Caristi’s Theorem and Extensions

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 . $$\varphi \left (x\right ) \leq r.$$