arXiv subjects
Publications and source records attributed to Pablo Amster.
A novel 2+1-dimensional extension of the solitonic Dym equation is shown to admit a Painlev\'e II symmetry reduction which permits the exact solution of a class of Stefan-type moving boundary problems.
A nonlinear time-delay model is proposed to describe the interaction dynamics between criminal and non-criminal populations, combining social influence mechanisms, saturation effects represented by a Holling type II functional response, and time-dependent law-enforcement actions. The delay accounts for the latency between exposure to criminal behavior and behavioral response, introducing memory effects that naturally lead to a delay differential equations framework. Fundamental analytical properties, including positivity, global existence, and invariance of the feasible region, are established to ensure the mathematical consistency of the population interpretation. In the autonomous setting, explicit threshold conditions governing the stability of the criminal-free equilibrium and the emergence of coexistence states are derived, while the delay is shown to induce stability switches and oscillatory regimes through characteristic root crossings. In the non-autonomous case, topological degree arguments guaranty the existence of strictly positive periodic solutions, indicating that long-term dynamics depend primarily on the averaged law enforcement intensity measures rather than on short-term fluctuations. These results identify time delay as a key structural mechanism underlying recurrent patterns and complex temporal behavior in crime dynamics.
Here, a novel 2+1-dimensional nonlinear evolution equation with temporal modulation is introduced which admits integrable Ermakov-Painlev\'e II symmetry reduction. Application is made to obtain exact solution to a class of Stefan-type moving boundary problems for this 2+1-dimensional nonlinear evolution equation. Involutory transformations with origin in autonomisation of certain Ermakov-type coupled systems are extended to 2+1-dimensions and applied to derive a wide 2+1-dimensional class with temporal modulation and which inherits the property of admittance of such hybrid Ermakov-Painlev\'e II symmetry reduction applicable to certain moving boundary problems.
In 1+1-dimensions, an extension of the canonical solitonic Dym equation has previously been derived both in a geometric torsion evolution context and in the analysis of peakon solitonic phenomena in hydrodynamics. Here, a novel 2+1-dimensional S-integrable extended Dym-type equation is introduced. As Lax pair is constructed and an associated $\bar{\partial}$-dressing scheme detailed. Integrable modulated versions of the 2+1-dimensional extended Dym equation are generated via application of a class of involutory transformations with genesis in classical Ermakov theory.
The main purpose of this paper is to analyze the dynamics of the system of time-delay differential equations (DDEs) \begin{equation*} \begin{split} \dot{T}(t)&=T(t) f(t,T(t))-\gamma E(t)T(t),\\ \dot{E}(t)&=\sigma+ \frac{pE(t)T(t-\tau_1)}{g+a T(t-\tau_1)}-\frac{mE(t)T(t-\tau_2)}{g+a T(t-\tau_2)}-\eta E(t), \end{split} \end{equation*} where $T=T(t)$ and $E=E(t)$ represent the concentrations of tumor and effector cells at the time $t$. The coefficients $\sigma$, $\mu$, $\gamma$, and $\eta$ are all positive, and $f(t, T)$ represents the relative growth rate of tumor cells, corresponding to a generalized logistic growth function that describes periodic time chemotherapeutic effects. The parameter $\tau_1 \in \mathbb{R}_{\ge 0}$ is the response time delay of the immune system (mediated by effector cells) to an invasion of tumor cells, while $\tau_2 \in \mathbb{R}_{\ge 0}$ represents the time delay of tumor cells in response to the appearance of effector cells.
Two-point boundary value problems for a discrete Ermakov-Painlev\'e II equation are analysed by means of topological methods. In addition, an alternative variational approach is detailed. Existence of solutions is established for appropriate choice of parameters.
We study an almost periodic version of a metapopulation model developed by Tilman \textit{et.al} and Nee \textit{et.al} in the nineties, which generalizes the classical Levins approach by considering several species in competition affected by habitat destruction. The novelty is to assume that the colonization and extinction rates are positive almost periodic functions whereas our main results show that the predominance of either colonization or extinction forces of a specific species is equivalent to the property of exponential dichotomy of a scalar linear differential equation. By using well known results of exponential dichotomy theory, we carry out a recursive and exhaustive description of persistence/extinction scenarios. In addition, we start a preliminary discussion describing a more elusive behavior when the colonization and extinction forces are similar in average.
An abstract version of the celebrated inequality is described by means of the spectral bound of an operator defined on a Banach lattice. As a consequence, uniqueness and continuous dependence results for the general semilinear problem $Lu=N(u)$ are established and a connection with the maximum principle is explored.
