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Leggett–Williams theorems for coincidences of multivalued operators. (English) Zbl 1152.47041

Let \(X,Y\) be Banach spaces, \(C\) be a cone in \(X\) and let \(\Omega_1,\Omega_2\) be open bounded subsets of \(X\) with \(\overline{\Omega}_1\subset\Omega_2\). For a pair \((L,N)\) consisting of a linear Fredholm operator \(L:\text{dom}\,L\subset X\to Y\) of zero index and an upper semicontinuous (compact convex)-valued multimap \(N:X\multimap Y\), the authors study the existence of a coincidence point \((Lx \in Nx)\) in \(C\cap(\overline{\Omega}_2\setminus\Omega_1)\). They use topological degree methods under some compactness or condensivity conditions. As application, the existence of positive periodic solutions for a differential inclusion is considered.

MSC:

47H10 Fixed-point theorems
34A60 Ordinary differential inclusions
34C25 Periodic solutions to ordinary differential equations
47H04 Set-valued operators
47H11 Degree theory for nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
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