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We study the half-linear differential equation $$ (r(t)\Phi(x'(t)))'+c(t)\Phi(x(\tau(t)))=0,\quad \Phi(x):=|x|^{p-2}x,\ p>1. $$ We formulate new oscillation criteria for this equation by comparing it with a certain ordinary linear or half-linear differential equation. Our proofs are based on a suitable estimate for the solution of the equation studied and on the modified Riccati technique, which, in ordinary case, appeared to be an effective replacement of the well known linear transformation formula.

In this paper we study the half-linear differential equation \begin{equation*} \bigl(r(t)\Phi_p(x')\bigr)'+c(t)\Phi_p(x)=0, \end{equation*} where $\Phi_p(x)=|x|^{p-2}x$, $p>1$. Using modified Riccati technique and suitable local estimates for terms in modified Riccati equation we derive new characterization of principal solution and new nonoscillation criteria.

In this article, we study a Riccati inequality that appears in the theory of partial differential equations with p-Laplacian. Our results allow to compare existence and nonexistence of positive solutions for Riccati type inequalities which are associated with equations with different powers p.

In the article, we extend the well-known Picone identity for half-linear partial differential equations to equations with anisotropic p-Laplacian.

We investigate the predator-prey model with cannibalism in the predator population, suggested by Magnusson [5] in 1995. We explore the model by a theory of bifurcations, based mainly on the results of Bautin. Among others, we show that the limit cycle appearing in the model due to the Andronov-Hopf bifurcation may be stable or unstable.

We study the positive solutions of equation \begin{equation*} \operatorname{div}(\|\nabla u\|^{p-2}\nabla u)+ \left\langle \vec b(x), \|\nabla u\|^{p-2}\nabla u\right\rangle + c(x)|u|^{q-2}u=0, \end{equation*} via the Riccati technique and prove an integral sufficient condition on the potential function $c(x)$ and the damping $\vec b(x)$ which ensures that no positive solution of the equation satisfies a lower (if $p>q$) or upper (if $q>p$) bound eventually.

Under suitable conditions on the coefficients of a partial differential equation, we prove a Riccati-type inequality. As an application of this result, we find oscillation criteria for second order damped half-linear partial differential equations. These criteria improve and complement earlier results on oscillation for partial differential equations. The main feature in our results is that the oscillation criteria are not radially symmetric and do not depend only on the mean value of the coefficients. We consider unbounded domains and state a special oscillation criterion for conic domains.

This paper presents sufficient conditions on the function $c(x)$ to ensure that every solution of partial differential equation $$ sum_{i=1}^{n}{partial over partial x_i} Phi_{p}({partial u over partial x_i})+B(x,u)=0, quad Phi_p(u):=|u|^{p-1}mathop{ m sgn} u. quad p>1 $$ is weakly oscillatory, i.e. has zero outside of every ball in $mathbb{R}^n$. The main tool is modified Riccati technique developed for Schrodinger operator by Noussair and Swanson [11].

The well known Hartman-Wintner oscillation criterion is extended to the PDE $$div(||\nabla u||^{p-2}\nabla u)+c(x)|u|^{p-2}u=0\quad p>1\tag{E}$$ The condition on the function $c(x)$ under which (E) has no solution positive for large $||x||$, i.e. $\infty$ belongs to the closure of the set of zeros of every solution defined on the domain $\Omega=\{x\in R^n: ||x||>1\}$, is derived.

Abstract Physically modelling the mechanical response of a tree by numerical simulation depends on having accurate data on the mechanical properties of green hardwood. Lacking such data, we developed and validated an orthotropic elasto-plastic (E–P) material model, based on the results of experiments performed on European beech (Fagus sylvatica L.) green wood, capable of including both the non-linearity and orthotropic properties of the material. We selected 655 clear samples with the special orthotropic structure of annual rings. All samples were prepared immediately after felling; their moisture content (MC) was 80% on average. The mechanical responses in normal directions and shear are represented by bi-linear stress–strain curves. The E–P model was validated by comparing the force–deflection response of three-point bending of green wood samples in a finite-element method (FEM) simulation (the average relative error was 4.6% for point-wise and 1.7% for integral-wise comparison). The output of this work was a consistent set of material constants for the E–P material model that is now available for the structural analysis of beech wood with MC above to fibre saturation point (FSP), especially green wood, subjected to relatively high loads (such that a plastic deformation appears) and that can very well predict a non-linear response above the proportional limits.

Tree risk assessment requires mechanical response studies, but simplification of the shape, material, or boundary conditions is necessary when dealing with such complex structures. To observe overall tree response, sub-structuring to several levels of detail can be used, enabled by recent developments in numerical methods and three-dimensional laser scanning (3D scan). This study aimed to determine an appropriate level of geometry and loading simplification allowed for high-order branches at the crown border, which is useful for the mechanical analysis of structured tree models. Four higher-order branches were pruned and experimentally tested by single-point loading. Beam and solid finite-element models (FEMs) were created based on measured geometric parameters and detailed 3D scans, respectively. The FEMs were used to analyze seven loading scenarios with force applied at (a) the center of gravity, (b) the top of side branches, (c) key discrete points, and (d) uniformly to the whole volume (to each finite element). Force was distributed by ratios weighted according to the mass, area, and diameter of side branches; or according to the mass of each finite element. The results showed no significant difference between the beam model and 3D scan-based model. The scenarios with finite elements’ mass-based force distribution deviated significantly from those of the other scenarios. The most simplified single-point loading caused a deviation in the deflection curve. The deviation of single-point loading in the case of the bending moment was related to force distribution ratios given by the branches architecture. Therefore, such loading simplification is not considered always appropriate. Consistency between the bending moment and branch deflection provided a representative mechanical response, recommended for further modeling of trees by sub-structuring.

In the paper, new comparison theorems for the half-linear difference equation Δ(RkΦ(Δzk)) + CkΦ(zk+1) = 0, Φ(u) = |u|p-2 u, p > 1,are derived. We show that if a solution of this equation has a generalized zero on the discrete interval [a, b], then the same holds for a solution of its majorant. The main tool used in the paper is the variational technique which relates nonexistence of a solution with a generalized zero with nonegativity of the p-degree functional defined on the suitable class of admissible functions.

We study the second-order neutral delay differential equation [r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, where Φα(t)=|t|α-1t, α≥1 and z(t)=x(t)+p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫∞r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

We study the half-linear differential equation (r(t)Φ(x′))′+c(t)Φ(x)=0, where Φ(x)=|x|p−2x, p>1. Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.