## 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 $$C \cap (\overline{\Omega}_2 \setminus \Omega_1) \neq \emptyset$$. 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 $$r \leq x(t) \leq R$$ for some $$0 < r < R < \infty$$.

### MSC:

 47J05 Equations involving nonlinear operators (general) 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations

### Citations:

Zbl 0508.34030; Zbl 0576.34018
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