We show that the composition operatorH, associated withh:[a,b]→ℝ, maps the spacesLip[a,b]on to the spaceκBVϕa,bof functions of bounded variation in Schramm-Korenblum's sense if and only ifhis locally Lipschitz. Also, verify that if the composition operator generated byh:[a,b]×ℝ→ℝmaps this space into itself and is uniformly bounded, then regularization ofhis affine in the second variable.

In this paper, we proof some properties of the space of bounded p(⋅)-variation in Wiener's sense. We show that a functions is of bounded p(⋅)-variation in Wiener's sense with variable exponent if and only if it is the composition of a bounded nondecreasing functions and hölderian maps of the () p 1 ⋅ variable exponent. We show that the composition operator H, associated with h   : → , maps the spaces () [ ] () p WBV a b , ⋅ into itself if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by [ ] h a b   : , × → maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable.

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.

In this paper we present the notion of the space of bounded p(⋅)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator H, associated with h → :  , maps the () [ ] () W p BV a b ⋅ , κ into itself, if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by [ ] h a b × → : ,   maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable, i.e. satisfies the Matkowski's weak condition.

The purpose of this paper is twofold. Firstly, we introduce the concept of boundedκΦ-variation in the sense of Schramm-Korenblum for real functions with domain in a rectangle ofR2. Secondly, we study some properties of these functions and we prove that the space generated by these functions has a structure of Banach algebra.