Singular \((p,n-p)\) focal and \((n,p)\) higher order boundary value problems
The authors study the focal \((p,n-p)\) boundary value problem \[ (-1)^{n-p}y^{(n)}=\phi (t)f(t,y,y',\dots ,y^{(p-1)}),\;0<t<1, \] \[ y^{(i)}(0)=0,\;0\leq i \leq p-1,\;y^{(i)}(1)=0,\;p\leq i \leq n-1, \] with \(n\geq 2\) and \(1\leq p \leq n-1\), and the \((n,p)\) boundary value problem \[ y^{(n)}+ \phi (t)f(t,y,y',\dots ,y^{(p-1)}),\;0<t<1, \] \[ y^{(i)}(0)=0,\;0\leq i \leq n-2,\;y^{(p)}(1)=0,\;1\leq p \leq n-1 \quad \text{is fixed}. \] The function \(f(t,y_0,\dots ,y_{p-1})\) may be singular at \(y_i, 0\leq i\leq p-1\) and \(\phi (t)\) may be singular at \(t=0\) and/or 1. Provided the right-hand sides of the differential equations are positive, they establish new results on the existence of positive solutions to the above problems belonging to \(C^{n-1}[0,1]\cap C^n(0,1)\). In the proofs they construct a sequence of auxiliary BVPs and then, using the nonlinear alternative by \textit{R. B. Guenther, J. W. Lee} and \textit{A. Granas} [J. Math. Pures Appl., IX. Sér. 70, No. 2, 153-196 (1991; Zbl 0687.34009)], they get a corresponding sequence of their solutions. Finally, they obtained solutions to the original BVPs by means of the Arzelà-Ascoli theorem as limits of such sequences.
scientific article; zbMATH DE number 1491647