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Zbl 1071.34014
Liu, Bing
Solvability of multi-point boundary value problem at resonance. IV. (English)
[J] Appl. Math. Comput. 143, No. 2-3, 275-299 (2003). ISSN 0096-3003

Summary: In this part, we consider the following second-order ordinary differential equation $$x''= f(t, x(t), x'(t))+ e(t),\quad t\in (0,1),\tag1$$ subject to one of the following boundary value conditions: $$\alignat 2 x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad &&x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag2\\ x(0) &= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\quad && x'(1)= \sum^{n-2}_{j=1} \beta_j x'(\eta_j),\tag3\\ x'(0) &= \sum^{m-2}_{i=1} \alpha_i x'(\xi_i),\quad && x(1)= \sum^{n-2}_{j=1} \beta_j x(\eta_j),\tag4\endalignat$$ with$\alpha_i\in \bbfR$, $1\le i\le m-2$, $\beta_j\in \bbfR$, $1\le j\le n-2$, $0< \xi_1< \xi_2<\cdots< \xi_{m-2}< 1$, 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.\par For the parts I, II and III see [the author and {\it 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 {\it J. Yu}, ibid. 129, 119--143 (2002; Zbl 1054.34033)].

MSC 2000:: *34B10 Multipoint boundary value problems

Keywords: Boundary value problems; Fredholm operator; resonance; coincidence; degree

Citations: Zbl 1021.34013; Zbl 1053.34016; Zbl 1054.34033

Cited in: Zbl 1127.34006

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