×

Commutativity of set-valued cosine families. (English) Zbl 1309.47048

Authors’ abstract: 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 \(\{F_t:t\geq 0\}\) is a regular cosine family of continuous additive set-valued functions \(F_t:K\to cc(K)\) such that \(x\in F_t(x)\) for \(t\geq 0\) and \(x\in K\), then \[ F_t\circ F_s(x)=F_s\circ F_t(x)\quad\text{for }s,t\geq 0\quad\text{and }x\in K. \]

MSC:

47D09 Operator sine and cosine functions and higher-order Cauchy problems
39B52 Functional equations for functions with more general domains and/or ranges
47H04 Set-valued operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[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; · Zbl 1257.47006
[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; · Zbl 0653.26001
[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; · Zbl 1276.26055
[5] Mainka-Niemczyk E., Multivalued second order differential problem, Ann. Univ. Paedagog. Crac. Stud. Math. 11 (2012), 53-67; · Zbl 1298.49024
[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; · Zbl 1245.26014
[7] Nikodem K., On concave and midpoint concave set-valued functions, Glasnik Mat. 22(42)(1987), 69-76; · Zbl 0642.39006
[8] Piszczek M., Integral representations of convex and concave set-valued functions, Demonstratio Math. 35 (2002), 727-742; · Zbl 1025.28005
[9] Piszczek M., Second Hukuhara derivative and cosine family of linear set-valued functions, Ann. Acad. Peadagog. Crac. Stud. Math. 5 (2006), 87-98; · Zbl 1156.26308
[10] Piszczek M., On multivalued cosine families, J. Appl. Anal. 14 (2007), 57-76; · Zbl 1131.26019
[11] Piszczek M., On multivalued iteration semigroups, Aequationes Math. 81 (2011), 97-108 http://dx.doi.org/10.1007/s00010-010-0034-1; · Zbl 1213.26030
[12] Sova M., Cosine operator functions, Dissertationes Math. (Rozprawy Mat.) 49 (1966), 1-47; · Zbl 0156.15404
[13] Smajdor A., Iteration of multivalued functions, Prace Naukowe Uniwersytetu Slaskiego w Katowicach Nr 759, Uniwersytet Slaski w Katowicach 1985; · Zbl 0595.20070
[14] Smajdor A., On regular multivalued cosine families, Ann. Math. Sil. 13 (1999), 271-280; · Zbl 0946.39013
[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; · Zbl 1026.39008
[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; · Zbl 1056.39036
[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; · Zbl 0388.34039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.