Third-order Three-point Boundary Value Problem Research Articles

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.

Existence of positive solution for a third-order three-point BVP with sign-changing Green's function

By using the Guo-Krasnoselskii fixed point theorem, we investigate the following thirdorder three-point boundary value problem

一类非线性四阶三点边值问题的可解性

研究了一类四阶三点边值问题解与正解的存在性。利用一类三阶三点边值问题的Green函数，建立了等价的积分方程组，得到了该类四阶三点边值问题解和正解存在的充分条件。 The existence of solutions and positive solutions for a class of fourth-order three-point boundary value problems is investigated in this paper. Employing the Green function of a third-order three-point boundary value problem, the equivalent integral equation system is established, and some sufficient conditions of existence of solutions and positive solutions are obtained for this class of fourth-order three point boundary value problems.

Positive Solutions of a Third-Order BVP Independent of the Sign of the Green’s Function

A third-order three-point boundary value problem (BVP) is studied, through this work. We derive sufficient conditions that guarantee the positivity of the solution of the corresponding linear boundary value problem (see Proposition 2). Then, based on the classical Guo-Krasnoselskii fixed point theorem, we obtain positive solutions of the nonlinear BVP. The proposed method may be useful, in cases where the Green’s function changes its sign. A BVP that falls in this category is studied through this paper.

On a Third-Order Three-Point Boundary Value Problem

We consider a third-order three-point boundary value problem. We introduce a generalized polynomial growth condition to obtain the existence of a nontrivial solution by using Leray-Schauder nonlinear alternative, then we give an example to illustrate our results.

Existence and Successive Iteration of Positive Solutions to a third-order three-point boundary value Problem

Existence and Successive Iteration of Positive Solutions to a third-order three-point boundary value Problem

EXISTENCE AND ITERATION OF MONOTONE POSITIVE SOLUTIONS FOR THIRD-ORDER THREE-POINT BVPS

This paper is concerned with the existence of monotone posi- tive solutions for a class of nonlinear third-order three-point boundary value problem. By applying iterative techniques, we not only obtain the existence of monotone positive solutions, but also establish iterative schemes for ap- proximating the solutions. An example is also included to illustrate the importance of the results obtained.

The upper and lower solution method for nonlinear third-order three-point boundary value problem

This paper is concerned with the following nonlinear third-order three-point boundary value problem � u ′′′ (t) + f (t, u(t) , u ′ (t)) = 0, t ∈ [0,1] , u(0) = u ′ (0) = 0, u ′ (1) = αu ′ (η) , where 0 < η < 1 and 0 ≤ α < 1. A new maximum principle is established and some existence

Existence and uniqueness of positive solutions for a third-order three-point problem on time scales

In this paper, a class of third-order three-point boundary value problem on time scales is considered. Using monotone iterative technique and cone expansion and compression fixed point theorem of norm type, we do not only obtain the existence and uniqueness of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

Positive solutions to singular third-order three-point boundary value problems on time scales

In this paper, we are concerned with singular third-order three-point boundary value problems on time scales. Some theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is also given to illustrate our results.

Positive solutions of singular third-order three-point boundary value problems

The positive solutions of a class of singular third-order three-point boundary value problems are considered by using the Guo–Krasnosel'skii fixed point theorem of cone expansion-compression type. In this class of problems, the nonlinear term is allowed to be singular. Main results show that this class of problems can have n positive solutions provided that the conditions on the nonlinear term on some bounded sets are appropriate.

Existence of triple positive solutions for a third-order three-point boundary value problem

In this paper we investigate the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem u ‴ ( t ) = a ( t ) f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , 0 < t < 1 , u ( 0 ) = δ u ( η ) , u ′ ( η ) = 0 , u ″ ( 1 ) = 0 , where δ ∈ ( 0 , 1 ) , η ∈ [ 1 / 2 , 1 ) are constants. f : [ 0 , 1 ] × [ 0 , ∞ ) × R 2 → [ 0 , ∞ ) , q : ( 0 , 1 ) → [ 0 , ∞ ) are continuous. First, Green’s function for the associated linear boundary value problem is constructed, and then, by using a fixed-point theorem due to Avery and Peterson, we establish results on the existence of triple positive solutions to the boundary value problem.

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.

Existence of positive solutions for nonlinear third-order three-point boundary value problems

This paper is concerned with the following nonlinear third-order three-point boundary value problem: u ‴ ( t ) + a ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = 0 , u ′ ( 1 ) = α u ′ ( η ) , where 0 < η < 1 and 1 < α < 1 η . First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when f is superlinear or sublinear.

Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function

We determine a Green’s function for the singular three-point third-order BVP u ‴ ( t ) = a ( t ) f ( t , u ( t ) ) , 0 < t < 1 u ( 0 ) = u ( 1 ) = u ″ ( η ) = 0 and then we apply the classical Krasnosel’skiĭ’s fixed point theorem on an also new cone. The emphasis is mainly that although this BVP does not admit a positive Green’s function, the solution obtained is still positive. In our approach we find out that η ∈ ( 17 24 , 1 ) . Our technique is a combination of the above fixed point theorem and properties of the corresponding vector field.

Discrete third-order three-point boundary value problem

We are concerned with the discrete boundary value problem Lx ( t ) ≔ Δ 3 x ( t ) = f ( t , x ( t + 1 ) ) , t 1 ⩽ t ⩽ t 3 - 1 , x ( t 1 ) = 0 , α x ( t 2 ) - β Δ x ( t 2 ) = 0 , γ x ( t 3 ) + δ Δ 2 x ( t 3 ) = 0 , and the eigenvalue problem Δ 3 x ( t ) = λ f ( t , x ( t + 1 ) ) with the same boundary conditions where t 1 < t 2 < t 3 are distinct integers. Under various assumptions on f and λ , we prove the existence of positive solutions of both problems by applying a fixed point theorem.

Positive solutions of singular third-order three-point boundary value problem

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular third-order three-point boundary value problem u ‴ ( t ) − λ a ( t ) F ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ′ ( η ) = u ″ ( 1 ) = 0 , where λ is a positive parameter and η ∈ [ 1 / 2 , 1 ) is a constant. By using a fixed point theorem of cone expansion-compression type due to Krasnosel'skii, we establish various results on the existence of single and multiple positive solutions to the boundary value problem. Under various assumptions on functions F and a, we give explicitly the intervals for parameter λ in which the existence of positive solutions is guaranteed. Especially, we allow the function a ( t ) of nonlinear term to have suitable singularities.

The Existence and Multiplicity of Positive Solutions for a Third-order Three-point Boundary Value Problem

The existence of n positive solutions for a class of third-order three-point boundary value problems is investigated, where n is an arbitrary natural number. The main tool is Krasnosel'skii fixed point theorem on the cone.

A remark on a third-order three-point boundary value problem

Let f be a real function defined on [0, 1] × R3 and let η ∈ (0, 1). Very recently, C.P. Gupta and V. Lakshimikantham, making use of the Leray-Schauder continuation theorem and Wirtinger-type inequalities, established an existence result for the problem(Theorem 1 and Remark 4 of [Nonlinear Anal. 16 (1991), 949–957]).The aim of the present paper is simply to point out how, bu means of a completely different approach, it is possible to improve that result not only by requiring much general on f, but also by giving a precise pointwise estimate on x″′

Existence and uniqueness theorems for a third-order three-point boundary value problem

Existence and uniqueness theorems for a third-order three-point boundary value problem