Variational approach for Sturm-Liouville boundary value problems on time scales

In this paper, we consider a class of second order Sturm-Liouville boundary value problems with positive parameter λ on time scales. By using variational method and critical point theory, we obtain that the boundary value problem has solutions for λ being in some different intervals. Recent results in the literature are generalized and improved.     

Observability inequality for a wave equation with elastic fastening in the case of critical time interval

Problems with one-sided boundary controls of three basic types and homogeneous boundary third conditions on uncontrollable ends are considered for a wave equation in classes of strong generalized solutions in time intervals of strict critical length. New constructive observability inequalities are obtained for dual problems in adjoint classes of weak generalized solutions.     

Estimates for normal solutions of problems with zone controls in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> for the wave equation

For the wave equation with variable coefficients and homogeneous boundary conditions of the first kind, we consider problems with regular zone controls and dual zone observation problems. For weak generalized solutions of the observation problem on sufficiently large time intervals, we obtain constructive estimates that imply the well-posed solvability of the observation operator. These estimates contain information that permits one to construct stable approximate solutions of both problems with the use of a variational method suggested earlier by the author for linear equations with nonuniformly perturbed operators.     

Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition

For the wave equation with variable coefficients, problems with one-side boundary controls of three basic types and a boundary condition of the third kind at the uncontrolled end are considered. For dual problems with one-side boundary observations in the classes of strong generalized solutions, new constructive observability inequalities are obtained that are superior to the earlier known ones in two respects. First, inequalities with an optimal value of the controllability-observability threshold are derived, and second, the value of the final evaluation constant is bounded away from zero on time intervals whose length is close to the critical length. This opens up a possibility of constructing stable approximate solutions to the indicated classes of dual control and observation problems on time intervals not only of an arbitrary supercritical but also of precisely critical length.     

Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier–Stokes equations

We review some results concerning the problem of global-in-time regularity for the initial boundary value problem for the Navier–Stokes equations in three-dimensional domains. In particular, we focus on sufficient conditions on the vorticity field which imply that strong (hence smooth) solutions exist on arbitrary time intervals.     

时标上动力方程在右局部边值条件下正解的存在性

主要研究时标上二阶动力学方程u~(△△)(t)+λ_p(t)f(t,u(σ(t)))=0在右局部边值条件u(0)=0=u~△(σ(1))下正解的存在性.应用格林函数和锥上Krasnoselskii不动点原理给出其正解存在的充分条件及正解存在的特征值区间.     

Decay of the energy for the Benjamin–Bona–Mahony equation posed on bounded intervals and on a half‐line

This paper concerns nonlinear dissipative initial boundary value problems for the Benjamin–Bona–Mahony equation posed on a half‐line and on bounded intervals. We prove the existence and uniqueness of global solutions and decay of the energy as time tends to infinity as well as convergence of solutions on bounded intervals to a solution on a half‐line. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimates for normal solutions in problems with irregular zonal controls for the wave equation

For the wave equation with variable coefficients and boundary conditions of the first kind, we consider mutually dual problems with irregular zonal controls and regular zonal observations. Constructive estimates of well-posed solvability are obtained for the observation problem with strong generalized solutions on sufficiently large time intervals. These estimates contain information necessary for the construction of stable approximations to solutions of both problems with the use of the earlier suggested variational method.     

带参数的非线性奇异三阶非局部边值问题的单调正解

运用Guo-Krasnoselskii不动点定理研究了带积分边界条件的奇异三阶非局部边值问题,在非线性项满足适当的条件下建立了关于参数的区间,得到了边值问题至少存在一个或两个单调正解的若干存在性结果.     

无穷区间上具有p-Laplacian算子边值问题的正解

研究了一类在无穷区间上具有p-Laplacian算子的边值问题的迭代正解.利用单调迭代方法得到问题的迭代正解存在性的充分条件,同时得到了解的相应迭代序列,最后给出例子证明所得结论.     

