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NOTICE: this is the author’s version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 235(2011), doi: 10.1016/j.cam.2011.04.041

Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} to analyze the stability of the differential operator $L_r$ associated with the $r$-th Newton tensor of $f$. This appears in the Jacobi operator for the variational problem of minimizing the $r$-mean curvature $H_r$. Two natural applications are found. The first one ensures that, under the mild condition that the integral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss map meets each equator of $\esse^m$ infinitely many times. The second one deals with hypersurfaces with zero $(r+1)$-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces $f_*T_pM$, $p\in M$, fill the whole $\erre^{m+1}$., Comment: 10 pages, corrected typos

Dynamical properties of laser models based on the Maxwell Bloch equation are studied. Instances of stability and chaotic behavior are investigated. Special solutions of the system one of which reduces to the Lotka Volterra system under simplifying assumptions are derived. Reasons for the absence of oscillating solutions in the modified systems are studied. © 2010 Springer Science+Business Media B.V.

For a class of second order nonlinear differential equations with damping, efficient oscillation criteria are derived by refining the standard integral averaging technique. Examples are provided to illustrate the relevance of new theorems.

Some oscillation criteria for solutions of a general perturbed second order ordinary differential equation with damping (r(t)x′ (t))′ + h(t)f (x)x′ (t) + ψ(t, x) = H(t, x(t), x′ (t)) with alternating coefficients are given. The results obtained improve and extend some existing results in the literature.

We establish the rate of decay to 0 and we study the oscillation properties of solutions to the scalar second order ODE : u″ + c|u′ |αu′ + |u|βu = 0 where c, α, β are positive constants. Various extensions (forced equation, system) are considered.

Second-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at infinity. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at infinity. Proceedings of the Edinburgh Mathematical Society

We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation $\ddot{x} + q(t) g(x) = 0$, where $g(x)$ has superlinear growth at infinity and $q(t)$ changes sign.

In questo lavoro si considerano equazioni differenziali ordinarie del secondo ordine di tipo singolare della forma $$[g(t)G_p (u,u')]' - g(t)G_u (u, u') + g(t)f(t, u) = 0$$ In particolare si studiano proprieta qualitative e di non esistenza delle soluzioni nel caso in cuiuf(t,u)≤0 per |u| sufficientemente piccolo et sufficientemente grande.

This paper deals with the existence, behaviour and approximation of oscillatory solutions of the equation y''+p(y, t)y'+s^2*y=f(y, t), s>0 is a real parameter, with the corresponding initial or boundary conditions.

We obtain all solutions of radial type for a class of singular partial differential equations of even order. The essential operators here are elliptic or ultrahyperbolic.

We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+P(m,n,y(m−k,n−l)) = Q (m,n,y(m − k,n −l)) , and y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+ ∑ i=1 τ P i (m,n,y(m − k i , n − l i )) = ∑ i=1 τ Q i (m,n,y(m − k i , n − l i )) . Several examples, which dwell upon the importance of our results, are also included.

The aim of this paper is twofold. First, we deal with the extension to the random framework of the piecewise Fröbenius method to solve Airy differential equations. This extension is based on mean square stochastic calculus. Second, we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process, but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario. This includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and Monte Carlo simulations. To conduct this comparative study, we have chosen the Airy random differential equation because it has highly oscillatory solutions. This feature allows us to emphasize differences between all the considered approaches. © 2011 Elsevier Ltd. All rights reserved., This work has been partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia grant PAID-06-09 (Ref. 2588).

An asymptotic approximation theorem is proved for the solutions of linear oscillatory three-term recurrence equations in a certain class. This result represents a discrete analogue of the well-known Liouville-Green (or WKBJ) theorem rigorously proved by Olver for second-order linear differential equations. As in Olver's theorem, precise estimates are obtained for the error terms, and the double asymptotic nature of the approximation is made clear. Applications to certain orthogonal polynomials are also given.

For the half-linear difference equation Δ[|Δy(k)|α−1Δy(k)]=∑i=1npi(k)|y(gi(k))|α−1y(gi(k)),k≥a where α > 0, we shall offer sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions. Several examples which dwell upon the importance of our results are also included.

Equation (1) has its origin from, e.g., the prescribed curvature problems in Riemannian geometry, and astrophysics (i.e., the Lane Emden Fowler equation and the Matukuma equation as special cases). The asymptotic behavior of the positive solutions to (1) has recently received much attention, see e.g., [L1, LN, Na]. However, it is well-known that the above equation (1) does not always have positive solutions or positive radial solutions. In other words, under suitable conditions, radial solutions to (1) must oscillate about the zero at infinite times (see e.g., [DCC, NY, N]). Thus it becomes very interesting to know as precisely as possible the asympototic behaviors of the oscillating periods, the amplitudes of the oscillatory solutions. In this paper, we restrict our attention to the study of radial solutions to (1). More precisely, we shall discuss the asymptotic behavior of oscillatory article no. DE963208

Consider the higher-order linear difference equation where m ≥ 1 is an odd integer, and { p n } is a sequence of nonnegative real numbers. We obtain several new sufficient conditions for the oscillation of all solutions of equation (*). Examples which dwell upon the importance of our results are also included.

The authors give sufficient conditions for all oscillatory solutions of a sublinear forced higher order nonlinear functional differential equation to converge to zero. They then prove a nonoscillation theorem for such equations. A few intermediate results are also obtained.

Nonlinear Analysis-Theory Methods & Applications

The paper deals with oscillatory solutions of equations of Sturm-Liouville type, with the initial and the boundary conditions. The results obtained refer to existence, approximation and asymptotic behaviour of some classes of solutions.

For the difference equation ▵(pn▵(yn+hnyn−k))+qn+1f(yn+1−e)=0, nϵZ={0,1,…}, the existence of solutions in the classes M+, M−, and WOS is established.