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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 $C \cap (\overline{\Omega}_2 \setminus \Omega_1),$ where $\Omega_1$ and $\Omega_2$ are open bounded subsets of $X$ with $\overline{\Omega}_1 \subset \Omega_2$ and $C \cap (\overline{\Omega}_2 \setminus \Omega_1) \ne \emptyset$. As an application, the periodic problem $$x'(t) = f(t,x(t)), \ t \in [0,1], \ x(0) = x(1)$$ is studied; the authors describe conditions under which this problem has a solution satisfying $r \le x(t) \le R$ for some $0 < r < R < \infty$.

47J05Equations involving nonlinear operators (general)
47H07Monotone and positive operators on ordered topological linear spaces
47H09Mappings defined by “shrinking” properties
34B18Positive solutions of nonlinear boundary value problems for ODE
34C25Periodic solutions of ODE
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