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The solution of a Fučik’s conjecture. (English) Zbl 0562.47049

Given an operator with jumping nonlinearity T on a Hilbert space H with the unit ball B, one can easily show, that the equation \((*)\;Tu=f\) has at least \(| \deg (T,0,B)|\) solutions for ”almost all” right-hand sides \(f\in H\). Having in mind the special type of non-linearity, we can ask, whether this number is exact, or at least, whether there exists an f such that (*) has no solution provided \(\deg (T,0,B)=0.\)
The result obtained in the article can be written as follows: The assertion \[ [\deg (T,0,B)=0\Rightarrow \exists f\in H \text{ s.t. (*) has no solution}] \] is true iff \(\dim H\leq 3.\) For \(\dim H=4\) an explicit counterexample is given. In order to obtain these results, a method of geometrical visualization of \({\mathbb{R}}^ 4\) was developed.

MSC:

47J05 Equations involving nonlinear operators (general)
55M25 Degree, winding number
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