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A class of retracts in $L^{p}$ with some applications to differential inclusion

100%

Bartuzel G., Fryszkowski A.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2002

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tom 22

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nr 2

213-224

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Filippov Lemma for matrix fourth order differential inclusions

100%

Bartuzel G., Fryszkowski A.

Banach Center Publications

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2014

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tom 101

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nr 1

9-18

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.

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Filippov Lemma for certain second order differential inclusions

100%

Bartuzel G., Fryszkowski A.

Open Mathematics

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2012

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tom 10

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nr 6

1944-1952

EN

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y′(0) = 0, where the matrix A ∈ ℝd×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y 0 ∈ W 2,1 be an arbitrary function fulfilling the above initial conditions and such that where p 0 ∈ L 1[0, 1]. Then there exists a solution y ∈ W 2,1 to the above differential inclusions such that a.e. in [0, 1], .

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Abstract differential inclusions with some applications to partial differential ones

100%

Bartuzel G., Fryszkowski A.

Annales Polonici Mathematici

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1991

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tom 53

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nr 1

67-78

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Banach Center Publications

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Open Mathematics

4 Bartuzel G.

4 Fryszkowski A.

1 2014

1 2012

1 2002

1 1991

A class of retracts in $L^{p}$ with some applications to differential inclusion

Filippov Lemma for matrix fourth order differential inclusions

Filippov Lemma for certain second order differential inclusions

Abstract differential inclusions with some applications to partial differential ones