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Non-simple turning points and cusps. (English) Zbl 0539.65043
The work described in this paper is motivated by the behaviour of solutions of boundary-value problems for nonlinear elliptic equations of the form $$Lx=\lambda.\exp(x(1+\mu x))$$ in dependence of ($$\lambda$$,$$\mu)$$. For $$\mu \leq \mu_ 0$$, there are two simple turning points w.r.t. $$\lambda$$ that produce a ”double” turning point at $$\mu =\mu_ 0$$. The turning points $$\lambda$$ in dependence of $$\mu$$ show a cusp behaviour.
After defining ”simple” and ”double” turning points, the authors first show how to characterize simple turning points of a one-parameter problem as isolated solutions of a suitably extended system of the usual type. Then they turn to a two-parameter system $$f(\lambda,\mu,x)=0$$ with the aim of computing points $$(\lambda_ 0,\mu_ 0,x_ 0)$$ that are double turning points of f w.r.t. $$\lambda$$. To this end, they consider the following extended system: $$f(\lambda,\mu,x)=0$$, $$f_ x(\lambda,\mu,x)\phi =0$$, $$\ell \phi =1$$, where $$\ell$$ is an element of the dual space with $$\ell \phi_ 0=1$$ where $$\phi_ 0$$ is in the kernel of $$f_ x(\lambda_ 0,\mu_ 0,x_ 0)$$. The main result of this paper states that a double turning $$(\lambda_ 0,\mu_ 0,x_ 0)$$ point of f w.r.t $$\lambda$$ corresponds to a simple turning point $$(\lambda_ 0,\mu_ 0,x_ 0,\phi_ 0)$$ w.r.t. $$\mu$$ ; this holds under a condition that is equivalent to $$(\lambda_ 0,\mu_ 0,x_ 0)$$ being a cusp point of f.
With this result, standard algorithms for computing simple turning points, applied to the extended equation w.r.t. the parameter $$\mu$$, can be used to compute double turning points of f w.r.t. $$\lambda$$. The authors illustrate this by giving a numerical example for the model problem mentioned above with $$Lx=x''$$ and zero boundary conditions (in discretized form).
Reviewer: H.Engl

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 47J25 Iterative procedures involving nonlinear operators
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