## 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

### Keywords:

positive solution; fixed-point index; cone
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