On tensor fields semiconjugated with torse-forming vector fields

summary:The paper deals with tensor fields which are semiconjugated with torse-forming vector fields. The existence results for semitorse-forming vector fields and for convergent vector fields are proved

Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation

Singular differential equation (p(t)u′)′=p(t)f(u) is investigated. Here f is Lipschitz continuous on ℝ and has at least two zeros 0 and L>0 . The function p is continuous on [0,∞) and has a positive continuous derivative on (0,∞) and p(0)=0. An asymptotic formula for oscillatory solutions is derived

Singular discrete problem arising in the theory of shallow membrane caps

Abstract. Assume that T ∈ (0, ∞), n ∈ N, n ≥ 2 and h = T n . We use the lower and upper functions method to prove the existence of a positive solution of the singular discrete problem where a 0 ≥ 0, b 0 > 0, γ > 1. We prove that for n → ∞ the sequence of solutions of the above discrete problems converges to a solution y of the corresponding continuous boundary value problem (t 3 y ) +

Barriers in impulsive antiperiodic problems

Some real world models are described by means of impulse control of nonlinear BVPs, where time instants of impulse actions depend on intersection points of solutions with given barriers. For $i=1,\dots, m$, and $[a,b]\subset \mathbb{R}$, continuous functions $\gamma_i:\mathbb{R} \to [a,b]$ determine barriers $\Gamma_i=\{(t,z): t=\gamma_i(z), z\in \mathbb{R} \}$. A solution $(x,y)$ of a planar BVP on $[a,b]$ is searched such that the graph of its first component $x(t)$ has exactly one intersection point with each barrier, i.e. for each $i\in \{1,\dots,m\}$ there exists a unique root $t=t_{ix}\in [a,b]$ of the equation $t=\gamma_i(x(t))$. The second component $y(t)$ of the solution has impulses (jumps) at the points $t_{1x},\dots, t_{mx}$. Since a size of jumps and especially the points $t_{1x},\dots,t_{mx}$ depend on $x$, impulses are called state-dependent. Here we focus our attention on an antiperiodic solution $(x,y)$ of the van der Pol equation with a positive parameter $\mu$ and a Lebesgue integrable antiperiodic function~$f$ \begin{equation*} x'(t) = y(t), \ y'(t) = \mu \left(x(t) - \frac{x^3(t)}{3}\right)' - x(t) + f(t)\quad \mbox{for a.e.}\ t \in \mathbb{R},\ t\not\in \{t_{1x},.\dots, t_{mx}\}, \end{equation*} where $y$ has impulses at the points from the set $\{t_{1x},\dots,t_{mx}\}$, \begin{equation*} y(t+)-y(t-) = \mathcal{J}_i(x), \quad t=t_{ix}, \quad i=1,\dots,m, \end{equation*} and $\mathcal{J}_i$ are continuous functionals defining a size of jumps. Previous results in the literature for this antiperiodic problem assume that impulse points are values of given continuous functionals. Such formulation is certain handicap for applications to real world problems where impulse instants depend on barriers. The paper presents conditions which enable to find such functionals from given barriers. Consequently the existence results for impulsive antiperiodic problem to the van der Pol equation formulated in terms of barriers are reached

Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation x(n+1)=x(n)+(n/(n+1))2(x(n)-x(n-1)+h2f(x(n))), n∈N, where h>0 is a parameter and f is Lipschitz continuous and has three real zeros L0<0<L. In particular we prove that for each sufficiently small h>0 there exists a solution {x(n)}n=0∞ such that {x(n)}n=1∞ is increasing, x(0)=x(1)∈(L0,0), and lim⁡n→∞x(n)>L. The problem is motivated by some models arising in hydrodynamics

Barriers in impulsive antiperiodic problems

Some real world models are described by means of impulse control of nonlinear BVPs, where time instants of impulse actions depend on intersection points of solutions with given barriers. For i = 1, . . . , m, and [a, b] ⊂ R, continuous functions γi : R → [a, b] determine barriers Γi = {(t, z) : t = γi(z), z ∈ R}. A solution (x, y) of a planar BVP on [a, b] is searched such that the graph of its first component x(t) has exactly one intersection point with each barrier, i.e. for each i ∈ {1, . . . , m} there exists a unique root t = tix ∈ [a, b] of the equation t = γi(x(t)). The second component y(t) of the solution has impulses (jumps) at the points t1x, . . . , tmx. Since a size of jumps and especially the points t1x, . . . , tmx depend on x, impulses are called state-dependent. Here we focus our attention on an antiperiodic solution (x, y) of the van der Pol equation with a positive parameter µ and a Lebesgue integrable antiperiodic function f x 0 (t) = y(t), y 0 (t) = µ x(t) − x 3 (t) 3 �0 − x(t) + f(t) for a.e. t ∈ R, t 6∈ {t1x, . . . . , tmx}, where y has impulses at the points from the set {t1x, . . . , tmx}, y(t+) − y(t−) = Ji(x), t = tix, i = 1, . . . , m, and Ji are continuous functionals defining a size of jumps. Previous results in the literature for this antiperiodic problem assume that impulse points are values of given continuous functionals. Such formulation is certain handicap for applications to real world problems where impulse instants depend on barriers. The paper presents conditions which enable to find such functionals from given barriers. Consequently the existence results for impulsive antiperiodic problem to the van der Pol equation formulated in terms of barriers are reached