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A note on a third-order multi-point boundary value problem at resonance. (English) Zbl 1229.34028
Summary: Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third-order multi-point boundary value problem at resonance
$x'''(t)=f(t,x(t),x'(t),x''(t)),\quad t\in (0,1),$
$x''(0)=\sum^m_{i=1}\alpha_ix''(\xi_i),\quad x'(0)=0,\;x(1)=\sum^n_{j=1}\beta_jx(\eta_j),$
where $$f:[0,1]\times \mathbb R^3\to\mathbb R$$ is a continuous function, $$0<\xi_1<\cdots<\xi_m < 1$$, $$\alpha_i\in\mathbb R$$, $$i=1,\dots,m$$, $$m\geq 1$$ and $$0<\eta_1<\eta_2<\cdots<\eta_n<1$$, $$\beta_j\in\mathbb R$$, $$j=1,2,\dots,n$$, $$n\geq 2$$. In this paper, the dimension of the linear space $$\text{Ker}\,L$$ (the linear operator $$L$$ is defined by $$Lx=x'''$$) is equal to 2. Since all the existence results for third-order differential equations obtained in previous papers are for the case $$\dim\text{Ker}\,L=1$$, our work is new.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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