Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space (γ,δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.     

Existence of limit cycles for Liénard-type systems with p-Laplacian

The purpose of this paper is to give sufficient conditions under which an equivalent system to the equation has at least one stable limit cycle, where is the one-dimensional p-Laplacian. The main results are proved by means of phase plane analysis with the Poincaré-Bendixson theorem. Sufficient conditions are also given for the origin to be unstable and for all solutions to be bounded in the future. Jitsuro Sugie: Supported in part by Grant-in-Aid for Scientific Research 16540152     

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation

     

On a parabolic problem with nonlinear term in a half space and global behavior of solutions

We take up the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation in (n?2) with boundary conditions u=0 on and u(x,0)=u₀(x). The nonlinear term is required to satisfy some conditions related to a functional class , which we introduce in this paper and will be called parabolic Kato class in the half space. Our approach is based on potential theory.     

Stability and instability criteria for Kaplan–Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions of Kaplan–Yorke type to the equation where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show. This research was supported by the Alexander von Humboldt Foundation (Germany) Received: March 14, 2005; revised: August 16, 2005     

Properties of theL p-solutions of linear impulsive equations in a Banach space

Summary In the paper conditions for the existence ofL _p-conditions (1 p ) of linear impulsive equations in a Banach space are found.     

On small solutions of delay equations in infinite dimensions

LetX be a Banach space and 1p<. LetL be a bounded linear operator fromL ^p ([–1,0],X) intoX. Consider the delay differential equationu(t)=Lu _t ,u(0)=x,u ₀=f on the state spaceL ^p ([–1,0],X). We prove that a mild solutionu(t)=u(t;x,f) is a small solution if and only if the Laplace transform ofu(t;x,f) extends to an entire function. The same result holds for the state spaceC([–1,0],X).This paper was written while the authors were affiliated with the University of Tübingen. It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. The authors warmly thank Professor Rainer Nagel and the AG Funktionalanalysis for the stimulating and enjoyable working environment.Support by DAAD is gratefully acknowledged.Support by an Individual Fellowship from the Human Capital and Mobility programme of the European Community is gratefully acknowledged.     

Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity

This is the second part of a series of study on the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm our theoretical study.     

The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form

     

Solving singular two-point boundary value problem in reproducing kernel space

In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n  -term approximation _un(x)$u_n(x)$ to the exact solution u(x)$u(x)$ is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.     

On existence of periodic solutions of Rayleigh equation of retarded type

Existence of periodic solutions for a kind of non-autonomous Rayleigh equations of retarded type

x^″(t)+f(t,x^′(t-σ))+g(t,x(t-τ(t)))=p(t)$x^{″}(t)+f(t,x^{\prime }(t-\sigma ))+g(t,x(t-\tau (t)))=p(t)$

Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument

By means of Mawhin’s continuation theorem, we study a class of p-Laplacian Duffing type differential equations of the form

(_φp(x^′(t)))^′=Cx^′(t)+g(t,x(t),x(t−τ(t)))+e(t).${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }=C{x}^{\prime }\left(t\right)+g(t,x\left(t\right),x(t-\tau \left(t\right)))+e\left(t\right)\text{.}$