## 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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### References:

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