Fixed Point Theorems on Spaces Endowed with Vector-Valued Metrics
© A.-D. Filip and A. Petruşel. 2010
Received: 2 July 2009
Accepted: 21 December 2009
Published: 17 January 2010
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.
The classical Banach contraction principle was extended for contraction mappings on spaces endowed with vector-valued metrics by Perov in 1964 (see ).
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.
For the multivalued operators we use the following notations:
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 ).
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 .
Some examples of matrix convergent to zero are
Main result for self contractions on generalized metric spaces is Perov's fixed point theorem; see .
Theorem 1.3 (Perov ).
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 . 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 , O'Regan et al. , M. Berinde and V. Berinde , and by Bucur et al. .
2. Main Results
We start our considerations by a local fixed point theorem for a class of generalized singlevalued contractions.
Indeed, we have the following estimation:
We show now the uniqueness of the fixed point.
We have also a global version of Theorem 2.1, expressed by the following result.
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 . 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:
As an application of the previous results we present an existence theorem for a system of operatorial equations.
Then, the system
We present another result in the case of a generalized metric space but endowed with two metrics.
In what follows, we will present some results for the case of multivalued operators.
We have also a global variant for the Theorem 2.8 as follows.
where are multivalued operators satisfying a contractive type condition (see also ).
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.
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:
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.
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 .
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 , Precup , Petruşel and Rus , and Feng and Liu . Notice also that the theorems presented here represent extensions of some results given Bucur et al. , O'Regan and Precup , O'Regan et al. , Perov , and so forth.
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 ). For other similar results, open questions, and research directions see [7, 11–13, 15–18].
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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