Baire's category and the bang-bang property for evolution differential inclusions of contractive type

Blasi, F. S. De; Pianigiani, Giulio

doi:10.1016/j.jmaa.2010.01.049

Cited by 4 publications

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References 34 publications

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“…This method was introduced in [8,10,11], starting from a generic type result due to Cellina [5]. For further information on the Baire method and differential inclusions see [6,12,17,25,30].…”

Section: Introductionmentioning

confidence: 99%

Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Blasi¹,

Pianigiani²

2011

Proceedings of the Edinburgh Mathematical Society

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The existence of mild solutions to the non-convex Cauchy problemis investigated. Here A is the infinitesimal generator of a C0-semigroup in a reflexive and separable Banach space $\mathbb{E}$, F is a Pompeiu–Hausdorff lower semicontinuous multifunction whose values are closed convex and bounded sets with non-empty interior contained in $\mathbb{E}$, and ∂F(t, x(t)) denotes the boundary of F(t, x(t)). Our approach is based on the Baire category method, with appropriate modifications which are actually necessary because, under our assumptions, the underlying metric space that naturally enters in the Baire method, i.e. the solution set of the convexified Cauchy problem (CF), can fail to be a complete metric space.

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Section: Introductionmentioning

confidence: 99%

Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Blasi¹,

Pianigiani²

2011

Proceedings of the Edinburgh Mathematical Society

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Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

Blasi

Pianigiani

2012

Nonlinear Differ. Equ. Appl.

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Abstract. We consider evolution inclusions, in a separable and reflexive Banach space E, of the form ( * )where A is the infinitesimal generator of a C0-semigroup, F is a continuous and bounded multifunction defined on [t0, t1] × E with values F (t, x) in the space of all closed convex and bounded subsets of E with nonempty interior, and ext F (t, x(t)) denotes the set of the extreme points of F (t, x(t)). For ( * ) and ( * * ) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of ( * * ) contains the set of all internal solutions of ( * ). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C0-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values F (t, x) ⊂ E, the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of ( * * ) is equal to the set of the mild solutions of ( * ).Mathematics Subject Classification. Primary 34G25; Secondary 34A60, 49J27.

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On the “bang-bang” principle for nonlinear evolution hemivariational inequalities control systems

Bin

Liu

2019

Journal of Mathematical Analysis and Applications

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Baire's category and the bang-bang property for evolution differential inclusions of contractive type

Abstract: The Baire category method is employed in order to establish that the bang-bang property holds for a class of evolution differential inclusions of contractive type in reflexive and separable real Banach spaces.

Cited by 4 publications

References 34 publications

Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

On the “bang-bang” principle for nonlinear evolution hemivariational inequalities control systems

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