Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.     

Positive solutions to a two-point higher order boundary value problem

We consider a higher order two-point boundary value problem, and establish some upper and lower type estimates for positive solutions of the problem. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. An example is included to illustrate the results.     

Positive solutions to a multi-point higher order boundary value problem

The authors consider a higher order multi-point boundary value problem. Some existence and nonexistence results for positive solutions of the problem are obtained by using Krasnosel'skii's fixed point theorem. Examples are included to illustrate the results.     

Multiple symmetric positive solutions for a second order boundary value problem

For the second order boundary value problem, , , , where growth conditions are imposed on which yield the existence of at least three symmetric positive solutions.

     

Sharp by order estimates of solutions of a simplest singular boundary value problem

We consider a boundary value problem ((0.1)) where f ∈ L_p (?), p ∈ [1, ∞] (L∞ (?) ? C (?)) and 0 ≤ q ∈ L^loc₁ (?). For a given p ∈ [1, ∞], for a correctly solvable problem (0.1) in L_p (?), we obtain minimal requirements to a positive, continuous function Θ(x) for x ∈ ? under which, regardless of f ∈ L_p (?), the solution y ∈ L_p (?) of problem (0.1) satisfies the equality . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性一阶边值问题正解的全局结构

微分方程的应用十分广泛,可以解决许多与导数有关的问题.物理中涉及变力的动力学问题,如空气的阻力为速度函数的落体运动等问题,很多可以用微分方程求解.运用Dancer全局分歧定理研究了一类非线性一阶微分方程边值问题正解的存在性,获得该问题正解存在的最优条件.     

Three positive solutions for higher order m-point boundary value problems

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem

     

Existence of positive solutions for the system of higher order two-point boundary value problems

In this paper, we establish the existence of at least one and two positive solutions for the system of higher order boundary value problems by using the Krasnosel’skii fixed point theorem.     

一类四阶边值问题的正解

§ 1　IntroductionThe deformations of an elastic beam are described by a fourth-order two-pointbound-ary value problem[1 ] .The boundary conditions are given according to the controls at theends of the beam. For example,the nonlinear fourth order problemu(4) (x) =λa(x) f(u(x) ) ,u(0 ) =u′(0 ) =u′(1 ) =u (1 ) =0 (1 .1 ) λdescribes the deformations of an elastic beam whose one end fixed and the other slidingclamped.The existence of solutions of (1 .1 ) λhas been studied by Gupta[1 ] . But …     

Three positive solutions for a second order difference equation with three-point boundary value problem

In this paper, we consider the solutions of discrete second-order boundary value problem $$\left\{\begin{array}{l}\Delta ^{2}y(k-1)+a(k)f(k,y(k))=0,\quad k\in \{1,\ldots,T\},\\[2pt]y(0)-\alpha\Delta y(0)=0,\quad y(T+1)=\beta y(n),\end{array}\right.$$ where f is continuous, T≥3 is a fixed positive integer, n∈{2,…,T?1}, constant α,β>0 such that T+1?β n+α(1?β)>0,T+1?β n>0. Under suitable conditions, we accomplish this by using the property of the associate Green’s function and Leggett-Williams fixed point theorem.     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

On positive solutions of a nonlocal fractional boundary value problem

In this paper, we investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation. Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.     

Three positive solutions of a nonlinear three-point boundary value problem

In this paper, existence criteria for three positive solutions of the nonlinear three-point boundary value problem

     

Positive solutions to a second order three-point boundary value problem

The existence, nonexistence, and multiplicity of nonnegative solutions are established for the three-point boundary value problem

     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

Uniform estimates for positive solutions of higher order quasilinear differential equations

For a higher order quasilinear differential equation, the existence of uniform estimates for positive solutions with common domain of definition is proved; these estimates depend on the estimates for the coefficients of the equation and do not depend on the coefficients themselves.     

Multiple positive solutions for a functional second-order boundary value problem

By using the Krasnoselskii fixed point theorem on cones in Banach spaces some existence results of positive solutions of a boundary value problem concerning a second-order functional differential equation are given.