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1

Three boundary value problems for second order differential inclusions in Banach spaces

Azzam D., Castaing C., Thibault L.

Control and Cybernetics

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2002

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Vol. 31, no 3

659-693

EN

The paper studies, in the context of Banach spaces, the problem of three boundary conditions for both second order differential inclusions and second order ordinary differential equations. The results are obtained in several new settings of Sobolev-type spaces involving Bochner and Pettis integrals. Some classes of second order multivalued evolution equations associated with m-accretive operators are also considered. Applications to some control problems are provided with the help of narrow convergence for Young measures.

2

The completeness in spaces of bounded pettis intergrable functions and in spaces of bounded functions satisfying the law of large numbers

Musiał K.

Demonstratio Mathematica

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2001

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Vol. 34, nr 2

339-344

EN

It has been proven already by Pettis [5] that the space P(, X) of Pettis integrable functions may be non-complete when endowed with the semivariation norm of the integrals. Then Thomas [9] proved that the space is almost always non-complete. In view of the Open Mapping Theorem in such a case no complete equivalent norm can be defined on P(p,,X). The question is now whether there are interesting linear subsets of P(/A, X) where a complete norm does exist. In this paper we consider two such subspaces: the space Poo (/^, X) of scalarly bounded Pettis integrable functions and the space LLNoo(^,X) of scalarly bounded functions satisfying the strong law of large numbers. We prove that in several cases these spaces are complete.

3

On the Fubini theorem for the Pettis integral for bounded functions

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 1

1-14

EN

We show that if X is a WGG Banach space and it does not contain any isomorphic copy of l1, then for every bounded Pettis integrable function f : [0, 1]^2 --> X* there exists a scalarly equivalent function for which the Fubini theorem for the Pettis integral holds. On the other hand, we show that for every bounded Pettis integrable function f : [0, 1]^2 --> l^2 (R) there exists a scalarly equivalent bounded function for which the Fubini theorem for the Pettis integral does not hold. We also show (assuming the Martin axiom) that there exists a bounded Pettis integrable function f : [0, 1]^2 --> L^[infinity](lambda) such that for each function g scalarly equivalent to f the function s --> g(t, s) is not weakly measurable for almost every t [belongs to] [0, 1].

4

On Pettis integrability of translations of functions in L^[infinity]

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

113-119

EN

Let K be a compact Hausdorff space, mi a positive Radon measure on K, and let G be a compact group with the Haar measure lambda. We consider properties of the following generalization of translations on groups: we associate with every bounded to mi x R[lambda]- measurable function f : K x G --> C the function T[sub f] : G --> L^[infinity] (mi x R[lambda]) given by T[sub f](t) = f[sub t] where f[sub t](r, s) = f (r, ts). We show that if K and G are metrizable and the class of f [belongs to] L^[infinity](mi x R[lambda]) contains a function g such that lambda({t [belongs to] G : (r, t) is a point of discontinuity of g}) = 0 for every r [belongs to] supp(mi), then T[sub f] is Pettis integrable with respect to lambda.