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.

We study the leading order behaviour of positive solutions of the equation $-\Delta u+ϵu-{\left|u\right|}^{p-2}u+{\left|u\right|}^{q-2}u=0,x\in {ℝ}^N$, where $N\ge 3$, $q>p>2$ and when $ϵ>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*}=\frac{2N}{N-2}$. For $p<2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>2^{*}$ the solution asymptotically...

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: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

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...

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 give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $\epsilon cₜ=k_c\Delta c-f\left(c\right)c+g(a,c)$, x ∈ Ω, t > 0, for $\Omega \subset {ℝ}^N$, completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^p$-estimates.

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.

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1<p<2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t\to \infty$.

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

We deal with the problem of practical uniform $h$-stability for nonlinear time-varying perturbed differential equations. The main aim is to give sufficient conditions on the linear and perturbed terms to guarantee the global existence and the practical uniform $h$-stability of the solutions based on Gronwall’s type integral inequalities. Several numerical examples and an application to control systems with simulations are presented to illustrate the applicability of the obtained results. ...

We consider the damped wave equation $\alpha u_{tt}+u_t=u_{xx}-V^{\text{'}}\left(u\right)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u(x,t)=h(x-st)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$. We show that, if the initial data are sufficiently close to the profile of a front for large $\left|x\right|$, the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t\to +\infty$. The proof of this...

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

We study the problem of simultaneous stabilization for the algebra $A_{ℝ}\left(\right)$. Invertible pairs $(f_j,g_j)$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ}\left(\right)$ has stable rank two, we are faced here with a different...

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $$\left(a+b\sum _{j=1}^p{\int }_M{\left|\nabla u_j\right|}^{2}d{v}_g\right){\Delta }_g{u}_i+\sum _{j=1}^p{A}_{ij}{u}_j={\left|U\right|}^{2^{☆}-2}{u}_i$$ for all $i=1,\cdots ,p$, where ${\Delta }_g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M_s^p\left(ℝ\right)$ of symmetric $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U=(u_1,\cdots ,u_p)$, $\left|U\right|:M\to ℝ$ is the Euclidean norm of $U$, $2^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees ${\lambda }_k$ of $f$ satisfy ${\lambda }_k^2\&gt;{\lambda }_{k-1}{\lambda }_{k+1}$. On the other hand, we give examples of monomial maps $f$, where...

We consider the equation $$x^{\text{'}\text{'}}+a^2\left(t\right)x=0,a\left(t\right):=a_k\text{if}{t}_{k-1}\le t<t_k,\text{for}k=1,2,...,$$ where $\left\{a_k\right\}$ is a given increasing sequence of positive numbers, and $\left\{t_k\right\}$ is chosen at random so that $\{t_k-t_{k-1}\}$ are totally independent random variables uniformly distributed on interval $[0,1]$. We determine the probability of the event that all solutions of the equation tend to zero as $t\to \infty$.