For a function $F: X \to Y$ between real Banach spaces, we show how continuation methods to solve $F(u) = g$ may improve from basic understanding of the critical set $C$ of $F$. The algorithm aims at special points with a large number of preimages, which in turn may be used as initial conditions for standard continuation methods applied to the solution of the desired equation. A geometric model based on the sets $C$ and $F^{-1}(F(C))$ substantiate our choice of curves $c \in X$ with abundant intersections with $C$. We consider three classes of examples. First we handle functions $F: R^2 \to R^2$, for which the reasoning behind the techniques is visualizable. The second set of examples, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special points admit a high number of solutions. Finally, we handle a semilinear elliptic operator, by computing the six solutions of an equation of the form $-\Delta - f(u) = g$ studied by Solimini.

翻译：对于实Banach空间之间的函数$F: X \to Y$，我们展示了如何利用对$F$的临界集$C$的基本理解来改进求解$F(u) = g$的延拓方法。该算法旨在寻找具有大量原像的特殊点，这些点可作为标准延拓方法求解目标方程的初始条件。基于集合$C$与$F^{-1}(F(C))$的几何模型证实了我们选择与$C$有丰富交集的曲线$c \in X$的合理性。我们考虑三类实例：首先处理函数$F: R^2 \to R^2$，其技术背后的推理过程可视化；第二类实例涉及15维空间之间的函数，通过对非线性Sturm-Liouville问题离散化得到，该问题的特殊点具有大量解；最后，我们处理半线性椭圆型算子，通过计算Solimini研究的形如$-\Delta - f(u) = g$方程的六个解来验证方法。