We design shifted $LR$ transformations based on the integrable discrete hungry Toda equation to compute eigenvalues of totally nonnegative matrices of the banded Hessenberg form. The shifted $LR$ transformation can be regarded as an extension of the extension employed in the well-known dqds algorithm for the symmetric tridiagonal eigenvalue problem. In this paper, we propose a new and effective shift strategy for the sequence of shifted $LR$ transformations by considering the concept of the Newton shift....

The author considers the convergence of quasilinear nonstationary multistep methods for systems of ordinary differential with parameters. Sufficient conditions for their convergence are given. The new numerical method is tested for two examples and it turns out to be a little better than the Hamming method.

In this paper we obtain existence conditions and a closed form of the general solution of higher order singular regular boundary value problems. The approach is based on the concept of co-solution of algebraic matrix equations of polynomial type that permits the treatment of the problem without considering an extended first order system as it has been done in the known literature.

A couple ($\sigma ,\tau$) of lower and upper slopes for the resonant second order boundary value problem $$x^{\text{'}\text{'}}=f(t,x,x^{\text{'}}),x\left(0\right)=0,x^{\text{'}}\left(1\right)={\int }_0^1{x}^{\text{'}}\left(s\right)\mathrm{d}g\left(s\right),$$ with $g$ increasing on $[0,1]$ such that ${\int }_0^{1}dg=1$, is a couple of functions $\sigma ,\tau \in C^1\left([0,1]\right)$ such that $\sigma \left(t\right)\le \tau \left(t\right)$ for all $t\in [0,1]$, $$\begin{array}{c}{\sigma }^{\text{'}}\left(t\right)\ge f(t,x,\sigma \left(t\right)),\sigma \left(1\right)\le {\int }_0^1\sigma \left(s\right)\mathrm{d}g\left(s\right),\\ {\tau }^{\text{'}}\left(t\right)\le f(t,x,\tau \left(t\right)),\tau \left(1\right)\ge {\int }_0^1\tau \left(s\right)\mathrm{d}g\left(s\right),\end{array}$$ in the stripe ${\int }_0^t\sigma \left(s\right)\mathrm{d}s\le x\le {\int }_0^t\tau \left(s\right)\mathrm{d}s$ and $t\in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

We show the location of so called critical points, i.e., couples of diffusion coefficients for which a non-trivial solution of a linear reaction-diffusion system of activator-inhibitor type on an interval with Neumann boundary conditions and with additional non-linear unilateral condition at one or two points on the boundary and/or in the interior exists. Simultaneously, we show the profile of such solutions.