## Bifurcation from simple eigenvalues.(English)Zbl 0219.46015

Let $$G$$ map a subset of a Banach space $$W$$ into a Banach space $$Y$$ and let $$C$$ be a curve in $$W$$ such that $$G(C) = \{0\}$$. A point $$p\in C$$ is called a bifurcation point for $$G$$ with respect to $$C$$ if every neighborhood of $$C$$ contains zeros of $$G$$ not on $$C$$. The main result of the paper gives sufficient conditions for $$c$$ to be a bifurcation point and for this case precisely characterizes $$G^{-1}(0)$$ near $$c$$. A stronger and useful specialization treats the case where $$C$$ is a line segment which arises often in practice. Several illustrative applications are given, in particular to ordinary and partial differential equations.

### MSC:

 46B99 Normed linear spaces and Banach spaces; Banach lattices 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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### References:

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