Geodesic
Existence of solutions for a fourth-order boundary-value problem
Electronic journal of differential equations, Tome 2008 (2008)
In this paper, we use the lower and upper solution method to obtain an existence theorem for the fourth-order boundary-value problem

$\displaylines{ u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 01,\cr u(0)=u'(1)=u''(0)=0,\quad u'''(1)=g\big(\int^1_0u''(t)d\theta(t)\big), }$

where $f : [0,1]\times \mathbb{R}^4 \to \mathbb{R}, g : \mathbb{R}\to \mathbb{R}$ are continuous and may be nonlinear, and $\int^1_0u''(t)d\theta(t)$ denotes the Riemann-Stieltjes integral.
Classification : 34B15
Keywords: fourth-order boundary-value problem, upper and lower solution, Riemann-stieltjies integral, Nagumo-type condition 
@article{EJDE_2008__2008__a69,
     author = {Liu,  Yang},
     title = {Existence of solutions for a fourth-order boundary-value problem},
     journal = {Electronic journal of differential equations},
     year = {2008},
     volume = {2008},
     zbl = {1172.34305},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a69/}
}
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Liu,  Yang. Existence of solutions for a fourth-order boundary-value problem. Electronic journal of differential equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a69/