Existence and continuation of periodic solutions of autonomous Newtonian systems

In this article, we study the existence and the continuation of periodic solutions of autonomous Newtonian systems. To prove the results we apply the infinite-dimensional version of the degree for SO(2)-equivariant gradient operators defined by the third author in Nonlinear Anal. Theory Methods Appl. 23(1) (1994) 83-102 and developed in Topol. Meth. Nonlinear Anal. 9(2) (1997) 383-417. Using the results due to Rabier [Symmetries, Topological degree and a Theorem of Z.Q. Wang, J. Math. 24(3) (1994) 1087-1115] and Wang [Symmetries and calculation of the degree, Chinese Ann. Math. 10 (1989) 520-536] we show that the Leray-Schauder degree is not applicable in the proofs of our theorems, because it vanishes.     

Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay

We consider a periodic Lotka-Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems)

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Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of nonconstant 2π-periodic solutions of the Hamilton equation

     

Existence and stability of positive almost periodic solutions and periodic solutions for a logarithmic population model

By means of the Cauchy matrix, we give sufficient conditions for the existence and exponential stability of almost periodic solutions and periodic solutions for the delay impulsive logarithmic population.     

On sequences of solutions for discrete anisotropic equations

Taking advantage of a recent critical point theorem, the existence of infinitely many solutions for an anisotropic problem with a parameter is established. More precisely, a concrete interval of positive parameters, for which the treated problem admits infinitely many solutions, is determined without symmetry assumptions on the nonlinear data. Our goal was achieved by requiring an appropriate behavior of the nonlinear terms at zero, without any additional conditions.     

Existence of positive periodic solutions for a functional differential equation with a parameter

In this paper, we study the existence, multiplicity and nonexistence of positive ω$\omega$-periodic solutions for a kind of nonautonomous functional differential equation by employing the fixed point theorem in cones. Some new results are obtained. As an application of our results, we discuss the existence of a positive periodic solution for a class of models of blood cell production with a parameter, which has not been investigated by using previous methods.     

Multiple solutions for strongly resonant periodic systems

We considered a semilinear, second order periodic system. We assumed that the differential operator x→−x^″−Ax$x\to -x^{″}-Ax$ has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F(t,x)$F(t,x)$. Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Existence and uniqueness of solutions for a class of initial value problems of fractional differential systems on half lines

In this article, we establish the existence and uniqueness results for solutions of a class of initial value problems of nonlinear fractional differential systems on half lines involving Riemann–Liouville fractional derivatives. Our analysis relies on the well-known fixed point theorem of Schauder. The novelty of this paper is that the problems discussed are defined on half lines, and the nonlinearities f and g   are allowed to be singular functions. Furthermore, we allow p∈(0,β)$p\in (0,\beta )$ and q∈(0,α)$q\in (0,\alpha )$.     

Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems

The purpose of this paper is two-fold. Firstly, we will give some parabolic-like conditions which improve the well-known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity. As an application, we then consider the second-order Hamiltonian systems

     

Stability of periodic solutions of state-dependent delay-differential equations

We consider a class of autonomous delay-differential equations

     

On the number of critical periods for planar polynomial systems of arbitrary degree

We construct a class of planar systems of arbitrary degree n having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with n. As far as we know, the previous results on this subject gave systems having linear growth.     

Existence of positive periodic solutions for a neutral population model with delays and impulse

In this paper, by using the theory of abstract continuous theorem of k$k$-set contractive operator and some analysis techniques, we obtain some sufficient conditions for the existence of positive periodic solutions for a neutral population model with delays and impulse.     

The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds

We investigate the kernels of the transformation operators for one-dimensional Schrödinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials.     

Zero separation results for solutions of second order linear differential equations

The oscillation of solutions of f^″+Af=0$f^{″}+Af=0$ is discussed by focusing on four separate situations. In the complex case A$A$ is assumed to be either analytic in the unit disc D$\mathbb{D}$ or entire, while in the real case A$A$ is continuous either on (−1,1)$(-1,1)$ or on (0,∞)$(0,\infty )$. In all situations A$A$ is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.     

Existence of multiple periodic solutions for an SIR model with seasonality

We study an SIR model with a seasonal contact rate and a staged treatment strategy, which is an extension of our previous work [Z. Bai, Y. Zhou, Existence of two periodic solutions for a non-autonomous SIR epidemic model, Appl. Math. Model. 35 (2011) 382-391]. It is proved that the persistence and extinction of the disease are determined by the basic reproductive number (R₀) and a threshold parameter (R_c). It is obtained that the model exhibits two different bistable behaviors under certain conditions: the stable disease-free state coexists with a stable endemic periodic solution, and three endemic periodic solutions coexist with two of them being stable. Numerical simulations are presented to illustrate theoretical results.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000