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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 $$\align &x^{(n)}(t) = f(t, x(t), x'(t), \dots, x^{(n-1)}(t)) + e(t), \qquad t \in (0, 1),\\&x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0, \qquad x(1) = \sum_{j=1}^{m-2} \beta_j x(\eta_j),\endalign$$ and $$\align &x^{(n)}(t) = f(t, x(t), x'(t), \dots, x^{(n-1)}(t)) + e(t), \qquad t \in (0, 1),\\ &x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0, \qquad x'(1) = \sum_{j=1}^{m-2} \beta_j x'(\eta_j).\endalign$$ 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:
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
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References:
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