A note on a third-order multi-point boundary value problem at resonance.

*(English)*Zbl 1229.34028Summary: 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.

\[ x'''(t)=f(t,x(t),x'(t),x''(t)),\quad t\in (0,1), \]

\[ x''(0)=\sum^m_{i=1}\alpha_ix''(\xi_i),\quad x'(0)=0,\;x(1)=\sum^n_{j=1}\beta_jx(\eta_j), \]

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 |

##### Keywords:

third-order differential equations; coincidence degree theory; multi-point boundary value problem; resonance
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\textit{X. Lin} et al., Math. Nachr. 284, No. 13, 1690--1700 (2011; Zbl 1229.34028)

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