Nonlinear fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with a class of fourth-order nonlinear elliptic equations with nonlocal boundary conditions, including a multi-point boundary condition in a bounded domain of Rn. Also considered is a second-order elliptic equation with nonlocal boundary condition, and the usual multi-point boundary problem in ordinary differential equations. The aim of the paper is to show the existence of maximal and minimal solutions, the uniqueness of a positive solution, and the method of construction for these solutions. Our approach to the above problems is by the method of upper and lower solutions and its associated monotone iterations. The monotone iterative schemes can be developed into computational algorithms for numerical solutions of the problem by either the finite difference method or the finite element method.     

On 2nth-order Lidstone boundary value problems

This paper is concerned with the solutions of a class of 2nth-order Lidstone boundary value problems. Sufficient conditions for the existence and uniqueness of a solution are given. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution. The approach to the problem is by the method of upper and lower solutions with a new maximum principle.     

Higher-order Lidstone boundary value problems for elliptic partial differential equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.     

Boundary value problem for first order impulsive functional integro-differential equations

This paper investigates extremal solutions of the boundary value problem for impulsive functional integro-differential equations with nonlinear boundary conditions and deviating arguments. In the presence of a lower solution u and an upper solution v with u≥v, existence of extremal solutions is proved by establishing a new comparison principle and using the monotone iterative technique.     

A necessary and sufficient condition for 2nth-order singular super-linear m-point boundary value problems

In this paper, we study the existence of positive solutions for singular super-linear m-point boundary value problems of 2nth-order ordinary differential equations. A necessary and sufficient condition for the existence of C2n−2[0,1] positive solutions as well as C2n−1[0,1] positive solutions is given by means of the fixed point theorems on cones.     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

Existence of positive solutions to n-point nonhomogeneous boundary value problem

In this paper, we are concerned with the existence of positive solutions to a n-point nonhomogeneous boundary value problem. By using the Krasnoselskii's fixed point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of positive solution is established for the n-point nonhomogeneous boundary value problem.     

Existence of positive solutions for 2nth-order singular sublinear boundary value problems

This paper investigates the existence of positive solutions for 2nth-order (n>1) singular sub-linear boundary value problems. A necessary and sufficient condition for the existence of C2n−2[0,1] as well as C2n−1[0,1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity f(t,x₁,x₂,…,xn) may be singular at xi=0, i=1,2,…,n, t=0 and/or t=1.     

Positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian

In this paper, we investigate the existence of positive solutions for a classes of m-point boundary value problems with p-Laplacian. By applying a monotone iterative technique, some sufficient conditions for the existence of twin positive solutions are established.     

Boundary value problems involving upper and lower solutions in reverse order

In this paper, we discuss the existence of extreme solutions of the boundary value problem for a class of first-order functional equations with a nonlinear boundary condition. In the presence of a lower solution α$\alpha$ and an upper solution β$\beta$ with β≤α$\beta \le \alpha$, we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique.     

二阶三点边值问题解的存在性与唯一性

利用单调迭代方法,研究了反序上、下解条件下的二阶微分方程三点边值问题解的存在性与唯一性,分别得到解存在与唯一的充分条件,在满足解的唯一性的条件下,给出了求解的迭代序列及误差估计式.     

Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order

This paper studies the existence and uniqueness of solutions of second-order three-point boundary value problems with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method, and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.     

On fourth-order elliptic boundary value problems with nonmonotone nonlinear function

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.     

On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations. We obtain some new results relative to the lower solution α and upper solution β with α?β.     

Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order

This article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. The sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method, meantime, the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given. Some results of previous literature are extended and improved. A numerical example is also included to illustrate the effectiveness of the proposed results.     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.     

ANTI-PERIODIC BOUNDARY VALUE PROBLEM FOR FUNCTIONAL DIFFENERTIAL EQUATION

By using upper and lower solutions coupled with monotone iterative technique, we discuss the anti-periodic boundary value problem for first order functional diffenertial equation. A sufficient condition for the existence of the solution is obtained.     

Exact solutions of a class of second-order nonlocal boundary value problems and applications

In this paper, solutions of a class of second-order differential equations with some multi-point boundary conditions are studied. We give exact expressions of the solutions for the linear m-point boundary problems by the Green’s functions. As applications, we study uniqueness and iteration of the positive solutions for a nonlinear singular second-order m-point boundary value problem.     

The Method of Upper and Lower Solutions for a Lidstone Boundary Value Problem

In this paper we develop the monotone method in the presence of upper and lower solutions for the 2nd order Lidstone boundary value problem

Existence of solutions for fractional differential equations with multi-point boundary conditions

We discuss the existence of solutions for a nonlinear multi-point boundary value problem of integro-differential equations of fractional order q ∈ (1, 2]. Our analysis relies on the contraction mapping principle and the Krasnoselskii’s fixed point theorem. Example is provided to illustrate the theory.