Intrinsic equations represent promising approach for the description of rotor blade dynamics. They are the system of non-linear partial differential equations. Stability of numeric solution by the finite difference method is described. The stability is studied for various numerical schemes with different methods for the computation of spatial derivatives from time level $n+0.5$ (i.e., mean values of old and new time step) to $n+1$ (i.e., only from new time step). Stable solution was obtained only...

A suitable Liapunov function is constructed for proving that the unique critical point of a non-linear system of ordinary differential equations, considered in a well determined polyhedron $K$, is globally asymptotically stable in $K$. The analytic problem arises from an investigation concerning a steady state in a particular macromolecular system: the visual system represented by the pigment rhodopsin in the presence of light.

A suitable Liapunov function is constructed for proving that the unique critical point of a non-linear system of ordinary differential equations, considered in a well determined polyhedron $K$, is globally asymptotically stable in $K$. The analytic problem arises from an investigation concerning a steady state in a particular macromolecular system: the visual system represented by the pigment rhodopsin in the presence of light.

In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders.

We investigate some nonlinear elliptic problems of the form $${\Delta }^{2}v+\sigma \left(x\right)v=h(x,v)\text{in}\Omega ,v=\Delta v=0\text{on}\partial \Omega ,\left(\mathrm{P}\right)$$ where $\Omega$ is a regular bounded domain in ${ℝ}^N$, $N\ge 2$, $\sigma \left(x\right)$ a positive function in $L^{\infty }\left(\Omega \right)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

Given a measurable set $A\subset {ℝ}^n$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y)=px+(1-p)y-{\sum }_{s=m}^{\infty }s(x,y)$ uniformly in a neighborhood of the origin, where m > 1, ${}_s(x,y)={\sum }_{i=0}^s{a}_{i,s}{x}^{s-i}{y}^i$; (c) $ₘ(1,1)={\sum }_{i=0}^m{a}_{i,m}>0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1}=f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill x^{\left(n\right)}\left(t\right)\in -A_1(t,x\left(t\right),...,x^{(n-1)}\left(t\right))x^{(n-1)}\left(t\right)-...-A_n(t,x\left(t\right),...,& x^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\text{a.a.}t\in [a,\infty ),\end{array}$$ where $a\in (0,\infty )$, and $A_i:[a,\infty )\times {ℝ}^n\to ℝ$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

We give sufficient conditions for the existence of a matrix of probabilities ${\left[p_{ik}\right]}_{i,k=1}^N$ such that a system of randomly chosen transformations ${\Pi }_k$, k = 1,...,N, with probabilities $p_{ik}$ is asymptotically stable.

Asymptotic forms of solutions of half-linear ordinary differential equation $\left(\right|u^{\text{'}}{|}^{\alpha -1}{u}^{\text{'}}{)}^{\text{'}}=\alpha (1+b\left(t\right)){\left|u\right|}^{\alpha -1}u$ are investigated under a smallness condition and some signum conditions on $b\left(t\right)$. When $\alpha =1$, our results reduce to well-known ones for linear ordinary differential equations.

In this paper, we consider solutions to the following chemotaxis system with general sensitivity $$\left\{\begin{array}{c}\tau u_t=\Delta u-\nabla ·\left(u\nabla \chi \left(v\right)\right)\text{in}\Omega \times (0,\infty ),\hfill \\ \eta v_t=\Delta v-v+u\text{in}\Omega \times (0,\infty ),\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }=\frac{\partial u}{\partial \nu }=0\text{on}\partial \Omega \times (0,\infty ).}\hfill \end{array}\right.$$ Here, $\tau$ and $\eta$ are positive constants, $\chi$ is a smooth function on $(0,\infty )$ satisfying ${\chi }^{\text{'}}(·)>0$ and $\Omega$ is a bounded domain of ${𝐑}^n$ ($n\ge 2$). It is well known that the chemotaxis system with direct sensitivity ($\chi \left(v\right)={\chi }_{0}v$, ${\chi }_0>0$) has blowup solutions in the case where $n\ge 2$. On the other hand, in the case where $\chi \left(v\right)={\chi }_{0}logv$ with $0<{\chi }_0\ll 1$, any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness...

We investigate the following quasilinear and singular problem, $$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\&gt;0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$ where $\Omega$ is an open bounded domain with smooth boundary, $1\&lt;p\&lt;\infty$, $p-1\&lt;q\le p^{*}-1$, $\lambda \&gt;0$, and $0\&lt;\delta \&lt;1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\&lt;p\&lt;N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\&gt;N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate theof the times, ${\overline{T}}_b\left(r\right)$ and $T_b^{*}\left(r\right)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (two-sided exit), $0\le blt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are...

Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.