Locally Lipschitz Composition Operators in Space of the Functions of BoundedκΦ-Variation

Abstract

We give a necessary and sufficient condition on a function ℎ : R → R under which the nonlinear composition operator , associated with the function ℎ, ( ) = ℎ( ( )), acts in the space Φ [ , ] and satisfies a local Lipschitz condition.

Locally Defined Operators and Locally Lipschitz Composition Operators in the Space <i>WBV</i><i>p</i>(·)([a, b])

“…In recent articles J. Appell, N. Merentes, J. L. Sánchez [10], N. Merentes, S. Rivas, J. L. Sánchez [11] and O. Mejía, N. Merentes, B. Rzepka [12], obtained several results of the Sobolevskij type. According to the authors mentioned above the importance of these results lies in the fact that in most applications to many nonlinear problems it is sufficient to impose a local Lipschitz condition, instead of a global Lipschitz condition.…”

The Spaces of Functions of Two Variables of BoundedκΦ-Variation in the Sense of Schramm-Korenblum

“…It is largely to the possession of these properties that functions of bounded variation owe their important role in the study of rectifiable curves, Fourier series, Walsh-Fourier series, and other series, Stieltjes integrals, Henstock-Kurzweil integral, and other integrals, and the calculus of variations [7]. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis [8,9]. Two well-known generalizations are the functions of bounded -variation and the functions of bounded Φvariation, due to N. Wiener and L. C. Young, respectively.…”