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Solvability of multi-point boundary value problem at resonance. IV. (English) Zbl 1071.34014
Summary: In this part, we consider the following second-order ordinary differential equation \[ x''= f(t, x(t), x'(t))+ e(t),\quad t\in (0,1),\tag{1} \] subject to one of the following boundary value conditions: \[ \begin{alignedat}{2} x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad &&x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag{2}\\ x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad && x'(1)= \sum^{n-2}_{j=1} \beta_j x'(\eta_j),\tag{3}\\ x'(0) &= \sum^{m-2}_{i=1} \alpha_i x'(\xi_i),\quad && x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag{4}\end{alignedat} \] with\(\alpha_i\in \mathbb{R}\), \(1\leq i\leq m-2\), \(\beta_j\in \mathbb{R}\), \(1\leq j\leq n-2\), \(0< \xi_1< \xi_2<\cdots< \xi_{m-2}< 1\), and \(0< \eta_1< \eta_2<\cdots< \eta_{n-2}< 1\). When all \(\alpha_i\) have no the same sign and all \(\beta_j\) have no the same sign, some existence result are given for (1) with boundary conditions (2)–(4) at resonance case. We give some examples to demonstrate our result, too.
For the parts I, II and III see [the author and J. Yu, Indian J. Pure Appl. Math. 33, 475–494 (2002; Zbl 1021.34013), the author, Appl. Math. Comput. 136, 353–377 (2003; Zbl 1053.34016) and the author and J. Yu, ibid. 129, 119–143 (2002; Zbl 1054.34033)].

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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