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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.

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 .

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.

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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