Existence of solutions for higher order ϕ−\phi-Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions

In this paper, we consider the following $(n+1)$st order bvp on the half line with a $\phi-$Laplacian operator $$\begin{cases} (\phi(u^{(n)}))'(t)=f(t,u(t),\ldots,u^{(n)}(t)),& a.e., \,t\in [ 0,+\infty ),\\& n\in \mathbb{N}\setminus\{0\}, \\ u^{(i)}(0)=A_{i},\, i=0,\ldots,n-2,\\ u^{(n-1)}(0)+au^{(n)}(0)=B,\\ u^{(n)}(+\infty )=C. \end{cases}$$ The existence of solutions is obtained by applying Schaefer's fixed point theorem under a one-sided Nagumo condition with nonordered lower and upper solutions method where $f$ is a $L^{1}$-Carath\'eodory function

Mobility Edge in Aperiodic Kronig-Penney Potentials with Correlated Disorder: Perturbative Approach

It is shown that a non-periodic Kronig-Penney model exhibits mobility edges if the positions of the scatterers are correlated at long distances. An analytical expression for the energy-dependent localization length is derived for weak disorder in terms of the real-space correlators defining the structural disorder in these systems. We also present an algorithm to construct a non-periodic but correlated sequence exhibiting desired mobility edges. This result could be used to construct window filters in electronic, acoustic, or photonic non-periodic structures.Comment: RevTex, 4 pages including 2 Postscript figure