Second-order Three-point Boundary Value Research Articles

Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

This work is concerned with the existence of unbounded positive solutions for a second-order nonlinear three-point boundary value problem on the positive half-line. The interesting points of the results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. New existence results of nontrivial single and multiple positive solutions are proved using recent fixed point theorems on cones in a special Banach space.

Multiple solutions of second-order three-point singular boundary value problems on the half-line

In this paper, the existence of multiple solutions of second-order three-point singular boundary value problems on the half-line is discussed by using the technique of upper and lower solutions and the theory of fixed point index in a cone. Moreover, it generalizes and includes some known results.

Positive solutions for a second-order three-point discrete boundary value problem

We establish the existence of at least three positive solutions for the second-order three-point discrete boundary value problem: $$\Delta ^{2}y(k-1)+f(k,y(k))=0,\quad k\in \{1,\ldots ,T\},$$ $$y(0)=0,\quad y(T+1)=\alpha y(n),$$ where f is continuous, T≥3 and n∈{2,…,T−1} are two fixed positive integers, constant α>0 such that α n<T+1. Under suitable conditions, we accomplish this by using the property of the associate Green’s function and Leggett-Williams fixed point theorem.