We deal with the existence of positive solutions for the following class of nonlinear equation $$u^{\prime \prime }(t)+Au^{\prime }(t)+g(t,u(t),v(t))=0$$ u ″ ( t ) + A u ′ ( t ) + g ( t , u ( t ) , v ( t ) ) = 0 a.e. in (0, 1), with boundary conditions $$u^{\prime }(0)=0$$ u ′ ( 0 ) = 0 , $$u^{\prime }(1)+Au(1)=0$$ u ′ ( 1 ) + A u ( 1 ) = 0 , where v is a functional parameter. The form of the problem is associated with the classical model described by Markus and Amundson. We show the existence of at least one positive solution of this problem and discuss its properties. Moreover we describe conditions that guarantee the continuous dependence of solution on parameter v also in the case of the lack of the uniqueness of a solution. The results are based on the clasical fixed point methods. Our approach allows us to consider both sub and superlinear nonlinearities which may be singular with respect to the first variable.

中文翻译：

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基于非线性反应堆动力学的边值问题解的存在性和性质

我们处理下面一类非线性方程$$u^{\prime \prime }(t)+Au^{\prime }(t)+g(t,u(t),v( t))=0$$ u ″ ( t ) + A u ′ ( t ) + g ( t , u ( t ) , v ( t ) ) = 0 ae in (0, 1)，有边界条件 $$u ^{\prime }(0)=0$$ u ′ ( 0 ) = 0 , $$u^{\prime }(1)+Au(1)=0$$ u ′ ( 1 ) + A u ( 1 ) = 0 ，其中 v 是函数参数。问题的形式与 Markus 和 Amundson 描述的经典模型有关。我们展示了这个问题的至少一个正解的存在，并讨论了它的性质。此外，我们还描述了在缺乏解决方案唯一性的情况下，保证解决方案对参数 v 的连续依赖的条件。结果基于经典不动点方法。