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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.
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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.
3
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On the topological dimension of the solutions sets for some classes of operator and differential inclusions

100%
Bader R., Gel'man B., Kamenskii M., Obukhovskii V.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 1
17-32
EN
In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $𝓕 = S 𝓟_F$ where $𝓟_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.
4
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An abstract Cauchy problem for higher order functional differential inclusions with infinite delay

100%
Ke T., Obukhovskii V., Wong N.-C., Yao J.-C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2011
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tom 31
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nr 2
199-229
EN
The existence results for an abstract Cauchy problem involving a higher order differential inclusion with infinite delay in a Banach space are obtained. We use the concept of the existence family to express the mild solutions and impose the suitable conditions on the nonlinearity via the measure of noncompactness in order to apply the theory of condensing multimaps for the demonstration of our results. An application to some classes of partial differential equations is given.
5
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On asymptotics of solutions for a class of functional differential inclusions

100%
Kornev S., Obukhovskii V., Yao J.-C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2014
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tom 34
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nr 2
219-227
EN
We define a non-smooth guiding function for a functional differential inclusion and apply it to the study the asymptotic behavior of its solutions.
6
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On some topological methods in theory of neutral type operator differential inclusions with applications to control systems

100%
Kamenskii M., Obukhovskii V., Yao J.-C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2013
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tom 33
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nr 2
193-204
EN
We consider a neutral type operator differential inclusion and apply the topological degree theory for condensing multivalued maps to justify the question of existence of its periodic solution. By using the averaging method, we apply the abstract result to an inclusion with a small parameter. As example, we consider a delay control system with the distributed control.
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