Discrete Filippov-Type Stability for One-Sided Lipschitzian Difference Inclusions

Abstract

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.

A Filippov approximation theorem for strengthened one-sided Lipschitz differential inclusions

“…A somewhat stronger (uniform) version of the SOSL condition appears earlier in [50,51] (see remarks, e.g., in [8,9]). First order convergence of the Euler scheme is derived in [49] for 1d case and in [50,51] for higher dimensions for the unique solution of a differential inclusion satisfying this condition.…”

“…Another example would be the Hölder continuous function of degree 1 3 from [30, Example 5.4] which is OSL with constant μ = 1 2 but not SOSL. More variants of Lipschitz-type or OSL-type set-valued maps and corresponding examples can be found in [9] and [7,8].…”

“…2.4], [3,Sec. 10.4], [6,9,24,42,58]), results on the asymptotic behavior of the solutions and others (e.g., [31][32][33]36]). That is why extending the scope of the Filippov theorem beyond the family of Lipschitzian, and beyond the one of continuous maps, is an attractive field of investigation.…”

“…First order convergence of the Euler scheme is derived in [49] for 1d case and in [50,51] for higher dimensions for the unique solution of a differential inclusion satisfying this condition. Later, first order of convergence of the solution set for the explicit/implicit Euler method is derived in [9,53] also for the wider class of SOSL maps as defined here.…”

“…Although the SOSL condition is weaker than the Lipschitz continuity, it is strong enough to provide approximation results for differential inclusions (see [9,53]), better than for OSL maps and often the same as for Lipschitz maps. This is exactly the case with the Filippov approximation theorem (Theorem 1.1) proved here for SOSL maps in the right-hand side.…”