Filippov Lemma for matrix fourth order differential inclusions

Grzegorz Bartuzel ; Andrzej Fryszkowski

Banach Center Publications, Tome 102 (2014), p. 9-18 / Harvested from The Polish Digital Mathematics Library

Résumé

In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices $A,B ∈ ℝ^{d×d}$ are commutative and the multifunction $F: [0,1] × ℝ^{d} ⇝ cl(ℝ^{d})$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that $F: [0,1] × ℝ^{d} ⇝ cl(ℝ^{d}) is measurable in t and integrably bounded. Let $y₀ ∈ W4,1$ be an arbitrary function satisfying (**) and such that $ $d_{H}(y₀(t),F(t,y₀(t))) ≤ p₀(t)$ a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*) with (**) such that |y(t)-y₀(t)| ≤ p₀(t) + l(Y₄(⋅,α,β)∗p₀)(t) |y(t)-y₀(t)| ≤ (Y₄(⋅,α,β)∗p₀)(t) a.e. in [0,1], where $Y₄(x,α,β) = (α^{-1}sinh(αx) - β^{-1}sinh(βx))/(α²-β²)$ and α,β depend on ||A||, ||B|| and l.

Détail

Publié le : 2014-01-01

Zbl 1300.34041

EUDML-ID : urn:eudml:doc:282548