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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 for nonlinear operators on topological linear spaces 34A60 Differential inclusions 34C25 Periodic solutions of ODE 47H04 Set-valued operators 47H11 Degree theory (nonlinear operators) 54H25 Fixed-point and coincidence theorems in topological spaces
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##### References:
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