Existence Results Of Periodic Solutions Research Articles

Fractional Differential Equations with Impulsive Effects

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems.

Periodic and Antiperiodic Solutions for a Second‐Order Hamiltonian System With Nonlinearity Depending on Derivative

Some existence results of periodic solution are obtained for a class of second‐order Hamiltonian systems with nonlinearity depending on derivative. We prove that there exists T0 > 0 such that, for any T < T0, the provided Hamiltonian system has a nontrivial T‐periodic and T/2‐antiperiodic solution via linking theorem and iteration method.

Existence Results of Periodic Solutions to First-Order Neutral Differential Equations on Time Scales

The purpose of this paper is to study the existence of periodic solutions for the first-order nonlinear neutral differential equation on time scales. Burton–Krasnoselskii’s fixed point theorem will be sufficiently general for application to the considered equation. An example has been carried out to show our results. It should be pointed out that the problem of periodic solutions is one of the current hot topics in the study of dynamic equations, which contains rich symmetry ideas and methods.

Existence of Periodic Solutions for Nonlinear Fully Third-Order Differential Equations

In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.

Global dynamics of delayed intraguild predation model with intraspecific competition

A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.

Time periodic solutions of Cahn-Hilliard systems with dynamic boundaryconditions

The existence problem for Cahn-Hilliard systems with dynamic boundary conditions andtime periodic conditions is discussed. We apply the abstract theory of evolution equations with viscosityapproach and the Schauder fixed point theorem in the level of approximate problems. One ofthe key points is the assumption for maximal monotone graphs with respect to their e ective domains.Thanks to this, we obtain the existence result of periodic solutions by using the passage to the limit.

Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space \(W^1L^\Phi([0,T])\). We employ the direct method of calculus of variations and we consider a potential function \(F\) satisfying the inequality \(|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)\), with \(b_1, b_2\in L^1\) and certain \(N\)-functions \(\Phi_0\).

Existence and asymptotic behavior results of periodic solution for discrete-time neutral-type neural networks

A generalized discrete neutral-type neural network with time-varying delays is studied. The existence results of periodic solution for the above neural networks are obtained by using Mawhin׳s continuation theorem of coincidence degree theory, and sufficient conditions are given to guarantee global exponential stability of periodic solution. Finally, a numerical example is given to show the effectiveness of the results in this paper.

Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays

This paper is concerned with the prescribed mean curvature Lienard type p-Laplacian equation with two arguments. By employing Mawhin’s coincidence degree theorem and the analysis techniques, some new existence results of periodic solutions are obtained. We also give an example to illustrate the application of our main results.

Periodic solutions of non-autonomous second order systems with (q(t), p(t))-Laplacian

Abstract Using the least action principle in critical point theory we obtain some existence results of periodic solutions for (q(t), p(t))-Laplacian systems which generalize some existence results.

Positive periodic solutions for planar differential systems with repulsive singularities on the axes

In this paper we provide an existence result of periodic solutions to planar systems having a repulsive singularity on both the axes. We require a nonresonance assumption, related to the spectrum of the periodic problem associated to the Steen's planar system{−y′=−cx3+μx,x′=−dy3+νy,x>0,y>0, which is isochronous (see [15]). As a particular case we obtain the existence of periodic solutions to a periodically perturbed Steen's planar system.

Periodic solutions for extended Fisher–Kolmogorov and Swift–Hohenberg equations obtained using a continuation theorem

This paper is concerned with an existence result for periodic solutions for a class of fourth-order ordinary differential equations involving extended Fisher–Kolmogorov and Swift–Hohenberg equations, where, under a suitable growth condition on the nonlinear term, one proves an existence result by applying Mawhin’s continuation theorem.

Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems

In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.

Existence results of periodic solutions for third‐order nonlinear singular differential equation

AbstractUsing Green's function for third‐order differential equation and some fixed‐point theorems, i.e., Leray‐Schauder alternative principle and Schauder's fixed point theorem, we establish three new existence results of periodic solutions for nonlinear third‐order singular differential equation, which extend and improve significantly existing results in the literature.

Periodic solutions for second-order discrete Hamiltonian system with a change of sign in potential

Some existence results of periodic solutions are obtained for nonautonomous second-order discrete Hamiltonian system with a change of sign in potential by the minimax methods in critical point theory.

Periodic Solutions of Second‐Order Differential Inclusions Systems with p(t)‐Laplacian

The existence of periodic solutions for nonautonomous second‐order differential inclusion systems with p(t)‐Laplacian is considered. We get some existence results of periodic solutions for system, a.e. t ∈ [0, T], , by using nonsmooth critical point theory. Our results generalize and improve some theorems in the literature.

Periodic Solutions For a Kind of Neutral Rayleigh Equations With Variable Parameter

Employing the coincidence degree theory of Mawhin we obtain some existence results of periodic solutions for a type of neutral Rayleigh equation with variable parameter $$((x(t) - c(t)x(t - \tau))'' + f(x'(t)) + g(x(t - \gamma(t))) = e(t).$$ It is worth noting that c(t) is no longer a constant which is different from the corresponding ones of past work. Furthermore, our results generalize corresponding work in the past.

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.

Existence results of periodic solutions for non-autonomous differential delay equations with asymptotically linear properties

In this paper, we study a class of non-autonomous differential delay equations which can be changed to Hamiltonian systems. By estimating Maslov-type index of the related Hamiltonian systems at infinity and at origin, we establish the existence of periodic solutions of the differential delay equations.