Third-order Three-point Boundary Value Problem Research Articles

Solvability of infinite systems of fractional differential equations in the double sequence space $$ 2^c(\triangle )$$

In this work, we first introduce the concept of double sequence space $$2^c(\triangle )$$ . Then, we construct a Hausdorff measure of noncompactness on this space. Furthermore, by employing this measure of noncompactness we discuss the existence of solutions for infinite systems of third-order three-point boundary value problem in the double sequence space $$2^c(\triangle )$$ . Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

Existence and Multiplicity of Positive Solutions for a Singular Third-Order Three-Point Boundary Value Problem with a Parameter

In this paper, we investigate the existence of positive solutions for a singular third-order three-point boundary value problem with a parameter. By using fixed point index theory, some existence, multiplicity and nonexistence results for positive solutions are derived in terms of different values of λ.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function

Abstract In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

For nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii's Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.

Existence of Multiple Positive Solutions for Third-Order Three-Point Boundary Value Problem

In this paper, we study the existence of positive solutions for a class of third-order three-point boundary value problem. By employing the fixed point theorem on cone, some new criteria to ensure the three-point boundary value problem has at least three positive solutions are obtained. An example illustrating our main result is given. Moreover, some previous results will be improved significantly in our paper.

Solvability for a nonlinear third-order three-point boundary value problem

In this article, the existence of positive solutions for a nonlinear third-order three-point boundary value problem with integral condition is investigated. By using Leray-Schauder fixed point theorem, sufficient conditions for the existence of at least one positive solution are obtained. Illustrative examples are also presented to show the applicability of our results.

Existence of solution for third-order three-point boundary value problem

Existence of solution for third-order three-point boundary value problem

Existence and uniqueness of monotone positive solutions for a third-order three-point boundary value problem

In this work we study a third-order three-point boundary-value problem (BVP). We derive sucient conditions that guarantee the positivity of the solution of the corresponding linear BVP Then, based on the classi- cal Guo-Krasnosel'skii's fixed point theorem, we obtain positive solutions to the nonlinear BVP. Additional hypotheses guarantee the uniqueness of the solution.

On some properties of positive solutions for a third-order three-point boundary value problem with a parameter

In this article, we discuss some properties of positive solutions for a third-order differential equation with three-point boundary conditions and a positive parameter. By using recent fixed point theory, we establish the existence and uniqueness of positive solutions for any given parameter, and we show that the positive solution is continuous, increasing with respect to the parameter. Moreover, we give some properties of limits for positive solutions. An example is provided to demonstrate the main result.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green’s function

In this paper, by using the Krasnosel’skii fixed point theorem in a cone, we discuss the existence of positive solutions to the discrete third-order three-point boundary value problem $$\left \{ \textstyle\begin{array}{l} \Delta^{3}u(t-1)=\lambda a(t)f(t,u(t)), \quad t\in[1,T-2]_{\mathbb{Z}}, \Delta u(0)=u(T)=\Delta^{2}u(\eta)=0, \end{array}\displaystyle \right . $$ where $T>4$ is an integer, $[1,T-2]_{\mathbb{Z}}=\{1,2,\ldots,T-2\}$ , $\lambda>0$ is a parameter, $\eta\in\{\frac{T-1}{2},\ldots,T-2\}$ for odd T, and $\eta\in\{\frac{T-2}{2},\ldots,T-2\}$ for even T. Despite the sign-changing Green’s function, we also give the explicit interval for λ to guarantee the existence of positive solutions of the problem when f satisfies different growth assumptions.

一类奇异三阶三点边值问题正解的存在性

讨论非线性三阶三点边值问题 其中 ，非线性项允许在t=0, t=1及u=0 处奇异，利用锥上的不动点定理在较弱的条件下得到了上述边值问题至少存在一个、两个、n个正解的存在性结果。 This paper is concerned with the following nonlinear third-order three-point boundary value problem where , the nonlinear term may be singular at t=0, t=1 and u=0 . By using fixed- point theorem in cone, the existence of one or two or n positive solutions is obtained with the weaker conditions.

