Pablo Amster

University of Buenos Aires | UBA · Department of Mathematics (FCEN)

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.

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 almos...

A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, i...

May the Gronwall Lemma be regarded as a maximum principle? An abstract version of the celebrated inequality is obtained by means of the spectral bound of a compact operator defined on a Banach lattice. As a consequence, uniqueness and continuous dependence results for the general semilinear problem $L(u)=N(u)$ are established.

Presentamos una versión divulgativa de nuestro artículo (Amster y Cid, 2022) en el que mostramos cómo la raíz cuadrada compleja nos permite demostrar de una forma muy sencilla diversos resultados topológicos en el plano, tan profundos como el Teorema de Brouwer y el Teorema de Invariancia del Dominio, así como del Teorema Fundamental del Álgebra.

Here, a proto-type Ermakov–Painlevé I equation is introduced and a homogeneous Dirichlet-type boundary value problem analysed. In addition, a novel Ermakov–Painlevé I system is set down which is reducible by an involutory transformation to the autonomous Ermakov–Ray–Reid system augmented by a single component Ermakov–Painlevé I equation. Hamiltonia...

We obtain sufficient conditions for the existence of $T$-periodic solutions for an electrostatic actuator modeled by the time delayed Duffing equation \[ \ddot{x}(t)+h_{D}(x(t),\dot{x}(t))+ x(t) =1- \dfrac{e \mathcal{V}^2(t,s_{1}(x(t),x_{d}(t),g_1),s_{2}(\dot{x}(t),\dot{x}_{d}(t),g_2))}{x^2(t)}, \] where $x=x(t)\in\,]0,\infty[$ and the position and...

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions depending continuously on a parameter , where is a convex and compact subset of . Here, the result is presented for a compact mapping where is a convex, closed, and bounded subset of an arbitrary normed space...

This article revisits and extends to the nonautonomous framework the results about the dynamics of a discrete and nonlinear matrix model describing the growth of a size-structured single microbial population in an autonomous chemostat, which has been introduced by T.B. Gage et.al and H.L. Smith. The first and the second result provide a threshold d...

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully un...

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 t...

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. A...

Using a Lyapunov-Krasovskii functional, new results concerning the global stability, boundedness of solutions, existence and non-existence of \begin{document}$ T $\end{document}-periodic solutions for a kind of delayed equation for a \begin{document}$ \varphi $\end{document}-Laplacian operator are obtained. An application is given for the well know...

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 conve...

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 t...

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...

An elementary proof of the no-retraction theorem in the plane is given by means of the complex square root.

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 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 -periodic solutions via topological degree techniques. Finally, we prove that, in some sense, t...

We extend to delay equations recent results obtained by G. Feltrin and F. Zanolin for second-order ordinary equations with a superlinear term. We prove the existence of positive periodic solutions for nonlinear delay equations − u ″( t ) = a ( t ) g ( u ( t ), u ( t − τ )). We assume superlinear growth for g and sign alternance for a . The approach...

We consider the \begin{document}$ (\omega,Q) $\end{document}-periodic problem for a system of delay differential equations, where \begin{document}$ Q $\end{document} is an invertible matrix. Existence and multiplicity of solutions is proven under different conditions that extend well-known results for the periodic case \begin{document}$ Q = I $\end...

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 co...

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.

This paper proposes an ω-periodic version of the Ellermeyer model of delayed chemostat. We obtain a sufficient condition ensuring the existence of a positive ω-periodic solution. Our proof is based on the application of the generalized continuation theorem. In addition, as a consequence of the implicit function theorem, we obtain a uniqueness resul...

This paper introduces a new consideration in the well known chemostat model of a one-species with a periodic input of single nutrient with period ω, which is described by a system of differential delay equations. The delay represents the interval time between the consumption of nutrient and its metabolization by the microbial species. We obtain a n...

En este artículo se presentan las ideas principales de la versión discreta del modelo SIR (Susceptibles, Infectados, Recuperados), que se emplea para describir las epidemias y se ha convertido en protagonista impensado en los tiempos actuales. Se muestran las propiedades básicas que rigen el comportamiento de las curvas de susceptibles e infectados...

We study the existence and multiplicity of periodic solutions for singular $\varphi$-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 $p(t)=p_0(t)+\bar p$, we prove the existence of a compact interval...

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...

