Sufficient conditions for the existence of heteroclinic solutions for Phi-Laplacian differential equations

In this paper we consider the second order discontinuous equation in the real line, (a(t)φ(u′(t)))′ = f(t,u(t),u′(t)), a.e.t∈R, u(-∞) = ν⁻, u(+∞)=ν⁺, with φ an increasing homeomorphism such that φ(0)=0 and φ(R)=R, a∈C(R,R\{0})∩C¹(R,R) with a(t)>0, or a(t)<0, for t∈R, f:R³→R a L¹-Carathéodory function and ν⁻,ν⁺∈R such that ν⁻<ν⁺. We point out that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities φ and f. Moreover, as far as we know, this result is even new when φ(y)=y, that is, for equation (a(t)u′(t))′=f(t,u(t),u′(t)), a.e.t∈R.

What are the conditions for good innovation results? A fuzzy-set approach for European Union

Innovation is one of the main concerns of European Union countries since the beginning of the century. Despite failing to reach their targets, innovation remains a priority because innovation enables countries to achieve better economic performance. This study analyzes the relation between the level of innovation and the economic effects and applies a fuzzy-set qualitative comparative analysis to study the relation between six conditions and two different outcomes. The data comes from the Union Innovation Scoreboard. The study finds that research systems, linkages and entrepreneurship, and intellectual assets are necessary conditions for the outcomes of a high level of innovation and positive economic effects. The main sufficient condition for both outcomes is a good research system.