergent and it has the limit Open image in new window , for all Open image in new window ;

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 Open image in new window . We suppose that

(1) Open image in new window is a matrix that converges toward zero;

(2)if Open image in new window is such that Open image in new window , then Open image in new window ;

In addition, if the matrix Open image in new window converges to zero, then Open image in new window .

Proof.

Using Open image in new window , we have Open image in new window .

Thus, by Open image in new window we get that Open image in new window and hence Open image in new window . Similarly, Open image in new window .

Since Open image in new window , by Open image in new window we get

Thus Open image in new window and hence Open image in new window .

Inductively, we construct the sequence Open image in new window in Open image in new window satisfying, for all Open image in new window , the following conditions:

(i) Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

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

Next, we show that Open image in new window

Indeed, we have the following estimation:

We show now the uniqueness of the fixed point.

which implies Open image in new window Taking into account that Open image in new window is nonsingular and Open image in new window we deduce that Open image in new window and thus Open image in new window

Remark 2.2.

for some matrices Open image in new window with Open image in new window 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 Open image in new window is a matrix that converges towards zero, then

(1) Open image in new window ;

(2)the sequence Open image in new window given by Open image in new window converges towards a fixed point of Open image in new window , for all Open image in new window ;

where Open image in new window

In addition, if the matrix Open image in new window converges to zero, then Open image in new window

Remark 2.4.

Any matrix Open image in new window , where Open image in new window and Open image in new window , satisfies the assumptions ( Open image in new window )-( Open image in new window ) 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 Open image in new window is a matrix convergent to zero is weaker than the contraction conditions for operators given in terms of the scalar norms on Open image in new window of the following type:

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

Theorem 2.6.

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

In addition, assume that the matrix Open image in new window converges to Open image in new window .

Then, the system

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

Proof.

Hence, Corollary 2.3 applies in Open image in new window , with Open image in new window .

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

Theorem 2.7.

Let Open image in new window be a nonempty set and let Open image in new window be two generalized metrics on Open image in new window . Let Open image in new window be an operator. We assume that

(1)there exists Open image in new window such that Open image in new window

(2) Open image in new window is a complete generalized metric space;

(3) Open image in new window is continuous;

If the matrix Open image in new window converges towards zero, then Open image in new window

In addition, if the matrix Open image in new window converges to zero, then Open image in new window

Proof.

Letting Open image in new window we obtain that Open image in new window . Thus Open image in new window is a Cauchy sequence with respect to Open image in new window .

On the other hand, using the statement Open image in new window , we get

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

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

The uniqueness of the fixed point Open image in new window is proved below.

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

Theorem 2.8.

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

(ii)there exists Open image in new window such that Open image in new window

(iii)if Open image in new window is such that Open image in new window , then Open image in new window .

If Open image in new window is a matrix convergent towards zero, then Open image in new window .

Proof.

By induction, we construct the sequence Open image in new window in Open image in new window such that, for all Open image in new window , we have

(3) Open image in new window .

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

Next we show that Open image in new window .

Using Open image in new window and the fact that Open image in new window , for all Open image in new window , we get, for each Open image in new window , the existence of Open image in new window such that

Letting Open image in new window , we get Open image in new window Hence, we have Open image in new window and since Open image in new window and Open image in new window is closed set, we get that Open image in new window .

Remark 2.9.

where Open image in new window is a fixed point for the multivalued operator Open image in new window , and the pair Open image in new window is arbitrary.

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

Corollary 2.10.

If Open image in new window is a matrix convergent towards zero, then Open image in new window .

Remark 2.11.

where Open image in new window are multivalued operators satisfying a contractive type condition (see also [9]).

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

Theorem 2.12.

Let Open image in new window be a complete generalized metric space and Open image in new window another generalized metric on Open image in new window . Let Open image in new window be a multivalued operator. One assumes that

(i)there exists a matrix Open image in new window such that Open image in new window , for all Open image in new window ;

(ii) Open image in new window has closed graph;

If Open image in new window is a matrix convergent towards zero, then Open image in new window .

Proof.

Let Open image in new window such that Open image in new window .

Consequently, we construct by induction the sequence Open image in new window in Open image in new window which satisfies the following properties:

(1) Open image in new window , for all Open image in new window ;

(2) Open image in new window , for all Open image in new window .

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

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

Using Open image in new window , we obtain that Open image in new window as Open image in new window . Thus, Open image in new window is a Cauchy sequence with respect to Open image in new window too.

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

Finally, we show that Open image in new window .

Since Open image in new window , for all Open image in new window and Open image in new window has closed graph, by using the limit presented above, we get that Open image in new window , that is, Open image in new window .

- (1)
Theorem 2.12 holds even if the assumption Open image in new window is replaced by

- (2)Letting Open image in new window in the estimation of Open image in new window , presented in the proof of Theorem 2.12, we get(2.34)

Theorem 2.14.

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

(i)there exists Open image in new window such that Open image in new window , for all Open image in new window ;

(ii) Open image in new window has closed graph;

(iv)if Open image in new window is such that Open image in new window , then Open image in new window ;

Then Open image in new window .

Proof.

which implies Open image in new window .

Since Open image in new window , there exists Open image in new window such that

which implies that Open image in new window , that is, Open image in new window .

For Open image in new window , there exists Open image in new window such that

and thus Open image in new window , that is, Open image in new window .

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

(1) Open image in new window , for all Open image in new window ;

(2) Open image in new window , for all Open image in new window .

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

(a) Open image in new window , for each Open image in new window and each Open image in new window ;

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

(c)there exists a continuous increasing function Open image in new window such that for all Open image in new window , each Open image in new window and each Open image in new window there exists Open image in new window such that Open image in new window ;

(d)if Open image in new window are such that Open image in new window , then Open image in new window ;

Then Open image in new window has a fixed point if and only if Open image in new window has a fixed point.

Proof.

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

Suppose Open image in new window . Choose Open image in new window with Open image in new window for each Open image in new window and Open image in new window such that Open image in new window , where Open image in new window . Since Open image in new window , by (c), there exists Open image in new window such that Open image in new window . Thus, Open image in new window .

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

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

Remark 2.16.

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

Open image in new window , for each Open image in new window and each Open image in new window .

Remark 2.17.

If in the above results we consider Open image in new window , 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 Open image in new window 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 Open image in new window -metric (or Open image in new window -normed) spaces (see Zabrejko [14]). For other similar results, open questions, and research directions see [7, 11, 12, 13, 15, 16, 17, 18].

## Notes

### 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".

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