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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 \(C \cap (\overline{\Omega}_2 \setminus \Omega_1) \neq \emptyset\). 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
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