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Discrete Filippov-type stability for one-sided Lipschitzian difference inclusions

Title data

Baier, Robert ; Farkhi, Elza:
Discrete Filippov-type stability for one-sided Lipschitzian difference inclusions.
Mathematisches Institut, Universität Bayreuth; School of Mathematical Sciences, Tel Aviv University
Bayreuth ; Tel Aviv , 2017 . - 24 p.

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Official URL: Volltext

Project information

Project financing: Andere
The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel

Abstract in another language

We state and prove Filippov-type stability theorems for discrete difference inclusions obtained by the Euler discretization of a differential inclusion with perturbations in the set of initial points, in the right-hand side and in the state variable. We study the cases in which the right-hand side of the inclusion is not necessarily Lipschitz, but satisfies a weaker one-sided Lipschitz (OSL) or strengthened one-sided Lipschitz (SOSL) condition. The obtained estimates imply stability of the discrete solutions for infinite number of fixed time steps if the OSL constant is negative and the perturbations are bounded in certain norms. We show a better order of stability for SOSL right-hand sides and apply our theorems to estimate the distance from the solutions of other difference methods, as for the implicit Euler scheme to the set of solutions of the Euler scheme. We also prove a discrete relaxation stability theorem for the considered difference inclusion, which also extends a theorem of G. Grammel (2003) from the class of Lipschitz maps to the wider class of OSL ones.

Further data

Item Type: Preprint, postprint, working paper, discussion paper
Additional notes: Accepted for publication in Lecture Notes in Economics and Mathematical Systems.
Dedicated to Vladimir Veliov's 65-th birthday.

Contents:
1. Introduction
2. Problem and Preliminaries
2.1 Preliminaries
2.2 Basic assumptions
3. Discrete Filippov-Type Theorems for One-Sided Lipschitz Maps
3.1 Outer perturbations
3.2 Inner perturbations
3.3 Both perturbations and applications
3.4 Discrete relaxation stability theorem
4. Discrete Filippov-Type Theorems for Strengthened One-Sided Lipschitz Maps
4.1 Both perturbations
4.2 Application
Keywords: one-sided Lipschitz condition; strengthened one-sided Lipschitz condition; set-valued Euler’s method; differential inclusions
Subject classification: Mathematics Subject Classification Code: 34A60 47H05 (39A30 54C60)
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 14 Oct 2017 21:00
Last Modified: 31 Oct 2018 12:38
URI: https://eref.uni-bayreuth.de/id/eprint/40030

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