Volume 39, Issue 4

The aim of this article is to form new existence theory for global solutions to nonlinear fractional differential equations. Traditional approaches to existence, uniqueness and approximation of global solutions for initial value problems involving fractional differential equations have been unwieldy or intractable due to the limitations of previously used methods. This includes, for example, certain invariance conditions of the underlying local fixed point strategies. Herein we draw on an alternative tactics, applying the more modern ideas of continuation methods for contractive maps to fractional differential equations. In doing so, we shed new light on the situation, producing these new perspectives through a range of novel theorems that involve sufficient conditions under which global existence, uniqueness, approximation and location of solutions are ensured.

In this paper we define and study rough convergence of triple sequences and the set of rough limit points of a triple sequence. We also investigate the relations between the set of cluster points and the set of rough limit points of Cauchy sequences of triple sequence spaces.

The Salem problem to verify whether Fourier–Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier–Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier–Stieltjes coefficients of a power m∈ℕ{m\in\mathbb{N}} of the Minkowski question mark function are derived.

We give a proof of the Euler’s infinite product for the sine using elementary trigonometric identities, and Tannery’s theorem for infinite products.