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 $\int_0^{2\pi}u(t+s)\,d\lambda(s)$, where $\lambda$ is a finite regular signed measure. Our main...

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...

We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a \(\varphi\)-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete \(p\)-Laplacian as well...

A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in the previous work \cite{AKR3} for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between t...

Using topological degree theory, we prove the existence of positive periodic solutions of a system of delay differential equations for models with feedback arising on regulatory mechanisms in which selfregulation is relevant, e.g. in cell physiology. We study different models based on the cycle of testosterone and generalizations. The method in the...

We apply a Pyragas-type control in order to synchronize the solutions of a glycolytic model that exhibits an aperiodic behavior. This delay control is used to stabilize the orbits of ordinary differential nonlinear equations systems. Inspired by several works that studied the chaotic behavior of diverse systems for the enzymatic reactions in the pr...

We prove the existence of multiple solutions for a second order ODE system under radiation boundary conditions. The proof is based on the degree computation of (Formula presented.), where K is an appropriate fixed point operator. Under a suitable asymptotic Hartman-like assumption for the nonlinearity, we shall prove that the degree is 1 over large...

TWe prove the existence of T−periodic solutions for the second order non-linear equation u⁰1 − u02⁰ = h(t)g(u), where the non-linear term g has two singularities and the weight function h changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The...

We prove the existence of T-periodic solutions for a family of second-order differential equations with two indefinite singularities, where the acceleration is modified by a singular phi-Laplacian operator.

A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in a previous work for a Painlev\'e II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $...

A generalized non‐linear nonautonomous model for the haematopoiesis (cell production) with several delays and an oscillating circulation loss rate is studied. We prove a fixed point theorem in abstract cones, from which different results on existence and uniqueness of positive almost periodic solutions are deduced. Moreover, some criteria are given...

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, s...

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, s...

A coupled Gompertz-like system of delay differential equations is considered. We prove the existence of T-periodic solutions under resonance assuming a Lazer–Leach type condition.

We consider the framework proposed by Burgard and Kjaer (2011) that derives the PDE which governs the price of an option including bilateral counterparty risk and funding. We extend this work by relaxing the assumption of absence of transaction costs in the hedging portfolio by proposing a cost proportional to the amount of assets traded and the tr...

In this article, we look at the diachronic changes in tango harmony with the methods of network science. We are able to detect some significant tendencies of harmonic discourse in the first half of the 20th century, among them an enrichment of harmonic transitions and power law frequency distribution of triadic chords with exponents compatible with...

We present a global bifurcation result for critical values of $C^1$ maps in Banach spaces. The approach is topological based on homotopy equivalence of pairs of topological spaces. For $C^2$ maps, we prove a particular global bifurcation result, based on the notion of spectral flow.

We present a global bifurcation result for critical values of $C^1$ maps in Banach spaces. The approach is topological based on homotopy equivalence of pairs of topological spaces. For $C^2$ maps, we prove a particular global bifurcation result, based on the notion of spectral flow.

Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small ball...

A second order ordinary differential equation with a superlinear term is studied under radiation boundary conditions. Employing the variational method and an accurate shooting-type argument, we prove the existence of at least three or five solutions, depending on the interaction of the nonlinearity with the spectrum of the associated linear operato...

We consider a Black-Scholes type equation arising on a pricing model for a multi-asset option with general transaction costs. The pioneering work of Leland is thus extended in two different ways: on the one hand, the problem is multi-dimensional since it involves different underlying assets; on the other hand, the transaction costs are not assumed...

En el curso del seminario 19, “... Ou pire”, el psicoanalista francés Jacques Lacan pronunció una de esas frases que iban a traer más de un dolor de cabeza a sus seguidores: No hay enseñanza más que matemática, el resto es broma. Pero esta aseveración no excluye la posibilidad certera de que también la matemática se trate, en el fondo, de una gran...

Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in the {next} gland, which successively generates another hormone to stimulate the next one and so on. In the end,...

Self-regulatory models are common in nature, as described e.g. in (\cite{mur}), (\cite{ha}) and (\cite{Gb}).\\ Let us consider a system made up of a number of glands as a motivation. Each gland secretes a hormone that allows secretion in the {next} gland, which successively generates another hormone to stimulate the next one and so on. In the end,...

