Geodesic
Positive and monotone solutions of an \(m\)-point boundary-value problem
Electronic journal of differential equations, Tome 2002 (2002)
We study the second-order ordinary differential equation

$ y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq 1, $

subject to the multi-point boundary conditions

$ \alpha y(0)\pm \beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)\,. $

We prove the existence of a positive solution (and monotone in some cases) under superlinear and/or sublinear growth rate in $f$. Our approach is based on an analysis of the corresponding vector field on the $(y,y')$ face-plane and on Kneser's property for the solution's funnel.
Classification : 34B10, 34B18, 34B15
Keywords: multipoint boundary value problems, positive monotone solution, vector field, sublinear, superlinear, kneser's property, solution's funel 
@article{EJDE_2002__2002__a40,
     author = {Palamides,  Panos K.},
     title = {Positive and monotone solutions of an \(m\)-point boundary-value problem},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {1010.34004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a40/}
}
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Palamides,  Panos K. Positive and monotone solutions of an \(m\)-point boundary-value problem. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a40/