Subclasses Determined by p-averages

Now, we deal with a condition that instead of arithmetical mean in the definition of α-lipschitzian mapping, uses average of order p ≥ 1.

Let (M,ρ) be a metric space, α = (α1, . . . , αn) be a multi-index and p ≥ 1. Definition 5.1 A mapping T : M → M is said to be (α, p)-lipschitzian for the constant k ≥ 0 (α and p as above) if, for every x, y ∈M , we have(

αiρ(T ix, T iy)p

≤ kρ(x, y). (5.1)

In fact, the above condition was suggested by Goebel and Japo´n Pineda in [33].