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Locally defined operators and a partial solution of a conjecture. (English) Zbl 1225.47079

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 \(A\subset \mathbb R^n\).

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Citations:

Zbl 0762.26015
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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
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