Existence results and iterative method for fully third order nonlinear integral boundary value problems

Abstract

We consider the boundary value problem

$$\begin{array}{*{20}{c}} {u'''(t) = f(t,u(t),u'(t),u''(t)),}&{0 < t < 1,} \end{array}$$

$$\begin{array}{*{20}{c}} {u(0) = u'(0) = 0,}&{u(1) = \int_0^1 {g(s)u(s)\text{d}s,} } \end{array}$$

where f: [0, 1] × ℝ³ → ℝ⁺, g: [0, 1] → ℝ⁺ are continuous functions. The case when f = f (u(t)) was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.