Caristi’s Theorem and Extensions

William Kirk and Naseer Shahzad
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William Kirk: University of Iowa, Department of Mathematics
Naseer Shahzad: King Abdulaziz University, Department of Mathematics

Chapter Chapter 2 in Fixed Point Theory in Distance Spaces, 2014, pp 7-18 from Springer

Abstract: 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 Caristi’s theorem [49]. Throughout we use ℝ $$\mathbb{R}$$ to denote the set of real numbers, ℕ $$\mathbb{N}$$ to denote the set of natural numbers, and ℝ + = [ 0 , ∞ ) . $$\mathbb{R}^{+} = [0,\infty ).$$ Recall that if X is a metric space, a mapping φ : X → ℝ + $$\varphi: X \rightarrow \mathbb{R}^{+}$$ is said to be (sequentially) lower semicontinuous (l.s.c.) if given any sequence x n $$\left \{x_{n}\right \}$$ in X, the conditions x n → x $$x_{n} \rightarrow x$$ and φ x n → r $$\varphi \left (x_{n}\right ) \rightarrow r$$ imply φ x ≤ r . $$\varphi \left (x\right ) \leq r.$$

Keywords: Caristi; Lower Semicontinuity; Khamsi; Order Principle; Theorem Turn (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-10927-5_2

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