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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 nth-order multipoint boundary value problems at resonance

x (n) (t)=f(t,x(t),x ' (t),,x (n-1) (t))+e(t),t(0,1),x(0)=x ' (0)==x (n-2) (0)=0,x(1)= j=1 m-2 β j x(η j ),

and

x (n) (t)=f(t,x(t),x ' (t),,x (n-1) (t))+e(t),t(0,1),x(0)=x ' (0)==x (n-2) (0)=0,x ' (1)= j=1 m-2 β j x ' (η j )·

The idea in Mawhin’s coincidence degree theory is to find a Fredholm map of index 0, L:domLYZ and two continuous projectors P:YY and Q:ZZ such that ImP=KerL,KerQ=ImL,Y=KerLKerP and Z=ImLImQ. Furthermore, one needs a map N:TL that is L-compact on a closed subset Ω ¯ of Y where domLΩ. 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 domL={xW n,1 (0,1): x(0)=x ' (0)==x (n-2) (0)=0, x(1)= j=1 m-2 β j x(η j )}, and define N:YZ by (Nx)(t)=f(t,x(t),x ' (t),,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:YY 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 2 x)(t)=(n-1)!x (n-1) (0)t n-1 (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