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• # Artykuł - szczegóły

## Banach Center Publications

2014 | 101 | 1 | 9-18

## Filippov Lemma for matrix fourth order differential inclusions

EN

### Abstrakty

EN
In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions
𝒟y = 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 ∈ W^{4,1}$be an arbitrary function satisfying (**) and such that$d_{H}(𝒟y₀(t),F(t,y₀(t))) ≤ p₀(t)$a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W^{4,1} of (*) with (**) such that |𝒟y(t)-𝒟y₀(t)| ≤ p₀(t) + l(Y₄(⋅,α,β)∗p₀)(t) |y(t)-y₀(t)| ≤ (Y₄(⋅,α,β)∗p₀)(t) a.e. in [0,1], where$Y₄(x,α,β) = (α^{-1}sinh(αx) - β^{-1}sinh(βx))/(α²-β²)\$
and α,β depend on ||A||, ||B|| and l.

9-18

wydano
2014

### Twórcy

autor
• Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
autor
• Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland