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

Paolo Cubiotti — 2001

Commentationes Mathematicae Universitatis Carolinae

We deal with the integral equation u ( t ) = f ( t , I g ( t , z ) u ( z ) d z ) , with t I : = [ 0 , 1 ] , f : I × n n and g : I × I [ 0 , + [ . We prove an existence theorem for solutions u L s ( I , n ) , s ] 1 , + ] , where f is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where f does not depend explicitly on the first variable t I .

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia ChinnìPaolo Cubiotti — 1996

Annales Polonici Mathematici

Let E be a Banach space. We consider a Cauchy problem of the type ⎧ D t k u + j = 0 k - 1 | α | m A j , α ( D t j D x α u ) = f in n + 1 , ⎨ ⎩ D t j u ( 0 , x ) = φ j ( x ) in n , j=0,...,k-1, where each A j , α is a given continuous linear operator from E into itself. We prove that if the operators A j , α are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions u C ( n + 1 , E ) whose derivatives are equi-bounded on each bounded subset of n + 1 .

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.

On generalized games in H -spaces

Paolo CubiottiGiorgio Nordo — 1999

Commentationes Mathematicae Universitatis Carolinae

We show that a recent existence result for the Nash equilibria of generalized games with strategy sets in H -spaces is false. We prove that it becomes true if we assume, in addition, that the feasible set of the game (the set of all feasible multistrategies) is closed.

Vector integral equations with discontinuous right-hand side

Filippo CammarotoPaolo Cubiotti — 1999

Commentationes Mathematicae Universitatis Carolinae

We deal with the integral equation u ( t ) = f ( I g ( t , z ) u ( z ) d z ) , with t I = [ 0 , 1 ] , f : 𝐑 n 𝐑 n and g : I × I [ 0 , + [ . We prove an existence theorem for solutions u L ( I , 𝐑 n ) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1 .

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