Abstract. We prove existence of solution to a local fractional nonlinear differential equation with initial condition. For that we introduce the notion of tube solution.

No abstract

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples. words and phrases. Hausdorff derivative of a function with respect to a fractal measure; structural and fractal derivatives; self-similarity; time scales; Hilger derivative of non-integer order.This paper is in final form and no version of it will be submitted for publication elsewhere.

We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and integral on time scales are proved. Linear conformable differential equations with constant coefficients are investigated, as well as hyperbolic and trigonometric functions.