Singular third-order \(m\)-point boundary value problems. (English) Zbl 1184.34033
Summary: This paper is concerned with the following boundary value problem
\[ \begin{aligned} u'''(t)&= f(t,u(t),u'(t),u''(t))+ e(t), \quad 0<t<1,\\ u(0)&= \sum_{i=1}^{m-2} k_i(\xi_i), \quad u'(0)=u'(1)=0, \end{aligned} \]
where \(f:(0,1)\times\mathbb R^3\to\mathbb R\) is a function satisfying Carathéodory’s conditions, \(e:(0,1)\to\mathbb R\) and \(t(1-t)e(t)\in L^1[0,1]\), \(0<\xi_1<\xi_2<\cdots< \xi_{m-2}<1\), \(k_i\in\mathbb R\) \((i=1,2,\dots,m-2)\) and \(\sum_{i=1}^{m-2} k_i\neq 1\). Some existence criteria of at least one solution are established by using the well-known Leray-Schauder continuation principle.
\[ \begin{aligned} u'''(t)&= f(t,u(t),u'(t),u''(t))+ e(t), \quad 0<t<1,\\ u(0)&= \sum_{i=1}^{m-2} k_i(\xi_i), \quad u'(0)=u'(1)=0, \end{aligned} \]
where \(f:(0,1)\times\mathbb R^3\to\mathbb R\) is a function satisfying Carathéodory’s conditions, \(e:(0,1)\to\mathbb R\) and \(t(1-t)e(t)\in L^1[0,1]\), \(0<\xi_1<\xi_2<\cdots< \xi_{m-2}<1\), \(k_i\in\mathbb R\) \((i=1,2,\dots,m-2)\) and \(\sum_{i=1}^{m-2} k_i\neq 1\). Some existence criteria of at least one solution are established by using the well-known Leray-Schauder continuation principle.
MSC:
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
