Pablo Amster, Rocío Balderrama
A general nonautonomous Mackey-Glass equation for the regulation of the hematopoiesis with several non-constant delays is studied. Using topological degree methods, we prove the existence and multiplicity of positive periodic solutions.
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We establish and prove a fixed point theorem from which some sufficient conditions on the existence of positive almost periodic solutions for a model of hematopoiesis with oscillatory circulation loss rate are deduced. Some particular assumption under the nonlinearity of the equation has been previously considered by authors as fundamental for the study of almost periodic solutions of the model. The aim of this paper is to establish results without such assumption. Some examples are given to illustrate our results.
In this paper, a generalized model of hematopoiesis with delays and impulses is considered. By employing the contraction mapping principle and a novel type of impulsive delay inequality, we prove the existence of a unique positive almost periodic solution of the model. It is also proved that, under the proposed conditions in this paper, the unique positive almost periodic solution is globally exponentially attractive. A numerical example is given to illustrate the effectiveness of the obtained results.
Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to carry out explanation on some blood diseases, characterized by oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is analyzed. The existence of a Hopf bifurcation for a positive steady-state is obtained through the study of an exponential polynomial characteristic equati…
A class of periodic differential $n$-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary. Criteria for the existence of at least one positive periodic solution are presented, extending and improving previous ones established for the scalar case. Applications to systems inspired in mathematical biology models, such as impulsive hematopoiesis and Nicholson-type systems, are also included.
We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such that the equation has exactly three ordered T-periodic solutions. Moreover, when h is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.
A multiplicity result for Hamiltonian systems with mixed periodic-type and Neumann-type boundary conditions
本文研究耦合两个子系统的 Hamiltonian 系统的多解问题，分别采用周期边界条件配合 Poincaré–Birkhoff 扭转假设，以及 Neumann 型两点边界条件。
In this paper, we deal with a problem of the type $$\cases{(ϕ(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\ ,\cr}$$ where, in particular, $ϕ$ is a homeomorphism from an open ball of ${\bf R}^n$ onto ${\bf R}^n$. Using the theory developed by Brezis and Mawhin in [1] jointly with our minimax theorem proved in [3], we obtain a general multiplicity result, under assumptions of qualitative nature only. Three remarkable corollaries are also presented.
We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - μ\, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity, $a(t)$ is a periodic sign-changing weight, $c\in\mathbb{R}$ and $μ>0$ is a real parameter. We prove the existence of $2^{m}-1$ positive solutions when $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a periodicity interval) and $μ$ is sufficiently large. The proof is based on the extension of Mawhin's coincidence degree defined in open (possibly …
We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function $n(x).$ Here we show that the number of positive periodic …
By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions.
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation \begin{equation*} u'' + cu' + \bigr{(} λa^{+}(x) - μa^{-}(x) \bigr{)} g(u) = 0, \end{equation*} where $λ,μ>0$ are parameters, $c\in\mathbb{R}$, $a(x)$ is a locally integrable $P$-periodic sign-changing weight function, and $g\colon\mathopen{[}0,1\mathclose{]}\to\mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(u)>0$ for all $u\in\mathopen{]}0,1\mathclose{[}$, with superlinear growth at zero. A typical example for $g(u)$, that is of interest in pop…
We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( λa^{+}(t) - μa^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $λ, μ$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is base…
This paper ascertains the global topological structure of the set of subharmonics of arbitrary order of the periodic predator-prey model introduced in López-Gómez, Ortega and Tineo in 1996. By constructing the iterates of the monodromy operator of the system, it is shown that the system admits subharmonics of all orders for the appropriate ranges of values of the parameters. Then, some sharp results of topological nature in the context of global bifurcation theory provide us with the fine topological structure of the components of subharmonics emanating from the T-periodic coexistence state.
The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.
Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for three classes of periodically forced equations with singularities, including the equations arising in micro-electro-mechanical systems (MEMS), the ones in condensed matter physics, as well as A.C. Lazer and S. Solimini's \cite{LS} problem.
