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Zbl 1172.34020
Zhang, Xingqiu
Existence of positive solutions for multi-point boundary value problems on infinite intervals in Banach spaces.
(English)
[J] Appl. Math. Comput. 206, No. 2, 932-941 (2008). ISSN 0096-3003

The author considers a singular boundary value problem in a Banach space: $$x''(t)+f(t,x(t),x'(t))=0, \quad t\in (0,\infty),$$ $$x(0)=\sum_{i=1}^{m-2}\alpha_i x(\xi_i),\ x'(\infty)=y_\infty,\text{ where }0<\xi_1<\cdots<\xi_{m-2}<\infty,\ \alpha_i\in [0,\infty)$$ with $\sum_{i=1}^{m-2}\alpha_i >0$, and $\sum_{i=1}^{m-2}\alpha_i \xi_i>1-\sum_{i=1}^{m-2}\alpha_i >0.$ Under some conditions on $f$, the existence of a positive solution is discussed by the fixed point method.
[Jin Liang (Shanghai)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34G20 Nonlinear ODE in abstract spaces
34B16 Singular nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: singular boundary value problem; positive solution; fixed point

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