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2014 | 12 | 12 | 1871-1881
Tytuł artykułu

Commutativity of set-valued cosine families

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Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.
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Bibliografia
  • [1] Aghajani M., Nourouzi K., On the regular cosine family of linear correspondences, Aequationes Math. 83 (2012), 215–221 http://dx.doi.org/10.1007/s00010-011-0112-z
  • [2] Edgar G.A., Measure, Topology and Fractal Geometry, Undergrad.Texts Math., Springer-Verlag New York Inc., New York, 1990 http://dx.doi.org/10.1007/978-1-4757-4134-6
  • [3] Łojasiewicz S., An Introduction to the Theory of Real Functions, Wiley, Chichester - New York - Brisbane - Toronto - Singapore 1988
  • [4] Mainka-Niemczyk E., Integral representation of set-valued sine families, J. Appl. Anal. 18(2) (2012), 243–258 http://dx.doi.org/10.1515/jaa-2012-0016
  • [5] Mainka-Niemczyk E., Multivalued second order differential problem, Ann. Univ. Paedagog. Crac. Stud. Math. 11 (2012), 53–67
  • [6] Mainka-Niemczyk E., Some properties of set-valued sine families, Opuscula Math. 32(1) (2012), 157–168 http://dx.doi.org/10.7494/OpMath.2012.32.1.159
  • [7] Nikodem K., On concave and midpoint concave set-valued functions, Glasnik Mat. 22(42)(1987), 69–76
  • [8] Piszczek M., Integral representations of convex and concave set-valued functions, Demonstratio Math. 35 (2002), 727–742
  • [9] Piszczek M., Second Hukuhara derivative and cosine family of linear set-valued functions, Ann. Acad. Peadagog. Crac. Stud. Math. 5 (2006), 87–98
  • [10] Piszczek M., On multivalued cosine families, J. Appl. Anal. 14 (2007), 57–76
  • [11] Piszczek M., On multivalued iteration semigroups, Aequationes Math. 81 (2011), 97–108 http://dx.doi.org/10.1007/s00010-010-0034-1
  • [12] Sova M., Cosine operator functions, Dissertationes Math. (Rozprawy Mat.) 49 (1966), 1–47
  • [13] Smajdor A., Iteration of multivalued functions, Prace Naukowe Uniwersytetu Slaskiego w Katowicach Nr 759, Uniwersytet Slaski w Katowicach 1985
  • [14] Smajdor A., On regular multivalued cosine families, Ann. Math. Sil. 13 (1999), 271–280
  • [15] Smajdor A., Hukuhara’s derivative and concave iteration semigrups of linear set-valued functions, J. Appl. Anal. 8 (2002), 297–305 http://dx.doi.org/10.1515/JAA.2002.297
  • [16] Smajdor A., Hukuhara’s differentiable iteration semigrups of linear set-valued functions, Ann. Polon. Math. 83(1) (2004), 1–10 http://dx.doi.org/10.4064/ap83-1-1
  • [17] Trevis C.C., Webb G.F., Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32(3–4) (1978), 75–96 http://dx.doi.org/10.1007/BF01902205
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