Filippov Lemma for matrix fourth order differential inclusions

Grzegorz Bartuzel, and Andrzej Fryszkowski. "Filippov Lemma for matrix fourth order differential inclusions." Banach Center Publications 101.1 (2014): 9-18. <http://eudml.org/doc/282548>.

@article{GrzegorzBartuzel2014,
abstract = {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.},
author = {Grzegorz Bartuzel, Andrzej Fryszkowski},
journal = {Banach Center Publications},
keywords = {differential inclusion; differential operator; Lipschitz multifunction; Filippov lemma},
language = {eng},
number = {1},
pages = {9-18},
title = {Filippov Lemma for matrix fourth order differential inclusions},
url = {http://eudml.org/doc/282548},
volume = {101},
year = {2014},
}

TY - JOUR
AU - Grzegorz Bartuzel
AU - Andrzej Fryszkowski
TI - Filippov Lemma for matrix fourth order differential inclusions
JO - Banach Center Publications
PY - 2014
VL - 101
IS - 1
SP - 9
EP - 18
AB - 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.
LA - eng
KW - differential inclusion; differential operator; Lipschitz multifunction; Filippov lemma
UR - http://eudml.org/doc/282548
ER -