Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem

Zili Wu

doi:10.1155/2010/970579

Abstract

With a recent result of Suzuki (2001) we extend Caristi-Kirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem in a complete metric space by replacing the distance with a -distance. In addition, these extensions are shown to be equivalent. When the -distance is l.s.c. in its second variable, they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds, which are in turn useful for extending some existing results such as the petal theorem.

Highlights

Let X, d be a complete metric space and f : X → −∞, ∞ a proper lower semicontinuous l.s.c. bounded below function
Caristi-Kirk fixed point theorem 1, Theorem 2.1 states that there exists x0 ∈ T x0 for a relation or multivalued mapping T : X → X if for each x ∈ X with infXf < f x there exists x ∈ T x such that d x, x f x ≤ f x, 1.1 see 2, Theorem 4.12 or 3, Theorem C while Ekeland’s variational principle EVP 4, 5 asserts that for each ∈ 0, ∞ and u ∈ X with f u ≤ infXf, there exists v ∈ X such that f v ≤ f u and f x d v, x > f v ∀x ∈ X with x / v
Kada et al have presented the concept of a w-distance in 10 to extend EVP, Caristi’s fixed point theorem, and Takahashi minimization theorem

Summary

Introduction

Let X, d be a complete metric space and f : X → −∞, ∞ a proper lower semicontinuous l.s.c. bounded below function. For the case where the τ-distance is l.s.c. in its second variable, we apply our generalizations to extend several existing results about the weak sharp minima and error bounds and demonstrate their equivalent relationship. P : X × X → 0, ∞ is said to be a τ-distance on X provided that τ1 p x, z ≤ p x, y p y, z for all x, y, z ∈ X × X × X and there exists a function η : X × 0, ∞ → 0, ∞ such that τ2 η x, 0 0 and η x, t ≥ t for all x, t ∈ X× 0, ∞ , and η is concave and continuous in its second variable; τ3 limn → ∞xn x and limn → ∞ sup{η zn, p zn, xm : n ≤ m} 0 imply p w, x lim inf p n→ ∞. For more properties of a τ-distance, the reader is referred to 11

Fixed Point Theorems

Extensions of Ekeland’s Variational Principle

Nonconvex Minimization Theorems

Generalized -Conditions of Takahashi and Hamel

Generalized Weak Sharp Minima and Error Bounds