Then (i) has at least one fixed point (ii) every Picard sequence of initial point converges to a fixed point of (iii) if are two distinct fixed points of, then. Let be a complete -metric space and let be a mapping such thatДЇor all. The proof that is a Cauchy sequence if ( 11) holds is the same.ДЄs a consequence of the previous result, we establish the following result of existence of fixed points. Since, then there exists such thatДЄnd by Lemma 6 we deduce again that is a Cauchy sequence. If, on taking limit as on both sides of ( 10), we obtain Thus there exists a nonnegative real number such that. If for all, from ( 10), we deduce that the sequence is decreasing. If for some, then is a fixed point of and so is a Cauchy sequence.ДЄssume that ( 10) holds for the sequence. Let be an arbitrary point, and let be a Picard sequence of initial point. Assume that a Picard sequence of initial point satisfies one of the following conditions: Let be a -metric space and let be a given mapping. In this section we consider a class of Picard sequences which are Cauchy. The sequence with for all is called a Picard sequence of initial point. Let be a -metric space, let, and let be a given mapping. Then for each and for all there exists such that. For any and any, we have the following: (i), for any (ii) (iii), for any (iv) (v) (vi) (vii). We recall the following properties from Czerwik et al. Note that is called the Hausdorff -metric induced by the -metric. Let be a -metric space and let be the collection of all nonempty closed bounded subsets of. Let be a sequence in a -metric space such thatДЇor some, , and each. ThenДЇrom Lemma 5, we deduce the following lemma. Let be a -metric space and let be a sequence in. The following results are useful for some of the proofs in the paper. We note that if are two nonnegative real numbers, then Is a -metric with constant and so is a -metric space. Next, we give some examples of -metric spaces.Д®xample 2. So the notions of convergent sequence, Cauchy sequence, and complete space are defined as in metric spaces.

We observe that a -metric space is a metric space if. A function is said to be a -metric if and only if for all the following conditions are satisfied: (1) if and only if (2) (3). Let be a nonempty set and let be a given real number. Let be the set of positive integers, the set of real numbers, and the set of nonnegative real numbers.Дefinition 1. The aim of this section is to collect some relevant definitions and results for our further use. Clearly, these results are a generalization of the results recently obtained from Khojasteh et al. The last part contains a fixed point result for multivalued mappings on complete -metric spaces. Then we use this result for establishing existence of fixed points for single-valued mappings. In the first part of this paper we consider a class of Picard sequences with a property of regularity and prove that these sequences are Cauchy.

For some results of fixed and common fixed point in the setting of -metric spaces, see. In dealing with the fixed point theory for single-valued and multivalued mappings in -metric spaces, seven years later, Khamsi and Khamsi and Hussain reintroduced such spaces under the name of metric-type spaces. The concept of -metric space appeared in some works, such as Bakhtin and Czerwik. The study of fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers see, for example.

Also its significance lies in its vast applicability in a number of branches of mathematics. It is widely considered as the source of metric fixed point theory. dissertation of Banach in 1920, which was published in 1922, is one of the most important theorems in classical functional analysis. It is well known that the contraction mapping principle, formulated and proved in the Ph.D. These results are generalizations of the analogous ones recently proved by Khojasteh, Abbas, and Costache. We obtain some fixed point results for single-valued and multivalued mappings in the setting of a -metric space.

Picard principal definition