Du Bois-Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)-Laplacian on a Bounded Time Scale.

This paper is devoted to study the existence of solutions and their regularity in the p(t)-Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois-Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.

Degree, quaternions and periodic solutions.

The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.