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1
Content available Necessary optimality conditions for a set valued fractional programming problem in terms of contingent epiderivatives
Gadhi A. N., Idrissi M. El.
Control and Cybernetics
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2018
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Vol. 47, no. 2
147--156
EN
In this paper, we are concerned with a multi-objective fractional extremal programming problem. Using the concept of subdifferential of cone-convex set valued mappings, introduced by Baier and Jahn (1999), together with the convex separation principle, we give necessary optimality conditions. An example illustrating the usefulness of our results is also provided.
2
Content available remote Support functions and subdifferentials
Grzybowski J., Pallaschke D., Urbański R.
Commentationes Mathematicae
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2016
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Vol 56, No. 1
45--53
EN
In this paper we study Minkowski duality, i.e. the correspondence between sublinear functions and closed convex sets in the context of dual pairs of vector spaces.
3
Content available remote A short proof of the separable reduction theorem
Penot J. P.
Demonstratio Mathematica
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2010
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Vol. 43, nr 3
653-663
EN
We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.
4
Content available remote On uniform differentiability
Rolewicz S.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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Vol. 56, no 3-4
231-237
EN
We introduce the notion of uniform Frechet differentiability of mappings between Banach spaces, and we give some sufficient conditions for this property to hold.
5
Content available remote Radiality and semismoothness
Jourani A.
Control and Cybernetics
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2007
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Vol. 36, no 3
669-680
EN
We provide sufficient conditions for radiality and semismoothness. In general Banach spaces, we show that calmness ensures Dini-radiality as well as Dini-convexity of solution set to inequality systems. In finite dimensional spaces, we introduce the concept of Clarke-radiality and semismoothness of order m and show that each subanalytic set satisfies these properties. Similar properties are obtained for locally Lipschitzian subanalytic functions.
6
Content available remote Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms
Barbu V., Lasiecka I., Rammaha A.
Control and Cybernetics
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2005
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Vol. 34, no 3
665-687
EN
In this article we focus on the global well-posedness of the differential equation u [...] in Omega x(O, T), where j' denotes the derivative of a C1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obt ains existence, regularity or finite time blow up of solutions. More specifically, when p [is less than or equal to] m + k global existence of generalized solutions in H1 x L2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H2 x H1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.
7
Content available remote Rotundity, smoothness and duality
Penot J.-P.
Control and Cybernetics
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2003
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Vol. 32, no 4
711-733
EN
The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.
8
Content available remote Periodic solutions for evolution inclusions with time-dependent subdifferentials
Matzakos N., Papageorgiu N. S.
Demonstratio Mathematica
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2002
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Vol. 35, nr 3
521-538
EN
In this paper we examine a periodic evolution equation driven by a time-dependent subdifferential and with a multivalued forcing term. Using a fixed point theorem for pseudo-acyclic multifunctions we prove the existence of periodic trajectories. This approach requires a study of the structure of the solution set of the Cauchy problem, which is also conducted in this paper.
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