We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In different cases, the lower bounds obtained for the number of solutions are related to the winding number of a curve in the plane and the homology of a space in ℝ³.

We consider a Nicholson type system for two species with mutualism and nonlinear harvesting terms. We give sufficient conditions for the existence of a positive periodic solution. We also provide a necessary condition; more precisely, we prove that if the harvesting rate is large enough, then 0 is a global attractor for the positive solutions and,...

A substitution is introduced which exploits the parameters of the high-order delayed linear non-autonomous models and consequently allows the use of the M-matrix methods for the stability analysis. To demonstrate the efficacy of the proposed algorithm we obtain explicit and practical stability conditions for the second and third order non-autonomou...

Two-point boundary value problems of Dirichlet type are investigated for a Ermakov-Painlevé II equation which arises out of a reduction of a three-ion electrodiffusion Nernst-Planck model system. In addition, it is shown how Ermakov invariants may be employed to solve a hybrid Ermakov-Painlevé II triad in terms of a solution of the single component...

We prove the existence of at least one positive θ-periodic solution of a system of delay differential equations for models with feedback arising on regulatory mechanisms in which self-regulation is relevant, e.g. in cell physiology .

A generalization of the nonautonomous Mackey–Glass equation for the regulation of the hematopoiesis with several non-constant delays is studied. Using topological degree methods we prove, under appropriate conditions, the existence of multiple positive periodic solutions. Moreover, we show that the conditions are necessary, in the sense that if som...

Given a Morse function defined in the complement of a knot K⊂R3K⊂R3 we obtain a lower bound for the number of its critical points, depending on a knot invariant t(K)t(K) known as the “tunnel number”. This lower bound is used to prove existence of many periodic solutions in a system of differential equations from celestial mechanics.

For some abstract classes of nonlinear non-autonomous systems with variable and state-dependent delays existence, non-existence and multiplicity of periodic solutions are discussed. To illustrate the efficiency of the method, we obtain some well-known results for applied systems as corollaries of our existence theorems.

We introduce one of the simplest topological methods, usually known as the shooting method, which basically consists in reducing a problem to a finite-dimensional equation for a certain parameter λ. Then, appropriate tools can be used, such as the Brouwer theorem or equivalent results. The chapter is intended to be self-contained and employs only c...

This chapter is devoted to applications of the topological degree theory to different boundary value problems. Starting from specific examples, we obtain some general continuation theorems that can be applied in many situations. In particular, most of the sections are devoted to the study of resonant problems, in connection with which we discuss so...

In this chapter, we construct the Brouwer topological degree and extended it for compact perturbations of the identity in a Banach space, namely, the Leray–Schauder degree. Some topological consequences are presented. Moreover, we give applications to some boundary value problems.KeywordsCompact OperatorDegree TheoryTopological DegreeCompact Pertur...

In this chapter, we prove the general version of Brouwer’s theorem and a well-known extension to Banach spaces: the Schauder theorem. Among other uses, this latter result allows us to give a full version of the method of upper and lower solutions introduced in the previous chapter, with applications to many different problems. As a corollary, we ob...

Some iterative methods, such as the monotone iterations method, Newton’s method, and some of its variants, are developed. The approach is accessible and requires only a basic knowledge of the theory of Banach spaces. Applications to different boundary value problems are given. Also, we introduce a Cantor diagonalization argument, which allows one t...

We present a model of style emergence based on flocking models. The system stabilizes in states with several non-interacting genres or style clusters, and the delayed dynamic yields more styles than the non-delayed ones. (Keywords: flocking, evolution, networks, musical genres).

Two-point boundary value problems of Dirichlet-type are investigated for a hybrid Ermakov–Painlevé IV equation. Existence and uniqueness results are established in terms of the Painlevé parameters. In addition, it is shown how Ermakov invariants may be used to systematically obtain solutions of a coupled Ermakov–Painlevé IV system in terms of seed...

Existence, uniqueness, and multiplicity properties are established via a variational formulation for a Painlevé II model subject to radiation boundary conditions in two-ion electrodiffusion. Numerical experiments using an adapted shooting method are also presented to support the theoretical results.

Introduction.- Shooting type methods.- The Banach Fixed Point Theorem.- Schauder's Theorem and applications.- Topological degree: an introduction.- Applications.- Basic facts on metric and normed spaces.- Brief review of ODE's.- Hints and Solutions to Selected Exercises.

