Abstract
We are concerned with a fourth-order two-point boundary value problem. We prove the existence of positive solutions and establish iterative schemes for approximating the solutions. The interesting point of our method is that the nonlinear term is involved with all lower-order derivatives of unknown function, and the iterative scheme starts off with a known cubic function or the zero function. Finally we give two examples to verify the effectiveness of the main results.
Highlights
The bending of an elastic beam can be described with some fourth-order boundary value problems
We prove the existence of positive solutions and establish iterative schemes for approximating the solutions
Almost all of the papers we mentioned focused their attention on the existence of solutions or positive solutions
Summary
The bending of an elastic beam can be described with some fourth-order boundary value problems. By the successively iterative technique, we study the existence and iteration of monotone positive solutions for the following fourth-order differential equation: u (t) = q (t) f (t, u (t) , u (t) , u (t) , u (t)) , (1). Using the Leggett-Williams fixed point theorem, Yang [18] established an existence criterion for triple positive solutions of the nonlinear fourth-order differential equation: u (t) = g (t) f (t, u (t) , u (t)) , 0 < t < 1, (4). By placing some restrictions on the nonlinear term f, Bai [4] obtained the existence results for the fourth-order two-point boundary value problem (1)-(2) via the lower and upper solution method. At the end of the paper, two examples are included to illustrate the main results
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