Fixed Points and Completeness in Metric and Generalized Metric Spaces
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FIXED POINTS AND COMPLETENESS IN METRIC AND GENERALIZED METRIC SPACES S. Cobza¸s
UDC 515.126.4
Abstract. The famous Banach contraction principle holds in complete metric spaces, but completeness is not a necessary condition: there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces, this is the case of Ekeland’s variational principle and of its equivalent, Caristi’s fixed point theorem. Other fixed point results having this property will also be presented in metric spaces, in quasi-metric spaces, and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.
All roads lead to Rome. An old saying All topologies come from generalized metrics. Ralph Kopperman Am. Math. Mon., 95, No. 2, 89–97 (1988)
Introduction The famous Banach contraction principle holds in complete metric spaces, but completeness is not a necessary condition – there are incomplete metric spaces on which every contraction has a fixed point (see, e.g., [54]). The aim of the present paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland’s variational principle (and of its equivalent, Caristi’s fixed point theorem) (see, for instance, [30,112,171]) but this is also true in quasimetric spaces [39, 89] and in partial metric spaces [3, 151]. Other fixed point results having this property will also be presented. Various order completeness conditions of some ordered structures implied by fixed point properties will be considered as well. Concerning proofs, in several cases we give proofs, mainly to the converse results, i.e., completeness implied by fixed point results. In Sec. 3, we give full proofs to results relating topology and order as well as in Sec. 5 in what concerns the properties of partial metric spaces. 1. Banach Contraction Principle in Metric Spaces The Banach contraction principle was proved by S. Banach in his thesis from 1920, published in 1922 [24]. Although the idea of successive approximations in some concrete situations (solving differential and integral equations) appears in some works of P. L. Chebyshev, E. Picard, R. Caccioppoli, et al., it was Banach who placed it in the right abstract setting, making it suitable for a wide range of applications (see the expository paper [99]). 1.1. Contractions and Weakly Contractive Mappings. Let (X, ρ) and (Y, d) be metric spaces. A mapping f : X → Y is called Lipschitz if there exists a number α ≥ 0 such that ∀ x, y ∈ X d f (x), f (y) ≤ αρ(x, y). (1.1) Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 127–215, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0475
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The number α is called a Lipschitz constant for f , and one says sometimes that the mapping f is α-Lipschitz. If α = 0, then the mapping f is constant f (x
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