This chapter is devoted to the Banach fixed point theorem and some of its immediate consequences. In particular, we shall prove the usual version of the implicit function theorem in Banach spaces and present some applications to boundary value problems. This requires knowing the basic notions of differentiation in Banach spaces, which, for the read...

The Wheldon model (1975) of a chronic myelogenous leukemia (CML) dynamics is modified and enriched by introduction of a time-varying microenvironment and time-dependent drug efficacies. The resulting model is a special class of nonautonomous nonlinear system of differential equations with delays. Via topological methods, the existence of positive p...

Theorems for the existence of periodic solutions for diverse models of population dynamics are obtained as corollaries of a few basic theorems, thus unifying the analysis of a broad class of scalar models in a single setting. The latter mechanism allows to obtain existence conditions for a broad class of nonlinear, non-autonomous models and models...

We generalize an existence result on second order systems with a nonlinear term satisfying the so-called Hartman-Nagumo condition. The generalization is based on the use of Gauss second fundamental form and continuation techniques. © 2013 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus Universit.

We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solu-tions are related to the winding number of a curve in the plane, the homology of a space in R 3 , the knot type of a curve and the link type of a se...

For a vector function u : R -> R-N we consider the system u ''(t) vertical bar del C(u(t)) = p(t) u(t) = u(t + T), where G : R-N -> R is a C-1 function. We are interested in finding all possible T-periodic forcing terms p(t) for which there is at least one solution. In other words, we examine the range of the semilinear operator S : H-per(2) -> L-2...

We study the existence of solutions for the nonlinear second order elliptic system Δu+g(u)=f(x)Δu+g(u)=f(x), where g∈C(RN∖S,RN)g∈C(RN∖S,RN) with S⊂RNS⊂RN bounded. Using topological degree methods, we prove an existence result under a geometric condition on gg. Moreover, we analyze the particular case of an isolated repulsive singularity: under a Ni...

Existence of solutions for a nonlinear fourth order ordinary differential equation arising in beam theory is considered. We obtain solutions by a degree argument under a non-asymptotic condition on the nonlinear terms of the problem. Moreover, assuming a potential Landesman-Lazer condition, we prove the existence of at least one solution by variati...

In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies ν + and ν - is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary va...

We study a Neumann problem for a nonlinear elliptic system. Unlike previous results in the literature of Landesman-Lazer type, our existence theorem allows rapid rotations on the nonlinear term. MSC 2010: 35J57, 35J60

We give sufficient and necessary conditions for the existence of at least one positive TT-periodic solution for a generalized Nicholson’s blowflies model with a nonlinear harvesting term. Our results extend those of the previous work Li and Du (2008) [1].

We study an integro-differential parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence of solutions for a multi-asset case in a general domain using the method of upper and lower solutions and a diagonal argument. We also model the jump in the related integro differential equation and give a solution...

To enrich the dynamics of mathematical models of angiogenesis, all mechanisms involved are time-dependent. We also assume that the tumor cells enter the mechanisms of angiogenic stimulation and inhibition with some delays. The models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit sufficient and necess...

A semilinear system of second order ODEs under Neumann conditions is studied. The system has the particularity that its nonlinear term de-pends on the (unknown) Dirichlet values y(0) and y(1) of the solution. Asymptotic and non-asymptotic sufficient conditions of Landesman-Lazer type for existence of solutions are given. We generalize our previous...

We study the existence of periodic solutions for a nonlinear system of second order ordinary differential equations. Assuming suitable conditions, we prove the existence of at least one solution applying topological degree methods. Instead of a Nirenberg type condition, we shall assume that each coordinate of the nonlinearity satisfies a one-side L...

A Newtonian equation in the plane is considered. There is a central force (attractive or repulsive) and an external force λh(t), periodic in time. The periodic second primitive of h(t) defines a planar curve and the number of periodic solutions of the differential equation is linked to the number of loops of this curve, at least when the parameter...

We obtain existence of T-periodic solutions to a second order system of ordinary differential equations of the form u′′+cu′+g(u)=p where c∈R, p∈C(R,R^{N}) is T-periodic and has mean value zero, and g∈C(R^{N},R^{N}) is e.g. sublinear. In contrast with a well known result by Nirenberg n , where it is assumed that the nonlinearity g has non-zero unifo...