Vol 89, No 4 (2025)

This paper is a continuations of the studies of [1] and [2] on existence of periodic bounded solutions for point-type functional differential equations, where deviations of the argument are defined in terms of a cyclic group of shifts on the real line. We prove an existence theorem for a bounded solution for equations in which the deviations of the argument are given by elements of a finitely generated group of orientation preserving diffeomorphisms of the real line.

The non-linear steepest descent method is employed to study the long-time asymptotics of solution to the non-local Lakshmanan–Porsezian–Daniel equation with step-like initial data$$q(x,0)=q_0(x)\to\begin{cases}0, &x\to-\infty,A, &x\to+\infty,\end{cases}$$where $A$ is an arbitrary positive constant. We first construct the basic Riemann–Hilbert (RH) problem. After that, to eliminate the influence of singularities, we use the Blaschke–Potapov factor to deform the original RH problem into a regular RH problem which can be clearly solved. Then different asymptotic behaviors on the whole $(x,t)$-plane are analyzed in detail. In the region $(x/t)^2<1/(27\gamma)$ with $\gamma>0$, there are three real saddle points due to which the asymptotic behaviors have a more complicated error term. We prove that the asymptotic solution constructed by the leading and error terms depends on the values of $\operatorname{Im}v(-\lambda_j)$, $j=1,2,3$, where $v(\lambda_j) =-(1/(2\pi))\ln|1+r_1(\lambda_j)r_2(\lambda_j)|-(i/(2\pi))\Delta(\lambda_j)$, $\Delta(\lambda_j)=\int_{-\infty}^{\lambda_j}d \arg(1+r_1(\zeta)r_2(\zeta))$, $r_i(\xi)$, $i=1,2$, are the reflection coefficients and $\lambda_j$ are the saddle points of thephase function $\theta(\xi,\mu)$. Besides, the leading term is characterized by parabolic cylinder functions and satisfies boundary conditions. In the region $(x/t)^2>1/(27\gamma)$ with $\gamma>0$, there are one real and two conjugate complex saddle points. Based on the positions of these points, we improve the extension forms of the jump contours and successfully obtain the large-time asymptotic results of the solution in this case.

Let $A$, $B$ be $n\times n$ normal matrices with eigenvalues $(a_1,…,a_n)$, $(b_1,…,b_n)$, respectively. We show that $\det(A+B)$ lies in the convex hull of$\bigcup_{\psi\in\mathcal{S}_n}\{\prod_{i=1}^n(a_i+b_{\psi_i})\}$if all eigenvalues of $A$, $B$ are real, except for three eigenvalues of $B$.