The Nemytskii operator in bounded (p,α)-variation space

Abstract

Abstract In this paper, we show that if the Nemytskii operator maps the ( p , α ) {(p,\alpha)} -bounded variation space into itself and satisfies some Lipschitz condition, then there are two functions g and h belonging to the ( p , α ) {(p,\alpha)} -bounded variation space such that f ⁢ ( t , y ) = g ⁢ ( t ) ⁢ y + h ⁢ ( t )   for all ⁢ t ∈ [ a , b ] , y ∈ ℝ . f(t,y)=g(t)y+h(t)\quad\text{for all }t\in[a,b],\,y\in\mathbb{R}.