Asymptotic behavior of third order functional dynamic equations with γ\gamma-Laplacian and nonlinearities given by Riemann-Stieltjes integrals

In this paper, we study the third-order functional dynamic equations with $\gamma$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals \begin{equation*} \left\{ r_{2}\left( t\right) \phi _{\gamma _{2}}\left( \left[ r_{1}\left( t\right) \phi _{\gamma _{1}}\left( x^{\Delta }\left( t\right) \right) \right] ^{\Delta }\right) \right\} ^{\Delta }+\int_{a}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left( x(g\left( t,s\right) )\right) d\zeta \left( s\right) =0, \end{equation*} on an above-unbounded time scale $\mathbb{T}$, where $\phi_{\gamma }(u):=\left\vert u\right\vert^{\gamma -1}u$ and $\int_{a}^{b}f\left( s\right) d\zeta \left( s\right)$ denotes the Riemann-Stieltjes integral of the function $f$ on $[a,b]$ with respect to $\zeta$. Results are obtained for the asymptotic and oscillatory behavior of the solutions. This work extends and improves some known results in the literature on third order nonlinear dynamic equations

Inverse Sturm–Liouville problems with finite spectrum

AbstractWe study inverse Sturm–Liouville problems of Atkinson type whose spectrum consists entirely of a finite set of eigenvalues. We show that given two finite sets of interlacing real numbers there exists a class of Sturm–Liouville equations of Atkinson type such that the two sets of numbers are the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. Parallel results are also obtained for real coupled boundary conditions. Our approach is to use the equivalence between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems and to apply our development of the well-known theory for inverse matrix eigenvalue problems

Oscillation of forced impulsive differential equations with pp-Laplacian and nonlinearities given by Riemann-Stieltjes integrals

In this article, we study the oscillation of second order forced impulsive differential equation with $p$-Laplacian and nonlinearities given by Riemann-Stieltjes integrals of the form \begin{equation*} \left( p(t)\phi _{\gamma }\left( x^{\prime }(t)\right) \right) ^{\prime}+q_{0}\left( t\right) \phi _{\gamma }\left( x(t)\right)+\int_{0}^{b}q\left( t,s\right) \phi _{\alpha \left( s\right) }\left(x(t)\right) d\zeta \left(s\right) =e(t), t\neq \tau _{k}, \end{equation*} with impulsive conditions \begin{equation*} x\left( \tau _{k}^{+}\right) =\lambda _{k}~x\left( t_{k}\right), x^{\prime }\left( \tau _{k}^{+}\right) =\eta _{k}~x^{\prime }\left( \tau_{k}\right), \end{equation*} where \phi _{\gamma }\left( u\right) :=\left\vert u\right\vert ^{\gamma } \mbox{{\rm sgn}\,}u, $\gamma, b\in \left( 0,\infty \right),$ $\alpha \in C\left[ 0,b\right)$ is strictly increasing such that $0\leq \alpha \left( 0\right) <\gamma <\alpha \left( b-\right)$, and $\left\{ \tau_{k}\right\}_{k\in {\mathbb{N}}}$ is the the impulsive moments sequence. Using the Riccati transformation technique, we obtain sufficient conditions for this equation to be oscillatory

Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

We study classes of $n$th order boundary value problems consisting of an equation having a sign-changing nonlinearity $f(t,x)$ together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of $f(t,x)/x$ near $0$ and $\pm\infty$ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples

Nodal solutions of nonlocal boundary value problems

We study the nonlinear boundary value problem consisting of the second order differential equation on [a, b] and a boundary condition involving a Riemann‐Stieltjes integral. By relating it to the eigenvalues of a linear Sturm‐Liouville problem with a two‐point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes. First published online: 14 Oct 201

Towards Global Earthquake Early Warning with the MyShake Smartphone Seismic Network Part 2 -- Understanding MyShake performance around the world

The MyShake project aims to build a global smartphone seismic network to facilitate large-scale earthquake early warning and other applications by leveraging the power of crowdsourcing. The MyShake mobile application first detects earthquake shaking on a single phone. The earthquake is then confirmed on the MyShake servers using a "network detection" algorithm that is activated by multiple single-phone detections. In part two of this two paper series, we report the first order performance of MyShake's Earthquake Early Warning (EEW) capability in various selected locations around the world. Due to the present sparseness of the MyShake network in most parts of the world, we use our simulation platform to understand and evaluate the system's performance in various tectonic settings. We assume that 0.1% of the population has the MyShake mobile application installed on their smartphone, and use historical earthquakes from the last 20 years to simulate triggering scenarios with different network configurations in various regions. Then, we run the detection algorithm with these simulated triggers to understand the performance of the system. The system performs best in regions featuring high population densities and onshore, upper crustal earthquakes M<7.0. In these cases, alerts can be generated ~4-6 sec after the origin time, magnitude errors are within ~0.5 magnitude units, and epicenters are typically within 10 km of true locations. When the events are offshore or in sparsely populated regions, the alerts are slower and the uncertainties in magnitude and location increase. Furthermore, even with 0.01% of the population as the MyShake users, in regions of high population density, the system still performs well for earthquakes larger than M5.5. For details of the simulation platform and the network detection algorithm, please see part one of this two paper series.Comment: 13 figures, submitted to Seismological Research Letter