Measurable multifunctions

Castaing, Charles; Valadier, Michel

doi:10.1007/bfb0087688

Cited by 134 publications

(133 citation statements)

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“…If G is a random set then there exists a random variable g with g(ω) ∈ G(ω) (see Castaing and Valadier [6] Chapter 3). A random set G is called tempered if the random variable…”

Section: Random Dynamical Systems and Stochastic Evolution Equationsmentioning

confidence: 99%

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

2004

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Abstract. We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semi-linear stochastic partial differential equations with Lipschitz continuous non-linearities.

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“…If G is a random set then there exists a random variable g with g(ω) ∈ G(ω) (see Castaing and Valadier [6] Chapter 3). A random set G is called tempered if the random variable…”

Section: Random Dynamical Systems and Stochastic Evolution Equationsmentioning

confidence: 99%

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

2004

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“…Then, the measurability of ω → dist x, ∪ t≥τ G (t, θ −t ω) B with respect to the P-completion of F follows from the projection theorem (see Castaing and Valadier [10], Theorem III.23). Taking…”

Section: Proofmentioning

confidence: 99%

Global attractors for multivalued random dynamical systems

Caraballo

Langa

Valero³

2002

Nonlinear Analysis: Theory, Methods & Applications

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“…(Here and in what follows the set-valued integral is the Lebesgue/Aumann integral, see, e.g., Castaing & Valadier [5].) Then we define the averaged inclusionẋ…”

Section: Averaging On a Finite Time Intervalmentioning

confidence: 99%

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Gama

Smirnov

2013

Set-Valued Var. Anal

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The averaging method is one of the most used tools to study dynamical systems. With the development of the theory of differential inclusions, a respective generalization of the averaging method followed the steps outlined in the theory of differential equations. Presently, it has been successfully applied to a wide range of problems involving differential inclusions, simplifying the study of the systems under consideration. In this work, the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions are surveyed. A detailed list of references is given and some examples of applications are presented.

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Measurable multifunctions

Cited by 134 publications

References 10 publications

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

Global attractors for multivalued random dynamical systems

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

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