This paper is concerned with the second-order nonlinear Robin problem involving the first-order derivative:

$$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0,\end{cases} $$

where \(f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})\) and \(\alpha \in ]0,1[\). Based on a priori estimates, we use fixed point index theory to establish some results on existence, multiplicity and uniqueness of positive solutions thereof, with the unique positive solution being the limit of of an iterative sequence. The results presented here generalize and extend the corresponding ones for nonlinearities independent of the first-order derivative.

中文翻译：

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涉及一阶导数的二阶非线性罗宾问题的正解

本文关注的是涉及一阶导数的二阶非线性Robin问题：

$$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0 ,\end{cases} $$

其中\(f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})\)和\(\alpha \in ]0 ,1[\)。在先验估计的基础上，我们利用不动点指数理论建立了一些关于其正解的存在性、多重性和唯一性的结果，唯一的正解是迭代序列的极限。这里给出的结果推广和扩展了与一阶导数无关的非线性的相应结果。