Some periodic and fixed point theorems on quasi-b-gauge spaces

The notions of a quasi-b-gauge space $(U,\textsl{Q}_{s ; \Omega })$ ( U , Q s ; Ω ) and a left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -family of generalized quasi-pseudo-b-distances generated by $(U,\textsl{Q}_{s ; \Omega })$ ( U , Q s ; Ω ) are introduced. Moreover, by using this left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -family, we define the left (right) $\mathcal{J}_{s ; \Omega }$ J s ; Ω -sequential completeness, and we initiate the Nadler type contractions for set-valued mappings $T:U\rightarrow Cl^{\mathcal{J}_{s ; \Omega }}(U)$ T : U → C l J s ; Ω ( U ) and the Banach type contractions for single-valued mappings $T: U \rightarrow U$ T : U → U , which are not necessarily continuous. Furthermore, we develop novel periodic and fixed point results for these mappings in the new setting, which generalize and improve the existing fixed point results in the literature. Examples validating our obtained results are also given.

Periodic and Fixed Points for Caristi-Type G -Contractions in Extended b -Gauge Spaces

In this paper, we introduce extended b -gauge spaces and the extended family of generalized extended pseudo- b -distances. Moreover, we define the sequential completeness and construct the Caristi-type G -contractions in the framework of extended b -gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.

Completeness in Quasi-Pseudometric Spaces—A Survey

The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

Characterizing Lie groups by controlling their zero-dimensional subgroups

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.(ii) An abelian topological groupGis a Lie group if and only ifGis locally minimal, locally precompact and all closed metric zero-dimensional subgroups ofGare discrete.(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

Completeness of Ordered Fields and a Trio of Classical Series Tests

This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of R. The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field.

Weak Sequential Completeness of 𝑲(X,Y)

For Banach spaces X and Y, we show that if X* and Y are weakly sequentially complete and every weakly compact operator from X to Y is compact, then the space of all compact operators from X to Y is weakly sequentially complete. The converse is also true if, in addition, either X* or Y has the bounded compact approximation property.