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Multivalued linear operators and differential inclusions in Banach spaces

100%
Baskakov A., Obukhovskii V., Zecca P.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2003
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tom 23
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nr 1
53-74
EN
In this paper, we study multivalued linear operators (MLO's) and their resolvents in non reflexive Banach spaces, introducing a new condition of a minimal growth at infinity, more general than the Hille-Yosida condition. Then we describe the generalized semigroups induced by MLO's. We present a criterion for an MLO to be a generator of a generalized semigroup in an arbitrary Banach space. Finally, we obtain some existence results for differential inclusions with MLO's and various types of multivalued nonlinearities. As a consequence, we give theorems on the existence of local, global and bounded solutions of the Cauchy problem for degenerate differential inclusions.
2
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Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator

100%
Benedetti I., Obukhovskii V., Zecca P.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2011
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tom 31
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nr 1
39-69
EN
We study a controllability problem for a system governed by a semilinear functional differential inclusion in a Banach space in the presence of impulse effects and delay. Assuming a regularity of the multivalued non-linearity in terms of the Hausdorff measure of noncompactness we do not require the compactness of the evolution operator generated by the linear part of inclusion. We find existence results for mild solutions of this problem under various growth conditions on the nonlinear part and on the jump functions. As example, we consider the controllability of an impulsive system governed by a wave equation with delayed feedback.
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On the topological structure of the solution set for a semilinear ffunctional-differential inclusion in a Banach space

100%
Conti G., Obukhovskiĭ V., Zecca P.
Banach Center Publications
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1996
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tom 35
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nr 1
159-169
EN
In this paper we show that the set of all mild solutions of the Cauchy problem for a functional-differential inclusion in a separable Banach space E of the form x'(t) ∈ A(t)x(t) + F(t,x_t) is an $R_δ$-set. Here {A(t)} is a family of linear operators and F is a Carathéodory type multifunction. We use the existence result proved by V. V. Obukhovskiĭ [22] and extend theorems on the structure of solutions sets obtained by N. S. Papageorgiou [23] and Ya. I. Umanskiĭ [32].
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