Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo–Katugampola derivative, is proven. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

The influence of photon excitation and proton irradiation on the luminescence properties of yttria stabilized zirconia doped with praseodymium ions

Lanthanide doped zirconia based materials are promising phosphors for lighting applications. Transparent yttria stabilized zirconia fibres, in situ doped with Pr3+ ions, were grown by the laser floating zone method. The single crystalline doped fibres were found to be homogeneous in composition and provide an intense red luminescence at room temperature. The stability of this luminescence due to transitions between the 1D2 → 3H4 multiplets of the Pr3+ ions (intra-4f2 configuration) was studied by photo- and iono-luminescence. The evolution of the red integrated photoluminescence intensity with temperature indicates that the overall luminescence decreases to ca. 40% of the initial intensity at 14 K when heated to room temperature (RT). RT analysis of the iono-luminescence dependence on irradiation fluence reveals a decrease of the intensity (to slightly more than ∼60% of the initial intensity after 25 min of proton irradiation exposure). Nevertheless the luminescence intensity saturates at non-zero values for higher irradiation fluences revealing good potential for the use of this material in radiation environments.

Nonlinear, nonhomogeneous parametric Neumann problems

We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator and with a Caratheodory reaction $f\left( t,x\right) $ which is $p-$superlinear in $x$ without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of the positive solutions on the parameter $\lambda>0,$ we show the existence of a smallest positive solution $\overline{u}_{\lambda}$ and investigate the properties of the map $\lambda\rightarrow\overline{u}_{\lambda}.$ Finally we also show the existence of nodal solutions.