Two-point higher-order BVPs with singularities in phase variables
This paper provides results on the existence of positive solutions to two kinds of singular boundary value problems. The first one is the singular Lidstone boundary value problem \((-1)^n x^{(2n)}(t)=f(t,x(t), \dots, x^{(2n-2)}(t)), x^{(2j)}(0)=0= x^{(2j)}(T), 0\leq j \leq n-1\). It is assumed that the function \(f\) satisfies suitable Carathéodory conditions; moreover, the function \(f(t,x_0,\dots, x_{n-2})\) may be singular at the points \(x_i=0, 0\leq i \leq 2n-2\). The second one is the singular \((n,p)\)-boundary value problem \(-x^{(n)}(t)=f(t,x(t), \dots, x^{(n-1)}(t)), x^{(i)}(0)=0, 0\leq i \leq n-2, x^{(p)}(T)=0\), with \(p \in {\mathbb N}, 0\leq p \leq n-1\). It is assumed that the function \(f\) satisfies suitable Carathéodory conditions; moreover, the function \(f(t,x_0,\dots, x_{n-1})\) may be singular at the points \(x_i=0, 0\leq i \leq n-2\). For the Lidstone problem, it is assumed that, on a suitable subset of the space variables, the function \(f\) satisfies a condition of the form \(f(t,x_0,\dots, x_{n-2}) \leq \phi(t) + \sum_{j=0}^{2n-2} q_j(t) \omega_j(| x_j| )+\sum_{j=0}^{2n-2} h_j(t) | x_j| \), where \(\phi, h_j,q_j\) satisfy suitable technical assumptions and \(\omega_j\) is nonincreasing. A similar condition is assumed for the singular \((n,p)\)-boundary value problem. The proof is performed by applying the Leray-Schauder continuation theorem. Several related results can be found in the book by \textit{R. P. Agarwal, D. O'Regan} and \textit{P. J. Y. Wong} [Positive solutions of differential, difference and integral equations. Dordrecht: Kluwer Academic Publishers, (1999; Zbl 1157.34301)]. Other related results for Dirichlet second-order BVPs can be found, e.g., in the paper by \textit{C. De Coster, M. R. Grossinho} and \textit{P. Habets} [Appl. Anal. 59, No. 1--4, 241--256 (1995; Zbl 0847.34022)]; see also \textit{C. De Coster} and \textit{P. Habets} [F. Zanolin (ed.), Nonlinear analysis and boundary value problems for ordinary differential equations. Wien: Springer. CISM Courses Lect. 371, 1--78 (1996; Zbl 0889.34018)].
singular higher-order boundary value problem
scientific article; zbMATH DE number 2093588
scientific article