Positive solutions for semipositone \((k,n-k)\) conjugate boundary value problems
The author establishes the existence of positive solutions to the \((k,n-k)\) conjugate boundary value problem \[ (-1)^{(n-k)}y^{(n)}=\lambda f(x,y),\quad x\in (0,1),\;y^{(i)}(0)=0,\quad 0\leq i\leq k-1, \] \[ y^{(i)}(1)=0, \quad 0\leq j\leq n-k-1, \] where \(k\) satisfies \(1\leq k<n-1\) is fixed. A fixed-point theorem of cone expansion/compression type is used to establish not only existence of positive solutions, but also multiplicity of positive solutions. Related results can be found in [\textit{V. Anuradha, D. D. Hai} and \textit{R. Shivaji}, Proc. Am. Math. Soc. 124, No. 3, 757-763 (1996; Zbl 0857.34032)] for second-order boundary value problems and in [\textit{P. W. Eloe} and \textit{J. Henderson}, Nonlinear Anal., Theory Methods Appl. 28, No. 10, 1669-1680 (1997; Zbl 0871.34015), and J. Differ. Equations 133, No. 1, 136-151 (1997; Zbl 0870.34031) and \textit{R. Ma} and \textit{H. Wang}, Appl. Anal. 59, No. 1-4, 225-231 (1995; Zbl 0841.34019)] for higher-order conjugate boundary-value problems.
scientific article; zbMATH DE number 1563457
scientific article