Cent. Eur. J. Math. • 7(3) • 2009 • 520-528 DOI: 10.2478/s11533-009-0027-2 Central European Journal of Mathematics Fixed point results for multivalued contractions on ordered gauge spaces Research Article Gabriela Petrușel 1∗ 1 Faculty of Business, Babeș-Bolyai University, Cluj-Napoca, Romania Received 9 February 2009; accepted 1 May 2009 Abstract: The purpose of this article is to present fixed point results for multivalued E ≤ -contractions on ordered complete gauge space. Our theorems generalize and extend some recent results given in M. Frigon [7], S. Reich [12], I.A. Rus and A. Petrușel [15] and I.A. Rus et al. [16]. MSC: 47H04, 47H10, 54H25, 54C60 Keywords: Gauge space • Multivalued operator • Fixed point • Data dependence © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Throughout this paper E will denote a gauge space endowed with a separating gauge structure D = {d α } α∈Λ , where Λ is a directed set (see [5] for definitions). Let N := {0, 1, 2, ···}. A sequence (x n ) of elements in E is said to be Cauchy if for every ε> 0 and α ∈ Λ, there is an N ∈ N with p α (x n ,x n+p ) ≤ ε, for all n ≥ N and p ∈ N ∗ . The sequence (x n ) is called convergent if there exists an x ∗ ∈ X such that for every ε> 0 and α ∈ Λ, there is an N ∈ N with p α (x ∗ ,x n ) ≤ ε, for all n ≥ N. A gauge space is called complete if any Cauchy sequence is convergent. A subset of E is said to be closed if it contains the limit of any convergent sequence of its elements. See also J. Dugundji [5] for other definitions and details. If D = {d α } α∈Λ is a separating gauge structure, then for r = {r α } α∈Λ ∈ (0, ∞) Λ and x 0 ∈ E, we will denote by B d (x 0 ,r) the closure of B d (x 0 ,r) in (E, D ), where B d (x 0 ,r)= {x ∈ E : d α (x 0 ,x ) <r α for all α ∈ Λ}. Let P(E) := {Y ⊆ E : Y = ∅}. ∗ E-mail: gabi.petrusel@tbs.ubbcluj.ro 520