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**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)].

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 |