Existence of Positive Solution to System of Nonlinear Third-Order Three-Point BVPs

We are concerned with the following system of third-order three-point boundary value problems:u′′′(t)+f(t,v(t))=0,t∈(0,1),v′′′(t)+g(t,u(t))=0,t∈(0,1),u(0)=u′′(0)=0,u′(1)=αu(η),v(0)=v′′(0)=0, andv′(1)=αv(η), where0<η<1and0<α<1/η. By imposing some suitable conditions onfandg, we obtain the existence of at least one positive solution to the above system. The main tool used is the theory of the fixed-point index.

Solvability of a Third-Order Three-Point Boundary Value Problem on a Half-Line

In this paper, we consider the solvability of a third-order three-point boundary value problem on a half-line of the form: $$\begin{aligned} {\left\{ \begin{array}{ll} x^{\prime \prime \prime }(t)=f\left( t,x(t),{x^{\prime }}(t),{x^{\prime \prime }}(t)\right) ,\quad 0< t<+\infty , \\ x(0)=\alpha x(\eta ),\quad \mathop {\lim }\nolimits _{t\rightarrow +\infty }x^{(i)}(t)=0,\quad i=1,2, \end{array}\right. } \end{aligned}$$ where $$\alpha \ne 1$$ and $$\eta \in (0,+\infty )$$ , while $$f:[0,+\infty )\times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$$ is $$S^2$$ —Caratheodory function. The existence and uniqueness of solutions for the boundary value problems are obtained by the Leray-Schauder continuation theorem. As an application, an example is given to demonstrate our results.

Solvability of Third-Order Three-Point Boundary Value Problems

We are interested in the existence theorems for a third-order three-point boundary value problem. In the nonresonant case, using the Krasnosel’skii fixed point theorem, we obtain some sufficient conditions for the existence of the positive solutions. In addition, we focus on the resonant case, the boundary value problem being transformed into an integral equation with an undetermined parameter, and the existence conditions being obtained by the Intermediate Value Theorem.

Existence and Uniqueness of Positive Solution for Third-Order Three-Point Boundary Value Problems

This paper is devoted to the study of the existence and uniqueness of the positive solution for a type of the nonlinear third-order three-point boundary value problem. Our results are based on an iterative method and the Leray-Schauder fixed point theorem.

Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

We are concerned with the following third-order three-point boundary value problem:u‴(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u″(η)+αu(0)=0, whereα∈[0,2)andη∈[2/3,1). Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions onfby applying iterative method. An example is also included to illustrate the main results obtained.

Positive Solutions of a Third-Order Three-Point BVP with Sign-Changing Green’s Function

We are concerned with the following third-order three-point boundary value problem:u′′′t=ft, ut, t∈0, 1, u′0=u1=0and u′′η-αu′1=0,whereα∈0, 1andη∈(14+α)/(24-3α),1. Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions onfby using the two-fixed-point theorem due to Avery and Henderson.

Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems

The existence of positive solutions to the discrete third-order three-point boundary-value problems (BVPs) was recently established in Ji and Yang [Positive solutions of discrete third-order three-point right focal boundary value problems, J. Differ. Equat. Appl. 15 (2009), pp. 185–195]. In this paper, we propose an algorithm for the computation of such positive solutions. The method is based on the power method for the dominant eigenvalue and the Crout-like factorization algorithm for the sparse system of linear equations. At each iteration of the method, it calls for a linear solver with linear computational complexity. The proposed method is extremely effective for large-scale problems. A numerical example is also included to demonstrate the effectiveness of the algorithm when applied to the third-order three-point BVPs of differential equation.

Eigenvalue for a singular third-order three-point boundary value problem

In this paper, the existence of positive solutions for a singular third-order three-point boundary value problem is investigated. By using Krasnoselskii’s fixed point theorem, several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained.