We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube [Formula: see text], [Formula: see text]. The procedure is based on the existence of special types of [Formula: see text]-dense curves (a generalization of the space-filling curves) in [Formula: see text] called [Formula: see text]-uniform curves.

We present some fixed point results in Banach algebras based on the so called degree of nondensifiability φ d . It is shown that φ d is an alternative method to measures of noncompactnes to obtain fixed point result. As an application of the usefulness of φ d it is proved the existence of solution for some quadratic integral equations.

In this paper we study the existence of solutions for an initial value problem, posed in a given Banach space, with a fractional differential equation via densifiability techniques. For our goal, we will prove a new fixed point result (not based on measures of noncompactness) which is, in forms, a generalization of the well-known Darbo’s fixed point theorem but essentially different. Some illustrative examples are given.

A novel technique to state the existence of solutions for certain infinite systems of differential equations is proposed. Our main tool will be the so called degree of nondensifiability, which seems to work under more general conditions than the measures of noncompactness. In fact, in our main result, the required conditions proposed for the existence of solutions of such system are more general than others required in most of the results based on such measures.

In 1905 Lebesgue showed that there is a sequence of continuous functions, put f n :such that f n (t) = a n for each positive integer n. This result was improved (in the sense of Theorem 1.1) in 1998 by Y. Benyamini. In this paper, we generalize the Benyamini's result in Theorem 4.1. The key tool for this goal are the so called α-dense curves. We apply our results to approach the solution of a certain infinite-dimensional linear program with a countable number of constraints.