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Existence of positive solutions for multi-point boundary value problems on infinite intervals in Banach spaces. (English) Zbl 1172.34020
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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34G20Nonlinear ODE in abstract spaces
34B16Singular nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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