Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

Problems of periodic solutions for a type of Duffing equation with state-dependent delay

Sufficient criteria are established for the existence of periodic solutions to a type of Duffing equation with state-dependent delay, which improve and generalize some related results in the literature. The approach is based on Mawhin’s continuation theorem. The significance of the present paper is that our results are relevant to delay by Lemma 2.1, which is different from the corresponding results of past work.     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Positive Periodic and Homoclinic Solutions for Nonlinear Differential Equations with Nonsmooth Potential

We study the existence of positive solutions and of positive homoclinic (to zero) solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and having a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory and our results extend the recent works of Korman–Lazer (Electronic JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).     

Generalized rational first integrals of analytic differential systems

In this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances. The main results extend some of the previous related ones, for instance the classical Poincaré?s one (Poincaré, 1891, 1897 [16]), the Furta?s one (Furta, 1996 [8]), part of Chen et al.?s ones (Chen et al., 2008 [4]), and the Shi?s one (Shi, 2007 [18]). The key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms.     

A comparison theorem and an inequality of redhefferf†

For equations of the form w″+B(e^z)w = 0, where B(ζ) is a rational function which is analytic on 0<|ζ|∞, we determine the regions where the bulk of the zeros of a solution must be located. In the special case of the general Mathieu equation, these results complement earlier results of E. Hille (1924) who considered the special case of real Mathieu equations     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Positive Solutions of Nonlinear Systems with <Emphasis Type="Italic">p</Emphasis>-Laplacian on Finite and Semi-infinite Intervals

Existence and localization results are established for systems of second-order differential equations with p -Laplacian on finite and semi-infinite intervals. For a compact interval, the compression and expansion conditions that are used are related to the first eigenvalue of the p-Laplacian.     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

Multiplicity of sign-changing solutions for some four-point boundary value problem

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f$f$.     

The Mackey–Glass model of respiratory dynamics: Review and new results

Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.     

Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices

In this paper a generalization of the delayed exponential defined by Khusainov and Shuklin (2003) [1] for autonomous linear delay systems with one delay defined by permutable matrices is given for delay systems with multiple delays and pairwise permutable matrices. Using this multidelay-exponential a solution of a Cauchy initial value problem is represented. By an application of this representation and using Pinto’s integral inequality an asymptotic stability results for some classes of nonlinear multidelay differential equations are proved.     

An inverse spectral problem for a nonnormal first order differential operator

We study the eigenvalues of two restrictions ofB _x +P whereB is the two-by-two matrix that is zero on the diagonal and one off the diagonal andP is a two-by-two matrix of Lipschitz functions on the unit interval. We establish asymptotic forms for their eigenvalues and associated root vectors and demonstrate that these root vectors constitute a Riesz basis inL ²(0, 1)². We show that our forward analysis makes rigorous the attack on the associated inverse problem by M. Yamamoto,Inverse spectral problem for systems of ordinary differential equations of first order, I, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 35, 1988, pp. 519–546. We apply these results to the recovery of the line resistance and leakage conductance of a nonuniform transmission line.Supported by NSF grant DMS-9258312.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.     

Effective Prüfer angles and relative oscillation criteria

We present a streamlined approach to relative oscillation criteria based on effective Prüfer angles adapted to the use at the edges of the essential spectrum.Based on this we provided a new scale of oscillation criteria for general Sturm-Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover and generalize the Gesztesy-Ünal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt.     

On camel-like traveling wave solutions in cellular neural networks

This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.     

Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions

Inverse spectral problems are considered for differential pencils with boundary conditions depending polynomially on the spectral parameter and with a finite number of transmission conditions. We give formulations of the associated inverse problems such as Titchmarsh–Weyl theorem, Hochstadt–Lieberman theorem and Mochizuki–Trooshin theorem, and prove corresponding uniqueness theorems. The obtained results are generalizations of the similar results for the classical Sturm–Liouville operator on a finite interval.     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.