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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

$\begin{array}{cc}& {x}^{\left(n\right)}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots ,{x}^{\left(n-1\right)}\left(t\right)\right)+e\left(t\right),\phantom{\rule{2.em}{0ex}}t\in \left(0,1\right),\hfill \\ & x\left(0\right)={x}^{\text{'}}\left(0\right)=\cdots ={x}^{\left(n-2\right)}\left(0\right)=0,\phantom{\rule{2.em}{0ex}}x\left(1\right)=\sum _{j=1}^{m-2}{\beta }_{j}x\left({\eta }_{j}\right),\hfill \end{array}$

and

$\begin{array}{cc}& {x}^{\left(n\right)}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots ,{x}^{\left(n-1\right)}\left(t\right)\right)+e\left(t\right),\phantom{\rule{2.em}{0ex}}t\in \left(0,1\right),\hfill \\ & x\left(0\right)={x}^{\text{'}}\left(0\right)=\cdots ={x}^{\left(n-2\right)}\left(0\right)=0,\phantom{\rule{2.em}{0ex}}{x}^{\text{'}}\left(1\right)=\sum _{j=1}^{m-2}{\beta }_{j}{x}^{\text{'}}\left({\eta }_{j}\right)·\hfill \end{array}$

The idea in Mawhin’s coincidence degree theory is to find a Fredholm map of index 0, $L:domL\subset Y\to Z$ and two continuous projectors $P:Y\to Y$ and $Q:Z\to Z$ such that $ImP=KerL,\phantom{\rule{4pt}{0ex}}KerQ=ImL,\phantom{\rule{4pt}{0ex}}Y=KerL\oplus KerP$ and $Z=ImL\oplus ImQ$. Furthermore, one needs a map $N:T\to L$ that is $L$-compact on a closed subset $\overline{{\Omega }}$ of $Y$ where $domL\cap {\Omega }\ne \varnothing$. For the first existence theorem, the authors take $Y={C}^{n-1}\left[0,1\right]$ and $Z={L}^{1}\left[0,1\right]$ and define the operator $L$ by $Lx={x}^{\left(n\right)}$, with $domL=\left\{x\in {W}^{n,1}\left(0,1\right)$: $x\left(0\right)={x}^{\text{'}}\left(0\right)=\cdots ={x}^{\left(n-2\right)}\left(0\right)=0$, $x\left(1\right)={\sum }_{j=1}^{m-2}{\beta }_{j}x\left({\eta }_{j}\right)\right\}$, and define $N:Y\to Z$ by $\left(Nx\right)\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots ,{x}^{\left(n-1\right)}\left(t\right)\right)+e\left(t\right)$. 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

$\left(Px\right)\left(t\right)={x}^{\left(n-1\right)}\left(0\right){t}^{n-1}·$

This $P$ is not a projector if $n>2$. Note that

${\left(Px\right)}^{\left(n-1\right)}\left(t\right)=\left(n-1\right)!{x}^{\left(n-1\right)}\left(0\right)$

and so,

$\left({P}^{2}x\right)\left(t\right)=\left(n-1\right)!{x}^{\left(n-1\right)}\left(0\right){t}^{n-1}\ne \left(Px\right)\left(t\right)·$

The theorems are valid in the case when $n=2$.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE