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Lipschitzian composition operators in some function spaces. (English) Zbl 0894.47052

There are several function spaces \(X\) with the property that, whenever the Nemytskij operator \(F\phi(x)= f(x,\phi(x))\) is Lipschitz continuous in the norm of \(X\), the generating function \(f\) must be affine, i.e. \(f(x, y)= g(x)y+ h(x)\) with \(g,h\in X\). For example, in case \(X= \text{Lip}\) by the author [Funkc. Ekvacioj Ser. Int. 25, 127-132 (1982; Zbl 0504.39008)], in case \(X= C^\alpha\) by A. Matkowska [Zeszyty Nauk. Polit. Łódz. Mat. 17, 81-85 (1984; Zbl 0599.46032)], in case \(X= C^\alpha_0\) by E. De Pascale, P. P. Zabrejko and the reviewer [Z. Anal. Anwendungen 6, 193-208 (1987; Zbl 0628.45003)], in case \(X= C^k\) by the author [Zeszyty Nauk. Polit. Łódz. Mat. 17, 5-10 (1984; Zbl 0599.46031)], and in case \(X=\text{BV}\) by the author and J. Mis [Math. Nachr. 117, 155-159 (1984; Zbl 0566.47033)]. Here the author proves the same for the space \(X= AC\) of absolutely continuous functions.

MSC:

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

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