In this paper, we prove the existence of two positive $T$-periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation $$\ddot{x}(t)+f_{D}(x(t),\dot{x}(t))+ x(t)=1- \dfrac{e \mathcal{V}^{2}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))}{x^2(t)}, \qquad x(t)\in\,]0,\infty[ $$ where $x_{d}(t)=x(t-d)$ and $\dot{x}_{d}(t)=\dot{x}(t-d),$ denote position and velocity feedback respectively, and $$ \mathcal{V}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))=V(t)+g_{1}(x(t)-x_{d}(t))+g_{2}(\dot{x}(t)-\dot{x}_{d}(t)),$$ is the feedback voltage with positive input voltage $V(t)\in C(\mathbb{R}/T\mathbb{Z})$ for $e\in \mathbb{R}^{+}, g_{1},g_{2}\in \mathbb{R}$, $d\in [0,T[$. The damping force $f_{D}(x,\dot{x})$ can be linear, i.e., $f_{D}(x,\dot{x}) = c\dot{x}$, $c\in\mathbb{R}^+$ or squeeze film type, i.e., $f_{D}(x,\dot{x}) = \gamma\dot{x}/x^{3}$, $\gamma\in\mathbb{R}^+$. The fundamental tool to prove our result is a local continuation method of periodic solutions from the non-delayed case $(d=0)$. Our approach provides new insights into the delay phenomenon on microelectromechanical systems and can be used to study the dynamics of a large class of delayed Li\'enard equations that govern the motion of several actuators, including the comb-drive finger actuator and the torsional actuator. Some numerical examples are provided to illustrate our results.
In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions $f_t:X\to X$ depending continuously on a parameter $t\in [0,1]$, where $X$ is a convex and compact subset of $\R^n$. Here, the result is presented for a compact mapping $f:A\times X\to X$ where $X$ is a convex, closed and bounded subset of an arbitrary normed space and $A$ is an arcwise connected topological space. Applications to nonlinear boundary value problems are given; specifically, we shall present new viewpoints of known results, introduce some novel results and exhibit some open problems.
An N-dimensional generalization of Nicholson's equation is analyzed. We consider a model including multiple delays, nonlinear coefficients and a nonlinear harvesting term. Inspired by previous results in this subject, we obtain sufficient conditions to guarantee strong and uniform persistence. Furthermore, under extra suitable hypotheses we prove the existence of T-periodic solutions and, reversing the prior conditions in a convenient manner, we show that the zero is a global attractor.
An abstract formulation of a duality principle established by Krasnoselskii is presented. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators share some topological properties.
We give an elementary proof of a Landesman-Lazer type result for systems by means of a shooting argument and explore its connection with the fundamental theorem of algebra.
We study the existence and multiplicity of periodic solutions for singular $φ$-laplacian equations with delay on time scales. We prove the existence of multiple solutions using topological methods based on the Leray-Schauder degree. A special case is the $T$-periodic problem for the forced pendulum equation with relativistic effects.
We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. The results in the present paper improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$.
The main purpose of this paper is to study the existence of periodic solutions for a nonautonomous differential-difference system describing the dynamics of hematopoietic stem cell (HSC) population under some external periodic regulatory factors at the cellular cycle level. The starting model is a nonautonomous system of two age-structured partial differential equations describing the HSC population in quiescent ($G_0$) and proliferating ($G_1$, $S$, $G_2$ and $M$) phase. We are interested on the effects of a periodically time varying coefficients due for example to circadian rhythms or to the periodic use of certain drugs, on the dynamics of HSC population. The method of characteristics reduces the age-structured model to a nonautonomous differential-difference system. We prove under appropriate conditions on the parameters of the system, using topological degree techniques and fixed point methods, the existence of periodic solutions of our model.
Motivated by Lazer-Leach type results, we study the existence of periodic solutions for systems of functional-differential equations at resonance with an arbitrary even-dimensional kernel and linear deviating terms involving a general delay of the form $\int_0^{2π}u(t+s)\,dλ(s)$, where $λ$ is a finite regular signed measure. Our main technique shall be the Coincidence Degree Theorem due to Mawhin.
We study semi-dynamical systems associated to delay differential equations. We give a simple criteria to obtain weak and strong persistence and provide sufficient conditions to guarantee uniform persistence. Moreover, we show the existence of non-trivial $T$-periodic solutions via topological degree techniques. Finally, we prove that, in some sense, the conditions are also necessary.