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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References:

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