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On the existence of positive solutions of nonlinear second order differential equations. (English) Zbl 0857.34036

The authors establish the following criterion for the existence of a positive solution to the boundary value problem (1) \(u''(t)+ f(t,u(t))=0\), \(0<t<1\), \(a\cdot u(0)-b\cdot u'(0)=0\), \(c\cdot u(1)+d\cdot u'(1)=0\).
Theorem. Suppose that \(f\in C([0,1]\times [0,\infty);[0,\infty))\); \(a,b,c,d\geq 0\); \(cb+ac+ab>0\); \(0<M:=\min\{ {{c+4d}\over{4(c+d)}}, {{a+4b}\over{4(a+b)}}\}<1\) and assume that there exist two distinct positive constants \(\lambda\), \(\eta\) such that \[ \begin{aligned} f(t,u) &\leq\lambda \biggl(\int^1_0 k(s,s)ds\biggr)^{-1} \qquad\text{on} \quad [0,1]\times [0,\lambda],\\ \text{and} f(t,u) &\geq\eta \biggl( \int^{3\over4}_{1\over4} k({\textstyle{1\over2}},s)ds) \biggr)^{-1} \qquad\text{on}\quad [{\textstyle{{1\over4},{3\over4}}}] \times[M\eta,\eta], \end{aligned} \] where \(k(t,s)\) is the Green’s function of the differential equation \(u''(t)=0\), \(t\in(0,1)\) with respect to the same boundary conditions as (1). Then (1) has at least one positive solution \(u\) such that \(|u|:= \sup_{t\in[0,1]}|u(t)|\) between \(\lambda\) and \(\eta\).
The authors apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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