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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