On extended versions of Dancs-Hegedüs-Medvegyev’s fixed-point theorem

Abstract

In this article we establish some fixed point (known also as critical point, invariant point) theorems in quasi-metric spaces. Our results unify and further extend in some regards the fixed point theorem proposed by Dancs et al. (1983), the results given by Quy (2010, 2011), the preorder principles established by , and the results obtained by . In addition, we provide examples to illustrate that the improvements of our results are significant.

Variational principles, completeness and the existence of traps in behavioral sciences

“…We conclude this section with an important observation clearly implying that the results obtained in Bao (2015), Bao et al (2015aBao et al ( , b, c, 2016, Bao and Théra (2015) are indeed established for quasimetric spaces in the sense of Definition 2.…”

“…We present and discuss definitions of generalized distances (known also as metrics) and of notions of closedness and completeness in topological spaces whose topologies are induced by these distances; (see, e.g., Bao and Théra 2015;Cobzaş 2011Cobzaş , 2013Kelly 1963;Reilly and Subrahmanyam 1982;Wilson 1931a).…”

Variational principles in set optimization with domination structures and application to changing jobs

“…Since we intend to apply our variational principles to worthwhile stay and change approach which might be viewed as a general model of human behavior, the domain space needs to be a quasi-metric space; see the last section for justifications of the chosen spaces. We follow notations in [4]; see also [6,11]. In contrast to metric spaces, there are several notions of convergence in (bitopological) quasimetric spaces; see [4,31] and the references therein.…”

“…We use the idea in the proof of extended versions of Dancs-Hegedüs-Medvegyev's fixedpoint theorem in [11]; for more details, see [6,7,10] that were used to establish versions of Ekeland's variational principles in set-valued optimization with ordering structure. In addition, we use marginal functions used in [30,54] associated with Gerstewitz' (nonlinear separation) scalarization function introduced in [21,22] (see also Krasnoselskii [27] for assertions in the context of operator theory and compare the scalar optimization problem by Pascoletti, Serafini [52]) is an important tool in nonconvex vector optimization.…”

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

“…The key tools used in the proofs of these results are Picard sequences for some special set-valued mappings corresponding to a function ϕ on a quasi-pseudometric space (see Subsection 2.5), which allow a unitary treatment of all these principles. The idea to use Picard sequences appeared in [13] and was subsequently exploited in [7] and [2].…”