The existence and multiplicity of positive periodic solutions for second order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our results provide a unified treatment for the problem and significantly improve several results in the literature. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.
We study the number of periodic solutions in two first order non-autonomous differential equations both of which have been used to describe, among other things, the mean magnetization of an Ising magnet in the time-varying external magnetic field. When the strength of the external field is varied, the set of periodic solutions undergoes a bifurcation in both equations. We prove that despite profound similarities between the equations, the character of the bifurcation can be very different. This results in a different number of coexisting stable periodic solutions in the vicinity of the bifurca…
By a nondegenerate $k$-parameterized family $K$ of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on $K$ system equals to $k.$ Bifurcation of asymptotically stable periodic solutions from $K$ is well studied in the literature and different conditions have been proposeddepending on whether the algebraic multiplicity of +1 is $k$ or not (by Malkin, Loud, Melnikov, Yagasaki). In this paper we assume that the later is unknown. Asymptotic stability can not be understood in this case, but we demonstrate that the information about…
Multiple non-collision periodic solutions for second order singular differential systems with parameters
该研究针对带参数的二阶奇异微分系统，利用乘积空间中Krasnoselskii-Guo不动点定理的向量版本，证明了混合次线性与超线性行为的系统最多存在四个正周期解，丰富了此类问题的解存在性结果。
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the mutiplicity is the same when the coefficients are polynomial functions of degree n or piecewise linear functions with n segments.
We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré-Birkhoff Theorem.
Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of periodic solutions. Our approach is suitable for numerical computations, and in fact we present some numerically computed bifurcation diagrams illustrating our results.
We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e., $kT$-periodic for some integer $k \geq 2$) to the equation \begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}} \Biggr{)}' + λa(t) g(u) = 0, \end{equation*} where $λ> 0$ is a parameter, $a(t)$ is a $T$-periodic sign-changing weight function and $g \colon \mathopen{[}0,+\infty\mathclose{[} \to \mathopen{[}0,+\infty\mathclose{[}$ is a continuous function having superlinear growth at zero. In particular, we prove that for both $g(u)=u^{p}$, with …
Global solution curves for first order periodic problems, with applications
本文通过连续方法和分支理论研究一阶周期问题的解的多重性及全局结构，并应用于种群模型和极限环的稳定性分析。
We study the set of T-periodic solutions of a class of T-periodically perturbed Differential-Algebraic Equations with separated variables. Under suitable hypotheses, these equations are equivalent to separated variables ODEs on a manifold. By combining known results on Differential-Algebraic Equations, with an argument about ODEs on manifolds, we obtain a global continuation result for the T-periodic solutions to the considered equations. As an application of our method, a multiplicity result is provided.
On existence of periodic solutions for Kepler type problems
本文在拓扑条件下证明了受迫二体问题周期解的存在性与多重性，并将结果推广至限制性 n 体问题。
In this paper, we consider the spectral theory of linear differential-algebraic equations (DAEs) for periodic DAEs in canonical form, i.e., \begin{equation*} J \frac{df}{dt}+Hf=λWf, \end{equation*} where $J$ is a constant skew-Hermitian $n\times n$ matrix that is not invertible, both $H=H(t)$ and $W=W(t)$ are $d$-periodic Hermitian $n\times n$-matrices with Lebesgue measurable functions as entries, and $W(t)$ is positive semidefinite and invertible for a.e. $t\in \mathbb{R}$ (i.e., Lebesgue almost everywhere). Under some additional hypotheses on $H$ and $W$, called the local index-1 hypotheses…
The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the perturbed Kepler system \[ \ddot x = -\frac{x}{|x|^3} + p(t), \quad x \in \mathbb{R}^d, \] where $d\geq 1$, and $p:\mathbb{R}\to\mathbb{R}^d$ is smooth and $T$-periodic, $T>0$. The existence of critical points for the action functional associated to the problem is proved via a non-local change of variables inspired by Levi-Civita and Kustaanheimo-Stiefel tech…
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