Rachůnková, Irena; Staněk, Svatoslav Sturm-Liouville and focal higher order BVPs with singularities in phase variables. (English) Zbl 1069.34029 Georgian Math. J. 10, No. 1, 165-191 (2003). This paper is devoted to the existence of solutions for two types of singular boundary value problems. The Sturm-Liouville boundary value problem \[ -x^n(t)=f\bigl(t,x(t),\dots ,x^{(n-1)}(t)\bigr),\quad n\geq2, \]\[ x^{(i)}(0)=0,\quad 0\leq i\leq n-3,\quad \alpha x^{(n-2)}(0)-\beta x^{(n-1)}(0)=0,\quad \gamma x^{(n-2)}(T)+\delta x^{(n-1)}(T)=0, \] where \(T\) is a positive constant, \(\alpha,\gamma>0,\;\beta,\delta\geq0\) and \(f\) satisfies the local Caratheodory conditions on \([0,T]\times D\) with \(D=\mathbb{R}_+^{n-1}\times \mathbb{R}_0\), \(\mathbb{R}_+=(0,\infty)\) and \(\mathbb{R}_0=\mathbb{R}\setminus\{0\}\), and the \((p,n-p)\) right focal BVP \[ (-1)^{n-p}x^{(n)}(t)=f\bigl(t,x(t),\dots .x^{(n-1)}(t)\bigr),\quad n\geq2, \]\[ x^{(i)}(0)=0,\;0\leq i\leq p-1,\quad x^{(i)}(T)=0,\quad p\leq i\leq n-1, \] where \(p\in \mathbb{N}\) is fixed, \(1\leq p\leq n-1\) and \(f\) satisfies the Carathéodory conditions on \([0,T]\times X,\;\) where \(X\) is a proper subset of \(\mathbb{R}^n\). In both problems the function \(f(t,x_0,\dots ,x_{n-1})\) may be singular at the points \(x_i=0,\;0\leq i\leq n-1,\) of all its phase variables \(x_0,\dots ,x_{n-1}.\) To prove the existence of solutions for each of the above problems, the authors construct a suitable sequence of regular problems. The construction of regular BVPs is based on a priori bounds. The proofs of the existence results for the auxiliary regular BVPs are based on the nonlinear Fredholm alternative. Reviewer: Petio S. Kelevedjiev (Sliven) Cited in 1 ReviewCited in 1 Document MSC: 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:singular higher-order differential equation; Sturm-Liouville boundary conditions; focal boundary conditions; existence; regularization PDFBibTeX XMLCite \textit{I. Rachůnková} and \textit{S. Staněk}, Georgian Math. J. 10, No. 1, 165--191 (2003; Zbl 1069.34029) Full Text: EuDML