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Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations. (English) Zbl 1081.34017

The authors attempt to use coincidence degree theory to study the \(n\)th-order multipoint boundary value problems at resonance \[ \begin{aligned} &x^{(n)}(t) = f(t, x(t), x'(t), \dots, x^{(n-1)}(t)) + e(t), \qquad t \in (0, 1),\\&x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0, \qquad x(1) = \sum_{j=1}^{m-2} \beta_j x(\eta_j),\end{aligned} \] and \[ \begin{aligned} &x^{(n)}(t) = f(t, x(t), x'(t), \dots, x^{(n-1)}(t)) + e(t), \qquad t \in (0, 1),\\ &x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0, \qquad x'(1) = \sum_{j=1}^{m-2} \beta_j x'(\eta_j).\end{aligned} \] The idea in Mawhin’s coincidence degree theory is to find a Fredholm map of index 0, \(L:\operatorname{dom} L \subset Y \to Z\) and two continuous projectors \(P : Y \to Y\) and \(Q: Z \to Z\) such that \(\operatorname{Im} P = \operatorname{Ker} L, \;\operatorname{Ker}Q = \operatorname{Im}L,\;Y = \operatorname{Ker}L \oplus \operatorname{Ker}P\) and \(Z =\operatorname{Im}L \oplus \operatorname{Im}Q\). Furthermore, one needs a map \(N:T \to L\) that is \(L\)-compact on a closed subset \(\overline {\Omega}\) of \(Y\) where \(\operatorname{dom}L \cap \Omega \neq \emptyset\). For the first existence theorem, the authors take \(Y = C^{n-1}[0, 1]\) and \(Z = L^1[0, 1]\) and define the operator \(L\) by \(Lx= x^{(n)}\), with \(\operatorname{dom}L = \{ x \in W^{n,1}(0, 1)\): \(x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0\), \(x(1) = \sum_{j=1}^{m-2} \beta_j x(\eta_j)\}\), and define \(N:Y \to Z\) by \( (Nx)(t) = f(t, x(t), x'(t), \dots, x^{(n-1)}(t)) + e(t)\). The first boundary value problem can then be written as \(Lx = Nx\). There is a subtle flaw in their arguments in the first existence theorem when \(n > 2\). They define the operator \(P: Y \to Y\) by \[ (Px)(t) = x^{(n-1)}(0) t^{n-1}. \] This \(P\) is not a projector if \(n > 2\). Note that \[ (Px)^{(n-1)}(t) = (n-1)!x^{(n-1)}(0) \] and so, \[ (P^2x)(t) = (n-1)!x^{(n-1)}(0) t^{n-1} \neq (Px)(t). \] The theorems are valid in the case when \(n = 2\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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