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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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