Harmonic oscillators at resonance,perturbed by a non-linear friction force

This note is an addendum to the results of Lazer and Frederickson [1], and Lazer [4] on periodic oscillations, with linear part at resonance. We show that a small modification of the argument in [4] provides a more general result. It turns out that things are different for the corresponding Dirichlet boundary value problem.     

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.     

Exact controllability of the suspension bridge model proposed by Lazer and McKenna

In this paper we give a sufficient condition for the exact controllability of the following model of the suspension bridge equation proposed by Lazer and McKenna in [A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990) 537-578]:

     

On permanence of Lotka–Volterra systems with delays and variable intrinsic growth rates

For competitive Lotka–Volterra systems, Ahmad and Lazer’s work [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Annali di Matematica 185 (2006) S47–S67] on total permanence of systems without delays has been extended to delayed systems [Z. Hou, On permanence of all subsystems of competitive Lotka–Volterra systems with delays, Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301]. In this paper, existence and boundedness of nonnegative solutions and permanence are considered for general Lotka–Volterra systems with delays including competitive, cooperative, predator–prey and mixed type systems. First, a condition is established for the existence and boundedness of solutions on a half line. Second, a necessary condition on the limits of the average growth rates is provided for permanence of all subsystems. Then the result for competitive systems is also proved for the general systems by using the same techniques. Just as for competitive systems, the eminent finding is that permanence of the system and all of its subsystems is completely irrelevant to the size and distribution of the delays.     

The permanence and global attractivity of a Kolmogorov system with feedback controls

In this paper, we study the permanence and global asymptotic behavior for a Kolmogorov system with feedback controls. By means of lower and upper averages of a function, the average conditions for permanence, global attractivity and extinction of this system are established respectively. The corresponding results given by Chen in [F. Chen, The permanence and global attractivity of Lotka–Volterra competition system with feedback controls, Nonlinear Anal. 7 (2006) 133–143] and Zhao in [J.D. Zhao, J.F. Jiang, A.C. Lazer, The permanence and global attractivity in a nonautonomous Lotka–Volterra system, Nonlinear Anal. Real World Appl. 5 (2004) 265–276] are extended and improved.     

Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations

We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman-Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer-Fu?ik spectrum.     

Solvability of Semilinear Hemivariational Inequalities at Resonance Using Generalized Landesman–Lazer Conditions

In this paper we examine semilinear hemivariational inequalities at resonance at any eigenvalue of the negative Laplacian with Dirichlet boundary condition. Our approach is variational based on the Nonsmooth Critical Point Theory and it uses a generalized Landesman–Lazer condition. We prove two existence theorems using two different forms of the generalized Landesman–Lazer condition. In the last section we show that our Landesman-Lazer condition is more general than the ones existing in the literature.     

Solvability of Semilinear Hemivariational Inequalities at Resonance Using Generalized Landesman–Lazer Conditions

In this paper we examine semilinear hemivariational inequalities at resonance at any eigenvalue of the negative Laplacian with Dirichlet boundary condition. Our approach is variational based on the Nonsmooth Critical Point Theory and it uses a generalized Landesman–Lazer condition. We prove two existence theorems using two different forms of the generalized Landesman–Lazer condition. In the last section we show that our Landesman-Lazer condition is more general than the ones existing in the literature.     

Multiple periodic solutions of scalar second order differential equations

We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987) [1] and Del Pino, Manasevich and Murua (1992) [2]. The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré-Birkhoff Fixed Point Theorem.     

Erratum to: “Travelling waves for a system of equations” [Nonlinear Anal. 68(12) (2008) 3909–3912]

We correct an oversight in the proof of Theorem 1.1 in [Shair Ahmad, Alan C. Lazer, Antonio Tineo, Traveling waves for a system of equations, Nonlinear Analysis 68 (12) (2008)], which, in addition to considerably reducing the assumptions made in [M.M. Tang, P.C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal. 73 (1980). [2]], gives a better derivation and a sharper estimate of the wave speed.     

On permanence of all subsystems of competitive Lotka–Volterra systems with delays

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system.     

Global solution curves for several classes of singular periodic problems

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 (Lazer and Solimini, 1987) problem.     

Periodic solutions to second order nonautonomous differential systems with gyroscopic forces

Existence of periodic solutions to second order differential systems with gyroscopic forces is considered via variational methods, where a generalized Ahmad–Lazer–Paul type condition is used. We do not impose the condition that the gyroscopic forces are small.     

吊桥方程非对称周期解的存在性

在本文中,利用Jabri Y和Moussaoui M在最近的文献中得到的一个临界点定理,我们在没有对称性假设的情况下,证明了Lezer A C和McKenna P J吊桥方程周期解的存在性.     

Lazer–Leach type conditions on periodic solutions of semilinear resonant Duffing equations with singularities

In this paper, we study the existence of positive periodic solutions of resonant Duffing equations with singularities. Some Lazer–Leach type conditions are given to ensure the existence of positive periodic solutions of singular resonant Duffing equations.     

Resonance at two consecutive eigenvalues for semilinear elliptic equations

The solvability of the Dirichlet problem for a semilinear elliptic equation is studied in some situations where the classical resonance conditions of Landesman and Lazer may fail.     

Landesman–Lazer resonance problems and saddle point methods

We show how saddle point techniques can be used to obtain new results for general resonance problems of the type considered by Landesman and Lazer.     

Periodic solutions to second order nonautonomous differential systems with unbounded nonlinearities

We investigate the existence of periodic solutions of differential systems with a skew‐symmetric matrix term and unbounded nonlinearities. Under new generalized Ahmad‐Lazer‐Paul conditions, some existence results are obtained and some results in the literature are improved.     

Large amplitude periodic bouncing for impact oscillators with damping

A result of A. Lazer and P. McKenna is extended to show the existence of large amplitude periodic bouncing for a damped linear impact oscillator with multiple impacts during one period.