Geodesic
Positive solutions of boundary value problems for \(2m\)-order differential equations
Electronic journal of differential equations, Tome 2003 (2003)
This article concerns the existence of positive solutions to the differential equation

$ (-1)^m x^{(2m)}(t)=f(t,x(t),x'(t),\dots,x^{(m)}(t)), \quad 0 less than t less than \pi, $

subject to boundary condition

$ x^{(2i)}(0)=x^{(2i)}(\pi)=0, $

or to the boundary condition

$ x^{(2i)}(0)=x^{(2i+1)}(\pi)=0, $

for $i=0,1,\dots,m-1$. Sufficient conditions for the existence of at least one positive solution of each boundary-value problem are established. Motivated by references [7,17,21], the emphasis in this paper is that $f$ depends on all higher-order derivatives.
Classification : 34B18, 34B15, 34B27
Keywords: higher-order differential equation, boundary-value problem, positive solution 
@article{EJDE_2003__2003__a69,
     author = {Liu,  Yuji and Ge,  Weigao},
     title = {Positive solutions of boundary value problems for \(2m\)-order differential equations},
     journal = {Electronic journal of differential equations},
     year = {2003},
     volume = {2003},
     zbl = {1042.34049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a69/}
}
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Liu,  Yuji; Ge,  Weigao. Positive solutions of boundary value problems for \(2m\)-order differential equations. Electronic journal of differential equations, Tome 2003 (2003). http://geodesic.mathdoc.fr/item/EJDE_2003__2003__a69/