Positive solutions for singular third-order nonhomogeneous boundary value problems

In this paper, we investigate the existence of positive solutions for singular third-order nonhomogeneous boundary value problems. By using a fixed point theorem of cone expansion-compression type due to Krasnosel??skii, we establish various results on the existence or nonexistence of single and multiple positive solutions to the singular boundary problems in the explicit intervals for the nonhomogeneous term. An example is also given to illustrate some of the main results.     

Continuous invertibility of a boundary control operator for the wave equation on subcritical intervals in classes of weak generalized solutions

For the wave equation, problems with two-sided boundary controls of three basic types are considered. Constructive estimates for the bounded invertibility of the control operator with known estimated constants are obtained in classes of weak generalized solutions on intervals of subcritical length. With the use of these estimates, stable numerical solutions of the considered problems on time intervals of strictly subcritical length can be found by applying a variational method. The main difference of the present estimated constants from those considered previously in the case of strong generalized solutions is that they degenerate as the time interval length approaches its critical value.     

Some results for fractional impulsive boundary value problems on infinite intervals

In this paper, we consider a fractional impulsive boundary value problem on infinite intervals. We obtain the existence, uniqueness and computational method of unbounded positive solutions.     

Multiple solutions of nonlinear elliptic equations for oscillation problems

We discussed oscillating equations with Neumann boundary value in [Nonlinear Anal. 54 (2003) 431-443] and [J. Math. Anal. Appl. 298 (2004) 14-32] and prove the existence of infinitely many nonconstant solutions. However, it seems difficult to find infinitely many disjoint order intervals for oscillating equations with Dirichlet boundary value. To get rid of this difficulty, in this paper, we build up a mountain pass theorem in half-order intervals and use it to study oscillating problems with Dirichlet boundary value in which we only have the existence of super-solutions (or sub-solutions) and obtain new results on the exactly infinitely many solutions.     

Boundary value problems for semi-continuous delayed differential inclusions on Riemannian manifolds

The existence of solutions to nonlinear second order boundary value problems for differential inclusions with time lags is investigated. The set-valued mappings generating the differential inclusions are semi-continuous with compact and convex values. It turns out that solutions exist for small time intervals and arbitrary time lags in the state variable. The proof relies on mapping degree methods.     

无穷区间上的二阶边值问题的多解性

讨论了无穷区间上的一类二阶两点边值问题,通过运用Leggett-Williams不动点定理,得到了新的三个正解的存在性结果。     

Existence and nonexistence results for a class of fractional boundary value problems

In this paper, we consider a fractional boundary value problem involving Riemann-Liouville fractional derivative and depending on a parameter. we obtain the existence and nonexistence results of positive solutions when the nonlinear term satisfies different requirements of superlinearity, sublinearity and the parameter lies in some intervals.     

Energetic BEM-FEM coupling for the numerical solution of the damped wave equation

Time-dependent problems modeled by hyperbolic partial differential equations can be reformulated in terms of boundary integral equations and solved via the boundary element method. In this context, the analysis of damping phenomena that occur in many physics and engineering problems is a novelty. Starting from a recently developed energetic space-time weak formulation for the coupling of boundary integral equations and hyperbolic partial differential equations related to wave propagation problems, we consider here an extension for the damped wave equation in layered media. A coupling algorithm is presented, which allows a flexible use of finite element method and boundary element method as local discretization techniques. Stability and convergence, proved by energy arguments, are crucial in guaranteeing accurate solutions for simulations on large time intervals. Several numerical benchmarks, whose numerical results confirm theoretical ones, are illustrated and discussed.     

一类四阶两点边值问题正解的存在性

运用上下解方法及拓扑度理论讨论了非齐次边界条件下四阶两点边值问题u″″(t)=f(u(t)),t∈(0,1),u(0)=u″(0)=u″(1)=0,u(1)=λ,其中λ＞0为参数,f∈C([0,+∞),[0,+∞)).在非线性项满足一定的增长条件下,获得了上述问题存在正解时λ的取值范围.     

Existence of multiple solutions for a second-order difference equation with a parameter

Applying the critical point theorem, we establish existence and multiple solutions for a second-order difference boundary value problem and show the explicit intervals of λ such that the equation has at least 2N distinct solutions.