Abstract
In Prager and Schwaiger (Grazer Math Ber 363:171–178, 2015) the classical notion of Banach limits was used to solve the inhomogeneous Cauchy equation \(f(x+y)-f(x)-f(y)=\phi (x,y)\) for real functions of one real variable. Here these methods are generalized to more general target spaces, namely Banach spaces which admit vector valued Banach limits.
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Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J.: On vector-valued banach limits. Funct. Anal. Appl. 47(4), 315–318 (2013)
Arendt, W., Nittka, R.: Lösungen zur Funktionalanalysis, Blatt 14 (2008/2009), Universität Ulm. https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/nittka/WS0809/FA/Blatt14sol.pdf. Accessed 24 June 2018
Banach, S.: Théorie des Opérations Linéaires. Monografje Matematyczne, Warszawa (1932)
Brézis, H.: Analyse fonctionnelle. Théorie et applications. (1983) Paris etc.: Masson. XIV
García-Pacheco, F.J., Pérez-Fernández, F.J.: Fundamental aspects of vector-valued banach limits. Izv. Math. 80(2), 316–328 (2016)
García-Pacheco, F.J., Pérez-Fernández, F.J.: Vector-valued banach limits and vector-valued almost convergence, (to appear)
Jessen, B., Karpf, J., Thorup, A.: Some functional equations in groups and rings. Math. Scand. 22, 257–265 (1968)
Lorentz, G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)
Prager, W., Reich, L.: Solutions of the inhomogeneous Cauchy equation. Result. Math. 54, 149–165 (2009)
Prager, W., Schwaiger, J.: On a conjecture of Ludwig Reich. Grazer Math. Ber. 363, 171–178 (2015)
Tim, B.: https://math.stackexchange.com/questions/2043262/weak-startopology-on-HrBa-finite-dimensional-spacecoincides-with-thestandart-topHrB. Accessed 24 June 2018
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Dedicated to Karol Baron on the occasion of his 70th birthday.
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Prager, W., Schwaiger, J. Vector valued Banach limits and generalizations applied to the inhomogeneous Cauchy equation. Aequat. Math. 93, 257–275 (2019). https://doi.org/10.1007/s00010-018-0589-9
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DOI: https://doi.org/10.1007/s00010-018-0589-9