## Multiple fixed points of positive mappings.(English)Zbl 0597.47034

The author proves several multiplicity results for fixed points of monotone nonlinear operators without using variational structure (hence non-potential operators are covered as well). He gives many interesting remarks and connections with related papers, in particular, with work of H. Hofer [Math. Ann. 261, 493-514 (1982; Zbl 0488.47034)]. A typical result reads as follows:
Let E be a Banach space and K a reproducing normal cone (with a certain very weak additional property) in E. Let A be a completely continuous $$C^ 1$$-map in E such that A’(x) is a demi-interior point for $$x\in E$$. Moreover, suppose that A has two fixed points $$x_ 0$$ and $$x_ 1$$ which are incomparable (i.e. $$x_ 0-x_ 1\not\in K\cup (-K))$$ and are both quasiminima (i.e. $$ind(I-A,x_ i)=1$$, $$A'(x_ i)$$ has no real eigenvalue $$\geq 1$$, and the kernel of $$I-A'(x_ i)$$ is at most one- dimensional). Finally, assume that there exist $$w_ 1\in (x_ 0,\infty)\cap (x_ 1,\infty)$$, $$w_ 2\in (-\infty,x_ 0)\cap (- \infty,x_ 1)$$ and $$R>0$$ such that $$x\neq \lambda A(x)+(1-\lambda)w_ 1$$ if $$\lambda\in (0,1)$$, $$\| x\| \geq R$$ and $$x\in [x_ 0,\infty)\cap [x_ 1,\infty)$$ and such that $$x\neq \lambda A(x)+(1-\lambda)w_ 2$$ if $$\lambda\in (0,1)$$, $$\| x\| \geq R$$ and $$x\in [-\infty,x_ 0]\cap (-\infty,x_ 1)$$. Then the equation $$x=A(x)$$ has at least 9 solutions.
An application to some second-order elliptic problem is also given which is a slight generalization of a previous result of P. Hess [Comm. Partial Differ. Equations 6, 951-961 (1981; Zbl 0468.35073)].
Reviewer: J.Appell

### MSC:

 47H10 Fixed-point theorems 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

### Citations:

Zbl 0488.47034; Zbl 0468.35073
Full Text: