Existence of solutions for fractional differential equations with three-point boundary conditions at resonance in Rn\mathbb{R}^n

In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for the fractional three-point boundary value problems in $\mathbb{R}^n$, where the dimension of the kernel of fractional differential operator with the boundary conditions can take any value in $\{1,2,\ldots,n\}$. This is our novelty. Several examples are presented to illustrate the result

Edge states in self-complementary checkerboard photonic crystals: Zak phase, surface impedance and experimental verification

Edge states of photonic crystals have attracted much attention for the potential applications such as high transmission waveguide bends, spin dependent splitters and one-way photonic circuits. Here, we theoretically discuss and experimentally observe the deterministic edge states in checkerboard photonic crystals. Due to the self-complementarity of checkerboard photonic crystals, a common band gap is structurally protected between two photonic crystals with different unit cells. Deterministic edge states are found inside the common band gap by exploiting the Zak phase analysis and surface impedance calculation. These edge states are also confirmed by a microwave experiment.Comment: 13 pages, 4 figure