×

zbMATH — the first resource for mathematics

General existence principle for singular BVPs and its application. (English) Zbl 1059.34016
First, the authors present a general existence principle for singular boundary value problems of the form \[ u^{(n)}(t)=f\bigl(t,u(t),\dots,u^{(n-1)}(t)\bigr),\quad u\in S. \] Here, the set \(S\subset C^{n-1}([0,T])\) is closed, \(f\in \text{Car}([0,T]\times D)\), where \(D\subset \mathbb{R}^n\) is not closed, and the function \(f(t,x_0,\dots,x_{n-1})\) has singularities on \(\partial D\) in all its phase variables. The existence principle is based on the construction of a proper sequence of auxiliary regular problems. Its convergence follows by Vitali’s convergence theorem, where the assumption about the existence of a Lebesgue integrable majorant function is replaced by a more general assumption on the uniform absolute continuity. Next, the general existence principle is applied to study the existence of positive solutions to the \((p,n-p)\) conjugate boundary value problem \[ (-1)^{p}u^{(n)}(t)=f\bigl(t,u(t),\dots,u^{(n-1)}(t)\bigr), \]
\[ u(0)=\dots=u^{(n-p-1)}(0)=0,\quad u(T)=\dots=u^{(p-1)}(T)=0. \] Here, \(n>2\) and \(1\leq p\leq n-1\) are fixed natural numbers, \(D=\mathbb{R}_+\times \mathbb{R}_0^{(n-1)}\), where \(\mathbb{R}_+=(0,\infty),\;\mathbb{R}_0=\mathbb{R}\setminus\{0\}\), and \(f\in \text{Car}([0,T]\times D)\) has singularities on \(\partial D\) in all its phase variables.

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite