## Uniqueness of positive solutions of quasilinear differential equations.(English)Zbl 0831.34028

The paper deals with differential equations of the form (1) $$|u' |m - 2u')' + p(t) f(u) = 0$$, $$t \in (a,b)$$, where $$m > 1$$, $$p$$ is continuous and positive on $$(a,b)$$, $$f$$ is continuous on $$0, \infty)$$, positive on $$(0, \infty)$$ and such that the function $${f(u) \over um - 1}$$ is decreasing in $$u \in (0, \infty)$$. In particular, the author proves uniqueness theorems for positive solutions of the equation (1) subjected to the boundary conditions $$u(a) = u'(b) = 0$$, $$u'(a) = u(b) = 0$$ and $$u(a) = u(b) = 0$$, respectively. To this aim the notion of a generalized Prüfer transformation is introduced.
Reviewer: M.Tvrdý (Praha)

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 47J05 Equations involving nonlinear operators (general)