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Leggett–Williams norm-type theorems for coincidences. (English) Zbl 1109.47051

The article deals with positive solutions to the operator equation $Lx=Nx$, where $L$ is a linear Fredholm mapping of index zero and $N$ a nonlinear operator between a Banach space $X$ ordered with a cone $C$ and a Banach space $Y$. The authors present some natural assumptions under which the operator equation $Lx=Nx$ has a solution in the set $C\cap \left({\overline{{\Omega }}}_{2}\setminus {{\Omega }}_{1}\right),$ where ${{\Omega }}_{1}$ and ${{\Omega }}_{2}$ are open bounded subsets of $X$ with ${\overline{{\Omega }}}_{1}\subset {{\Omega }}_{2}$ and $C\cap \left({\overline{{\Omega }}}_{2}\setminus {{\Omega }}_{1}\right)\ne \varnothing$. As an application, the periodic problem

${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right)\right),\phantom{\rule{4pt}{0ex}}t\in \left[0,1\right],\phantom{\rule{4pt}{0ex}}x\left(0\right)=x\left(1\right)$

is studied; the authors describe conditions under which this problem has a solution satisfying $r\le x\left(t\right)\le R$ for some $0.

MSC:
 47J05 Equations involving nonlinear operators (general) 47H07 Monotone and positive operators on ordered topological linear spaces 47H09 Mappings defined by “shrinking” properties 34B18 Positive solutions of nonlinear boundary value problems for ODE 34C25 Periodic solutions of ODE