Wróbel, Małgorzata Locally defined operators and a partial solution of a conjecture. (English) Zbl 1225.47079 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 495-506 (2010). Summary: We prove a weaker form of conjecture concerning the representation theorem for locally defined operators acting between some spaces of differentiable functions presented by K. Lichawski, J. Matkowski and J. Miś [Bull. Pol. Acad. Sci., Math. 37, No. 1–6, 315–325 (1989; Zbl 0762.26015)]. As a corollary, we obtain the representation formula for continuous locally defined operators mapping the Banach space \(C^m(A)\) into \(C^k(A)\) for \(k\geq 2\), where \(C^m(A)\) is the space of \(m\)-times Whitney continuously differentiable functions on a perfect set \(A\subset \mathbb R^n\). Cited in 7 Documents MSC: 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) Keywords:locally defined operator; \(m\)-times Whitney continuously differentiable functions; superposition operator; Nemytskii operator Citations:Zbl 0762.26015 PDFBibTeX XMLCite \textit{M. Wróbel}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, 495--506 (2010; Zbl 1225.47079) Full Text: DOI References: [1] Appell, J.; Zabrejko, P. P., Nonlinear Superposition Operators (1990), Cambridge University Press: Cambridge University Press Cambridge, Port Chester, Melbourne, Sydney · Zbl 0701.47041 [2] Lichawski, K.; Matkowski, J.; Miś, J., Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math., 37, 315-325 (1989) · Zbl 0762.26015 [3] Matkowski, J.; Wróbel, M., Locally defined operators in the space of Whitney differentiable functions, Nonlinear Anal. TMA, 68, 2873-3232 (2008) [4] Matkowski, J.; Wróbel, M., Representation theorem for locally defined operators in the space of Whitney differentiable functions, Manuscripta Math., 129, 437-448 (2009) · Zbl 1173.47044 [5] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36, 63-89 (1934) · JFM 60.0217.01 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.