Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Effects of mixing rules on phase equilibrium calculations

The density dependence in the mixing rule of the cohesion parameter of the cubic equation of state is found not to be necessary in vapour—liquid equilibrium (VLE) calculations. The effects of the functional form and the number of the adjustable parameters of the mixing rules on VLE calculations are examined. It is found that the functional form is less influential in VLE calculations than the number of adjustable parameters and that two adjustable parameters in the mixing rules are sufficient for practical applications.     

Rotational isomerism as studied by nuclear quadrupole coupling.: Theoretical and experimental 14NX tensors for hydrazine,methylhydrazine, and 1,2-dimethylhydrazine

The nuclear quadrupole coupling constants of the ¹⁴N nuclei for hydrazine, the inner and outer conformers of methylhydrazine, and the inner-outer and the outer-outer conformers of 1,2-dimethylhydrazine are calculated by an ab initio SCF method, and also by a CI calculation for hydrazine. The results are compared with available experimental values. Characteristic dependence of the X tensors on the conformational structure is demonstrated. An application of theoretical hyperfine structures to a spectral analysis is discussed.     

Mass spectra of the bitter principles fromPicrasma ailanthoides Planchon

The mass spectra of several bitter principles isolated from Picrasma ailanthoides Planchon (= P. quassioides Bennett) have been measured and characteristic fragmentation patterns useful for structure elucidation shown. The mass spectra of some related compounds are also described.     

Nonoscillation criteria for second‐order nonlinear differential equations with decaying coefficients

This paper is concerned with the problem of deciding conditions on the coefficient q (t) and the nonlinear term g (x) which ensure that all nontrivial solutions of the equation (|x ′|^α–1x ′)′ + q (t)g (x) = 0, α > 0, are nonoscillatory. The nonlinear term g (x) is not imposed no assumption except for the continuity and the sign condition xg (x) > 0 if x ≠ 0. In our problem, it is important to examine the relation between the decay of q (t) and the growth of g (x). Our main result extends some nonoscillation theorem for the generalised Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase‐plane analysis. A simple example is includes to show that the monotonicity of g (x) is not essential in our problem. Finally, elliptic equations with the m ‐Laplacian operator are discussed as an application to our results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global asymptotic stability of nonautonomous systems of Liénard type

This paper deals with nonautonomous Liénard-type systems. Sufficient conditions are given for the zero solution of the systems to be globally asymptotically stable. The main result is proved by means of phase plane analysis with a Liapunov function. Examples are included to contrast our theorem with results which were presented by Hatvani and Cantarelli. Some global phase portraits are also attached.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

A necessary and sufficient condition for the global asymptotic stability of damped half-linear oscillators

We are concerned with the global asymptotic stability of the equilibrium of the half-linear differential equation with a damped term $$\left(\phi_p(x')\right)' + h(t)\phi_p(x') + (p-1)\phi_p(x) = 0. $$ A necessary and sufficient growth condition on h(t) is obtained by using the generalized Prüfer transformation and the generalized Riccati transformation. Equivalent planar systems to the above-mentioned equation are also considered in the proof of our main result.     

Global Asymptotic Stability for Oscillators with Superlinear Damping

A necessary and sufficient condition is established for the equilibrium of the damped superlinear oscillator $$x^{\prime\prime} + a(t)\phi_q(x^{\prime}) + \omega^2x = 0$$ to be globally asymptotically stable. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation $$u^{\prime} + \omega^{q-2}a(t)\phi_q(u) + 1 = 0$$ is divergent or convergent. Since this nonlinear differential equation cannot be solved in general, it can be said that the presented result is expressed by an implicit condition. Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, it is proved that a certain growth condition of a(t) guarantees the global asymptotic stability for the equilibrium of the damped superlinear oscillator.     

Lebesgue piecewise affine approximation of nonlinear systems

This paper addresses a piecewise affine (PWA) approximation problem, i.e., a problem of finding a PWA system model which approximates a given nonlinear system. First, we propose a new class of PWA systems, called the Lebesgue PWA approximation systems, as a model to approximate nonlinear systems. Next, we derive an error bound of the PWA approximation model, and provide a technique for constructing the approximation model with specified accuracy. Finally, the proposed method is applied to a gene regulatory network with nonlinear dynamics, which shows that the method is a useful approximation tool.