Multiple Positive Periodic Solutions Research Articles

Existence and bifurcation of periodic solutions to the Lp-Minkowski problem with indefinite weight

This paper is devoted to the existence and bifurcation of positive periodic solutions for Lp-Minkowski problem with indefinite weight. We provide new sufficient conditions for the existence of at least one positive periodic solution. The main tools are Leray-Schauder alternative principle and a global continuation theorem by Manásevich-Mawhin. Using the numerical bifurcation theory, we study the dynamic behaviors of periodic solutions in the cases of indefinite and positive weight, where the weight term is a simple sinusoidal function. In the positive weight case, the multiplicity of positive periodic solutions is detected for the first time in the Lp-Minkowski problem, which is generated by a saddle-node bifurcation.

Multiplicity of positive periodic solutions to third-order variable coefficients singular dynamical systems

In this paper, by applying a nonlinear alternative principle of Leray–Schauder and Guo–Krasnosel’skii fixed point theorem on compression and expansion of cones, together with truncation technique, we study the existence of multiplicity noncollision periodic solutions to third-order singular dynamical systems. By combining the analysis of the sign of Green’s function for a linear equation, we consider the systems where the potential has a repulsive singularity at origin. The so-called strong force condition is not needed, and the nonlinearity may have sign changing behavior. Recent results in the literature, even in the scalar case, are generalized and improved.

Multiplicity of positive periodic solutions for a discrete impulsive blood cell production model

<abstract><p>In this paper, we investigate the multiplicity of positive periodic solutions of a discrete blood cell production model with impulse effects. This model is described by periodic coefficients and time delays, as well as nonlinear feedback with exponential terms. By employing the Krasnosel'skii fixed point theorem, we establish a sufficient condition for the existence of at least two positive periodic solutions. To this end, we construct solution transformation between an impulsive delay difference equation and the corresponding nonimpulsive delay difference equation. Aditionally, a solution representation of the positive periodic solution of the blood cell production model is presented. Moreover, a numerical example and its simulations are given to illustrate the main result.</p></abstract>

Multiple Positive Periodic Solutions to Minkowski-Curvature Equations with a Singularity of Attractive Type

Multiple Positive Periodic Solutions to Minkowski-Curvature Equations with a Singularity of Attractive Type

A Generalized Approach of the Gilpin–Ayala Model with Fractional Derivatives under Numerical Simulation

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we showed that the classical nonlinear fractional model is bounded. The Banach contraction mapping principle was used to prove that the model has a unique positive asymptotical periodic solution. We provide an example and numerical simulation to inspect the correctness and availability of our essential outcomes.

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Global bifurcation for a class of nonlinear ODEs

We briefly survey global bifurcation techniques, and illustrate their use by finding multiple positive periodic solutions to a class of second order quasilinear ODEs related to the Yamabe problem. As an application, we give a bifurcation-theoretic proof of a classical nonuniqueness result for conformal metrics with constant scalar curvature, that was independently discovered by Kobayashi and Schoen in the 1980s.

Existence and multiplicity of positive periodic solutions to Minkowski-curvature equations without coercivity condition

We study the existence, non-existence and multiplicity of positive periodic solutions to a class of Minkowski-curvature equations with indefinite attractive singularities. A multiplicity result of Ambrosetti-Prodi type is established using a new method of construction of lower functions and some properties of Leray-Schauder degree.

含参系统正周期解的存在性及多解性

含参系统正周期解的存在性及多解性

Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations

Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations

Multiplicity of positive periodic solutions of Rayleigh equations with singularities

The existence of positive periodic solutions of the Rayleigh equations $ x''+f(x')+g(x) = e(t) $ with singularities is investigated in this paper. Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, the multiple periodic solutions of the singular Rayleigh equations can be determined under the weak conditions of the term $ g $. We discuss both the repulsive singular case and the attractive singular case. Some results in the literature are generalized and improved. Moreover, some examples and numerical simulations are given to illustrate our theoretical analysis.

EXISTENCE AND MULTIPLICITY OF POSITIVE PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER DAMPED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

<abstract> By using the Krasnoselskii fixed point theorem, sufficient conditions are obtained for the existence and multiplicity of positive periodic solutions for a class of second order damped functional differential equations with multiple delays. Our results are a further expansion of the previous research results. </abstract>

Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations

In this paper, we study the existence of positive periodic solutions for a class of non-autonomous second-order ordinary differential equations $$\begin{aligned} x''+\alpha x' +a(t)x^{n}-b(t)x^{n+1}+c(t)x^{n+2}=0, \end{aligned}$$ where $$\alpha \in {\mathbb {R}} $$ is a constant, n is a finite positive integer, and a(t), b(t), c(t) are continuous periodic functions. By using Mawhin’s continuation theorem, we prove the existence and multiplicity of positive periodic solutions for these equations.

Existence of multiple positive periodic solutions for discrete hematopoiesis models with a unimodal production function

This paper is devoted to elucidating a sufficient condition under which Mackey-Glass type discrete hematopoiesis models have at least two positive periodic solutions. This model has periodic coefficients and time delays, and includes several production function terms that act as feedback. Our result is obtained by applying the Krasnosel’skii fixed point theorem and is represented by a relationship between period, coefficients and the production function. Example and its simulations are attached to show how to apply our result. In this example, there are exactly two positive 3-periodic solutions in the hematopoiesis model. Simulation shows that one periodic solution is stable and the other is unstable. It also shows that our result can be improved by weakening assumptions about the production function.

Positive periodic solutions to an indefinite Minkowski-curvature equation

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≥2) to the equation(u′1−(u′)2)′+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ is a continuous function having superlinear growth at zero. In particular, we prove that for both g(u)=up, with p>1, and g(u)=up/(1+up−q), with 0≤q≤1<p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a “small” one and a “large” one) for λ large enough. Moreover, in both cases the “small” T-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of T-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré–Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.

Multiple positive periodic solutions of a Gause-type predator-prey model with Allee effect and functional responses

This paper deals with a Gause-type predator-prey model with Allee effect and Holling type III functional response. We also consider the influence of predator competition and the artificial harvesting on predator-prey system. The existence of multiple positive periodic solutions of the predator-prey model is established by using the Mawhin coincidence degree theory.

Positive periodic solutions of functional discrete systems with a parameter

The existence of multiple positive periodic solutions of the system of difference equations with a parameter x(n + 1) = A(n, x(n))x(n) + λf(n, xn), is studied. In particular, we use the eigenvalue problems of completely continuous operators to obtain our results. We apply our results to a well-known model in population dynamics.

On positive periodic solutions of second order singular equations

Using the fixed point theorem, we study the existence and multiplicity of positive periodic solutions for the second order differential equations \t\t\t{x¨+a(t)x=f(x),x(0)=x(T),x˙(0)=x˙(T).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} \\ddot{x}+a(t) x=f(x),\\\\ x(0)=x(T),\\qquad \\dot{x}(0)=\\dot{x}(T). \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} For given nonnegative constants 0<beta_{1}<beta_{2}<cdots<beta_{N}, the function f may be singular at x=beta_{i}.

Multiplicity of Positive Periodic Solutions to Second Order Singular Dynamical Systems

The existence and multiplicity of non-collision periodic solutions for second order singular dynamical systems are discussed in this paper. Using the Green’s function of linear differential equation, we consider general singularity and do not need any kind of strong force condition. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones.

Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.