Caristi’s Theorem and Extensions

  • William Kirk
  • Naseer Shahzad
Chapter
First Online:

Abstract

Much of the material immediately following is taken from [115]. We begin with two “equivalent” facts. The first is a well-known variational principle due to Ekeland [70, 71] and the second is the well-known Caristi Theorem [49]. Throughout we use \(\mathbb{R}\) to denote the set of real numbers, \(\mathbb{N}\) to denote the set of natural numbers, and \(\mathbb{R}^{+} = [0,\infty ).\) Recall that if X is a metric space, a mapping \(\varphi: X \rightarrow \mathbb{R}^{+}\) is said to be (sequentially) lower semicontinuous (l.s.c.) if given any sequence \(\left \{x_{n}\right \}\) in X, the conditions \(x_{n} \rightarrow x\) and \(\varphi \left (x_{n}\right ) \rightarrow r\) imply \(\varphi \left (x\right ) \leq r.\)

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • William Kirk
    • 1
  • Naseer Shahzad
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

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