Uniqueness of positive solutions for Sturm-Liouville boundary value problems

Abstract

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems (BVP) { ( E ) ( | u ′ | m − 2 u ′ ) ′ + f ( t , u , u ′ ) = 0 , in ( θ 1 , θ 2 ) , m ≥ 2 , ( B C ) { α 1 u ( θ 1 ) − β 1 u ′ ( θ 1 ) = 0 , α 2 u ( θ 2 ) + β 2 u ′ ( θ 2 ) = 0 , \begin{equation*} \begin {cases} (\mathrm E) (|u’|^{m-2}u’)’+f(t,u,u’)=0,\quad \text {in} (\theta _1,\theta _2),m\ge 2,\ (\mathrm {BC})\begin {cases} \alpha _1u(\theta _1)-\beta _1u’(\theta _1)=0,\ \alpha _2u(\theta _2)+\beta _2u’(\theta _2)=0, \end{cases} \end{cases} \tag {BVP} \end{equation*} where α i , β i ≥ 0 \alpha _i,\beta _i\ge 0 and α i 2 + β i 2 ≠ 0 \alpha _i^2+\beta _i^2\not =0 ( i = 1 , 2 ) (i=1,2) , are established.