TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.Comment: 14 page

The Results on Coincidence and Common Fixed Points for a New Type Multivalued Mappings in b-Metric Spaces

In this paper, we obtain the results of coincidence and common fixed points in b-metric spaces. We work with a new type of multivalued quasi-contractive mapping with nonlinear comparison functions. Our results generalize and improve several recent results. Additionally, we give an application of the obtained results to dynamical systems

Some Fixed Points Results in b-Metric and Quasi b-Metric Spaces

We present a fixed point result in quasi b-metric spaces. Our result generalizes recent fixed point results obtained by Aleksit et al., Dung and Hang, Jovanovit et al., Sarwar, and Rahman and classical results obtained by Hardy, Rogers, and Cirie. Also, we obtain a common fixed point result in b-metric spaces. As a special case, we get a result of Cirie and Wong

Extension of two minimax theorems of S. Park

In this paper we prove two general minimax theorems which gen- eralize famous classical saddle point theorems of M. Sion [6] and J. von Neu- mann. Our theorems also include some results of S. Park [3]-[5]. Results of this type have many applications in the Game theory, because they gives existence of solution of zero sums games

A Common Fixed Point Theorem for Nonlinear Quasi-Contractions on b-Metric Spaces with Application in Integral Equations

In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies quasi-contractive inequality with nonlinear comparison functions. An application in solving a class of integral equations will support our results

On d*-Complete Topological Spaces and Related Fixed Point Results

In this paper, we introduce the concept ofd*-complete topological spaces, which include earlier defined classes of complete metric spaces and quasib-metric spaces. Further, we prove some fixed point results for mappings defined ond*-complete topological spaces, generalizing earlier results of Taskovic, Ciric and Presic, Presic, Bryant, Marjanovic, Yen, Caccioppoli, Reich and Bianchini

On d*-Complete Topological Spaces and Related Fixed Point Results

In this paper, we introduce the concept ofd*-complete topological spaces, which include earlier defined classes of complete metric spaces and quasib-metric spaces. Further, we prove some fixed point results for mappings defined ond*-complete topological spaces, generalizing earlier results of Taskovic, Ciric and Presic, Presic, Bryant, Marjanovic, Yen, Caccioppoli, Reich and Bianchini

Semi-metric spaces and fixed points of α - φ -contractive maps

A negative answer to an open problem is provided. Fixed point results for α -φ -contractive mappings in semi-metric spaces are proved. To show the generality of this results, examples are given. Finally, an application of this result to probabilistic spaces is derived

On the convergence of Ishikawa iterates defined by nonlinear quasi-contractions

In this study, we establish the convergence of Ishikawa iterates defined by nonlinear quasicontractive mappings on TVS-cone metric space. Further, our results generalize many existing results in the literature

Fifth European Dirofilaria and Angiostrongylus Days (FiEDAD) 2016

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