# Relationship between T V S-partial cone metric spaces, dislocated

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## Partial metric type structures and

Lipschitzian mappings The notion of metric type structure was originally developed and studied by Khamsi  in 2010 as a generalization of metric spaces. Inspired and motivated by this notion we introduce partial metric type structure as a generalization of partial metric spaces and metric type structures and present some fixed point results of Lipschitzian mappings in this setting. The chapter is aligned as follows: Preliminaries and some basic notions are recalled in Section 3.1. In Section 3.2 we present some fundamental properties of partial metric type structures and further show that every partial metric type space gives rise to a metric type space. Lipschitzian mappings and some fixed point results on metric type structures and partial metric type structures are discussed in Section 3.3. Most results in this chapter can be found in the papers  and .

Completeness in symmetric spaces

and some fixed point results In the literature the notion of completeness for metric spaces is discussed in terms of Cauchy sequences. Unlike in the classical case (metric spaces), in symmetric spaces not every convergent sequence is a Cauchy sequence. Motivated by the notion of absolute closure , the author in  defined a new notion of completeness for symmetric spaces. It should be observed that this new notion is equivalent to completeness when restricted to the class of metric spaces. A similar study was done for quasi pseudo metric spaces  and for symmetric spaces . The reader should also note that using the classical notion of (Cauchy) completeness for symmetric spaces, analogous fixed point results are presented in the literature see for example  for single valued maps; , , ,  and  for multivalued maps. We note that the paper  presents a fixed point theory result of a single valued maps as presented in , without appealing to Cauchy sequences.

### Properties of T V S-cone metric spaces

In what follows we recall basic properties of T V S-cone metric spaces and refer the reader to ,  and  for more details. In this section by (X, σ) we refer to (X, P, E, σ) where X is a nonempty set, E is a normed topological vector space, P is a normal cone in E with normal constant K and σ is a T V S-cone metric on X. Definition 1.4.1  A topological vector space (T V S) E is a vector space over a topological field K that is endowed with a topology such that the vector addition and scalar multiplication are continuous functions.

Relationship between T V S-partial

cone metric spaces, dislocated metric spaces and metric spaces In the literature, it has been established that T V S-cone metric spaces and metric spaces are equivalent see,  and . Also, it is shown that every partial metric space gives rise to a metric space . In this chapter we discuss the relationship between T V S-partial cone metric spaces, dislocated metric spaces and metric spaces. In particular, we show that a T V S-partial cone metric space does not gives rise to a partial metric space, unlike in the case where a T V S-cone metric space give rise to an equivalent metric space as seen in Chapter 1 and , . In fact, T V S-partial cone metric space gives rise to a dislocated metric space but the two are not equivalent. The chapter shall unfold as follows: Section 2.1 present some relationship on T V Spartial cone metric spaces.

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#### Completeness in symmetric spaces

and some fixed point results In the literature the notion of completeness for metric spaces is discussed in terms of Cauchy sequences. Unlike in the classical case (metric spaces), in symmetric spaces not every convergent sequence is a Cauchy sequence. Motivated by the notion of absolute closure , the author in  defined a new notion of completeness for symmetric spaces. It should be observed that this new notion is equivalent to completeness when restricted to the class of metric spaces. A similar study was done for quasi pseudo metric spaces  and for symmetric spaces . The reader should also note that using the classical notion of (Cauchy) completeness for symmetric spaces, analogous fixed point results are presented in the literature see for example  for single valued maps; , , ,  and  for multivalued maps. We note that the paper  presents a fixed point theory result of a single valued maps as presented in , without appealing to Cauchy sequences. The notion of a complete metric space and symmetric space is very important and so, is the completion of such structures; recently a completion for the dislocated metric spaces is presented in .

Contents :

• Acknowledgements
• Abstract
• Summary
• Declaration
• 1 Introduction
• 1.1 Metric spaces and some fixed point results
• 1.2 Quasi metric spaces and their properties
• 1.3 Partial metric spaces and some Lipschitzian mappings
• 1.4 Properties of T V S-cone metric spaces
• 1.5 T V S-partial cone metric spaces and their properties
• 2 Relationship between T V S-partial cone metric spaces, dislocated
• metric spaces and metric spaces
• 2.1 More properties of T V S-partial cone metric spaces
• 2.2 T V S-partial cone metric spaces and T V S-quasi cone metric spaces
• 2.3 T V S-partial cone metric spaces and T V S-cone metric spaces
• 3 Partial metric type structures and Lipschitzian mappings
• 3.1 Metric type structures and Lipschitzian mappings
• 3.2 Dislocated metric type structures
• 3.2.1 T V S-partial cone metric spaces and partial metric type spaces
• 3.3 Lipschitzian mappings and some fixed point results in partial metric
• type structures
• 4 Completeness in symmetric spaces and some fixed point results
• 4.1 Some properties on symmetric spaces
• 4.2 Completeness in symmetric spaces
• 4.3 Products of symmetric spaces
• 4.4 Symmetric spaces and fixed point results

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ON COMPLETENESS OF PARTIAL METRIC SPACES, SYMMETRIC SPACES AND SOME FIXED POINT RESULTS