We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.
翻译:我们引入了一种新的 Krasnoselskii 的结果类型。 使用简单的差异性条件, 我们放松了 Krasnoselskii 理论中的非扩展性条件。 更清楚的是, 我们根据某种差异性条件 $g$, 我们根据某种差异性条件 $g$x_ x_ wrac{ x_ n+g( n_n)\\\\n) 2} 来分析美元序列的趋同。 我们引入了一些迭接序列, 对于任何真正的差异性功能, $g$和任何起始点 $x_ 0\ in\mathb[ a, b], 我们使用单调的单调方式将美元集中到 $[ $[ b] 最接近的根, $$[ $[ b] 和 美元。 基于这个方法, 我们提出了一个找到真实功能的真正根源的有效和新颖的方法。 我们证明我们的方法不会丢失任何根。 值得一提的是, 我们的迭接合方法可以不受衍生方法的影响, 与许多其他方法相比, 都可被视为该方法的优势。 最后, 我们用数字的例子来说明我们的结果。