Positive solutions of a second-order nonlinear Robin problem involving the first-order derivative

Abstract

Abstract This paper is concerned with the second-order nonlinear Robin problem involving the first-order derivative: { u ″ + f ( t , u , u ′ ) = 0 , u ( 0 ) = u ′ ( 1 ) − α u ( 1 ) = 0 , $$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0,\end{cases} $$ where f ∈ C ( [ 0 , 1 ] × R + 2 , R + ) $f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})$ and α ∈ ] 0 , 1 [ $\alpha \in ]0,1[$ . 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.

Keywords

• Robin problem
• Positive solution
• A priori estimate
• Fixed point index
• Iterative sequence
• Super- and sub-solution