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.

中文翻译：

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完全三阶非线性积分边值问题的存在结果和迭代方法

我们考虑边值问题

$$ \ begin {array} {* {20} {c}} {u'''（t）= f（t，u（t），u'（t），u''（t）），}＆ {0 <t <1，} \ end {array} $$ $$ \ begin {array} {* {20} {c}} {u（0）= u'（0）= 0，}＆{u（ 1）= \ int_0 ^ 1 {g（s）u（s）\ text {d} s，}} \ end {array} $$

其中˚F：[0，1]×ℝ ³ →ℝ ⁺，克：[0，1]→ℝ ⁺是连续函数。当的情况下˚F = ˚F（û（吨））的混合物在2018由Guendouz等人研究。他们使用锥的定点理论建立了正解的存在性。在这里，通过我们自己最近开发的方法，我们在容易验证的条件下确定了该溶液的存在性，唯一性和正性，并提出了一种迭代的方法来寻找该溶液。一些实例证明了所获得的理论结果的有效性和迭代方法的有效性。