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The strong Ekeland variational principle in quasi-pseudometric spaces

S. Cobzaş

Abstract

Roughly speaking, Ekeland's Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324--353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. \textbf{131} (1988), no.~1, 1--21) and Tomonari Suzuki (J. Math. Anal. Appl. \textbf{320} (2006), no.~2, 787--794 and Nonlinear Anal. \textbf{72} (2010), no.~5, 2204--2209)), proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key-role played by reflexivity. In the last years an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, that is, to quasi-metric spaces (see the references). Applications to optimization, behavioral sciences, and others, were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev and Suzuki versions, to the quasi-pseudometric case. At the end we ask for the possibility to extend it to asymmetric normed spaces (i.e., the extension of Suzuki's results).

The strong Ekeland variational principle in quasi-pseudometric spaces

Abstract

Roughly speaking, Ekeland's Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324--353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. \textbf{131} (1988), no.~1, 1--21) and Tomonari Suzuki (J. Math. Anal. Appl. \textbf{320} (2006), no.~2, 787--794 and Nonlinear Anal. \textbf{72} (2010), no.~5, 2204--2209)), proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key-role played by reflexivity. In the last years an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, that is, to quasi-metric spaces (see the references). Applications to optimization, behavioral sciences, and others, were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev and Suzuki versions, to the quasi-pseudometric case. At the end we ask for the possibility to extend it to asymmetric normed spaces (i.e., the extension of Suzuki's results).
Paper Structure (10 sections, 16 theorems, 52 equations)

This paper contains 10 sections, 16 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Ekeland and the strong Ekeland variational principles in metric and Banach spaces
  3. Ekeland principle
  4. The strong Ekeland variational principle
  5. The case of quasi-pseudometric spaces
  6. Quasi-pseudometric spaces
  7. Ekeland principle in quasi-pseudometric spaces
  8. The strong Ekeland principle -- Georgiev's version
  9. The strong Ekeland principle -- Suzuki's versions
  10. Conclusions

Key Result

Theorem 2.1

Let $(X,d)$ be a complete metric space and $f:X\to \mathbb R\cup\{+\infty\}$ a lsc bounded below function. Let $\varepsilon > 0$ and $x_0\in \operatorname{dom} f.$ Then given $\lambda > 0$ there exists $\,z=z_{\varepsilon,\lambda}\in X\,$ such that If, further, $f(x_0) \leq \inf f(X) + \varepsilon,$ then

Theorems & Definitions (29)

  • Theorem 2.1: Ekeland Variational Principle
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4: Strong Ekeland Variational Principle
  • Remark 2.5
  • Theorem 2.6: suzuki06
  • Remark 2.7
  • Theorem 2.8: suzuki06
  • Theorem 2.9
  • Theorem 2.10
  • ...and 19 more