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REMARKS ON A LASALLE CONJECTURE ON GLOBAL ASYMPTOTIC STABILITY.

Authors :
RUS, IOAN A.
Source :
Fixed Point Theory. 2016, Vol. 17 Issue 1, p159-172. 14p.
Publication Year :
2016

Abstract

In this paper we present some remarks on the following problem: Let X be a (real or complex) Banach space, Ω ⊂ X be an open convex subset and f: Ω → Ω be an operator. We suppose that: (i) f ∈ C¹ (X, X); (ii) the differential of f at x, df(x): X → X is a Picard operator for all x ∈ Ω; (iii) the fixed point set of f, Ff ≠ ∅. The problem is in which conditions f is a Picard operator? In the case X:= Rm or X:= Cm, this problem is in connection with a LaSalle Conjecture (J.P. LaSalle, The Stability of Dynamical Systems, SIAM, No. 25, 1976) and with the Belitskii-Lyubich Conjecture (G.R. Belitskii and Yu.I. Lyubich, Matrix Norms and their Applications, Birkhäuser, 1988). We also formulate the following conjecture: Let X be a Banach space, Ω ⊂ X be an open convex subset and f: Ω → Ω be an operator. We suppose that: (i) f ∈ C¹ (Ω, X); (ii) dfk (x) is a Picard operator, ∀ x ∈ Ω, ∀ k ∈ N∗; (iii) Ff ≠ ∅. Then f is a Picard operator. Some research directions are also presented. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
15835022
Volume :
17
Issue :
1
Database :
Academic Search Index
Journal :
Fixed Point Theory
Publication Type :
Academic Journal
Accession number :
116803075