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Covering dimension and differential inclusions

Giovanni Anello — 2000

Commentationes Mathematicae Universitatis Carolinae

In this paper we shall establish a result concerning the covering dimension of a set of the type { x X : Φ ( x ) Ψ ( x ) } , where Φ , Ψ are two multifunctions from X into Y and X , Y are real Banach spaces. Moreover, some applications to the differential inclusions will be given.

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

Giovanni AnelloGiuseppe Cordaro — 2003

Colloquium Mathematicae

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ - Δ p u + λ ( x ) | u | p - 2 u = μ f ( x , u ) in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω N is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, λ L ( Ω ) with e s s i n f x Ω λ ( x ) > 0 and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

Giovanni AnelloPaolo Cubiotti — 2004

Annales Polonici Mathematici

We consider a multifunction F : T × X 2 E , where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

Non-autonomous implicit integral equations with discontinuous right-hand side

Giovanni AnelloPaolo Cubiotti — 2004

Commentationes Mathematicae Universitatis Carolinae

We deal with the implicit integral equation h ( u ( t ) ) = f ( t , I g ( t , z ) u ( z ) d z ) for a.a. t I , where I : = [ 0 , 1 ] and where f : I × [ 0 , λ ] , g : I × I [ 0 , + [ and h : ] 0 , + [ . We prove an existence theorem for solutions u L s ( I ) where the contituity of f with respect to the second variable is not assumed.

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