Positive solutions and \(J\)-focal points for two point boundary value problems
The paper deals with boundary value problems of the form (1) \(y^{(n)}=p_ j(x)y^{(j)}+\cdots+p_ 0(x)y\), \(x\in[\alpha,\beta]\); \((2_ b)\) \(y^{(j)}(\alpha)=0\), \((i=0,\dots,k-1)\); \(y^{(i)}(b)=0\) \((i=j,\dots,n-k+j-1)\), where the integers \(k\), \(1\leq k\leq n-1\) and \(j\), \(0\leq j\leq k\) are given \(b\in(\alpha,\beta]\), \(p_ i\in C[\alpha,\beta]\), \((-1)^{n-k}p_ i(x)\geq 0\) on \([\alpha,\beta]\) for \(i=0,\dots,j\) and \(p_ 0\) does not vanish identically on each compact subinterval of \([\alpha,\beta]\). Making use of the cone theory the authors show that the following three assertions are equivalent: (i) \(b_ 0\) is the \(j\)-focal point of (1) with respect to \((2_ b)\) (i.e. \(b_ 0=\inf\{b>\alpha|(1),(2_ b)\) possesses a nontrivial solution\}), (ii) there exists a nontrivial solution \(y\) of (1), \((2_{b_ 0})\) such that \((-1)^{n-k}y^{(j)}(x)\geq 0\) on \([\alpha,b_ 0]\), (iii) the spectral radius of an associated integral operator generated by the Green function of the boundary value problem \(y^{(n)}=0\), \((2_{b_ 0})\) equals 1.
scientific article; zbMATH DE number 219334
scientific article