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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.

MSC:
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
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References:
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