Fixed point index and solutions of differential equations in Banach spaces
In reflexive Banach spaces by constructing homotopies, the author presents a formula for the computation of the fixed point index for completely continuous operators by the solutions of the corresponding ordinary differential equations. The main result is the following: Assume that \(\Omega\subset X\) is open bounded, \(A: P\to P\) is completely continuous, locally Lipschitzian and \(Ax\neq x\) for every \(x\in \partial\Omega\cap P.\) Then \(i(A,\Omega\cap P,P)=\lim_{t\to 0+}i(S(t),\Omega\cap P,P).\) The formula is then used to obtain fixed points for completely continuous operators on cones.
scientific article; zbMATH DE number 2161541