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Loss of positivity in a nonlinear scalar heat equation. (English) Zbl 1112.34017

This paper deals with a boundary value problem of the form
\[ -u''=f(t,u), \, 0<t<1, \, u'(0)=0,\quad u(\eta)+\beta u'(1)=0, \]
where \(\eta \in [0,1], \beta >0\) and \(f\) is a nonnegative function. It is investigated the existence of positive solutions; in particular, the authors are concerned with the problem of the lack of positive solutions when \(\beta\) decreases.
It is first proved a result on the existence of one or two solutions, positive in the interval \([0,b]\), being \(\eta < b < \eta + \beta < 1\). Secondly, sufficient conditions on \(f\) and \(\beta, \eta\) are given which guarantee that all solutions are positive in \([0,1]\). Finally, a uniqueness result is obtained in case that \(f\) is strictly decreasing in the second variable. The proof is performed in the framework of fixed-point index theory in cones; in particular, the given BVP is treated as a Hammerstein integral equation with sign-changing kernel.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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