Positive solutions for system of 2n-th order Sturm–Liouville boundary value problems on time scales

Intervals of the parameters λ and μ are determined for which there exist positive solutions to the system of dynamic equations $$ \begin{array}{lll} && (-1)^nu^{\Delta^{2n}}(t)+\lambda p(t)f(v(\sigma(t)))=0,\quad t\in[a, b], \\ &&(-1)^n v^{\Delta^{2n}}(t)+\mu q(t)g(u(\sigma(t)))=0, \quad t\in[a, b], \end{array} $$ satisfying the Sturm–Liouville boundary conditions $$ \begin{array}{lll} &&\alpha_{i+1} u^{\Delta^{2i}}(a)-\beta_{i+1} u^{\Delta^{2i+1}}(a)=0,\;\gamma_{i+1} u^{\Delta^{2i}}(\sigma(b))+\delta_{i+1} u^{\Delta^{2i+1}}(\sigma(b))=0,\\ &&\alpha_{i+1} v^{\Delta^{2i}}(a)-\beta_{i+1} v^{\Delta^{2i+1}}(a)=0,\; \gamma_{i+1} v^{\Delta^{2i}}(\sigma(b))+\delta_{i+1} v^{\Delta^{2i+1}}(\sigma(b))=0, \end{array} $$ for 0?≤?i?≤?n???1. To this end we apply a Guo–Krasnosel’skii fixed point theorem.     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

Asymptotic behaviour of dynamic systems on time scales†

This paper is concerned with asymptotic behaviour of solutions of perturbed dynamic systems on time scales. A time scale version of the Hartman–Wintner theorem is established for a class of time scales.     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

Inhomogeneous boundary value problem for nonstationary compressible Navier–Stokes equations

The existence of global weak renormalized solutions to the evolution flow problems for compressible Navier–Stokes equations is established. The in/out flow problem in a bounded domain in three spatial dimensions is considered. A general mathematical theory for the flow problem is developed. Bibliography: 15 titles.     

Positive solutions of singular Emden–Fowler boundary value problem with negative exponent and multiple impulses

In this paper, we first give the method of lower and upper solutions for the existence of positive solutions to a singular impulsive boundary value problem. Then we obtain a necessary and sufficient condition for the existence of positive solutions of Emden–Fowler singular impulsive boundary value problem by means of the method of lower and upper solutions.     

Oscillation criteria for third order neutral Emden–Fowler delay dynamic equations on time scales

This paper is concern with a class of third-order neutral Emden–Fowler dynamic equation

$$\begin{aligned} (a((rz^\Delta )^\Delta )^\alpha )^\Delta (t)+q(t)x^\alpha (\delta (t))=0, \end{aligned}$$

where \(z(t):=x(t)+p(t)x(\tau (t)), \alpha \) is a quotient of odd positive integers. By generalized Riccati transformation and comparison principles, some new criteria which ensure that every solution is oscillatory are established, which improve and supplement some known results in literatures.

     

Mixed initial–boundary value problem for equations of motion of Kelvin–Voigt fluids

The initial–boundary value problem for equations of motion of Kelvin–Voigt fluids with mixed boundary conditions is studied. The no-slip condition is used on some portion of the boundary, while the impermeability condition and the tangential component of the surface force field are specified on the rest of the boundary. The global-in-time existence of a weak solution is proved. It is shown that the solution is unique and depends continuously on the field of external forces, the field of surface forces, and initial data.     

Asymptotic behavior of solutions for initial–boundary value problems for strongly damped nonlinear wave equations

We study initial–boundary value problems for strongly damped nonlinear wave equations. By using improved integral estimates, it is proven that the solutions of the problems decay to zero exponentially as time t$t$ approaches infinity, under a very simple and general assumption regarding the nonlinear term.     

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)² is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L _V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V ₀ ≤ V ₁ and ${\mathcal{L}}$ a second order elliptic operator.     

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

     

Pontryagin’s maximum principle for dynamic systems on time scales

In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained.     

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals

Unbounded upper and lower solutions theories are established for the Sturm–Liouville boundary value problem of a second order ordinary differential equation on infinite intervals. By using such techniques and the Schäuder fixed point theorem, the existence of solutions as well as the positive ones is obtained. Nagumo conditions play an important role in the nonlinear term involved in the first-order derivatives.