Electron. J. Diff. Eqns., Vol. 2008(2008), No. 52, pp. 1-7.

Existence of solutions for a fourth-order boundary-value problem

Yang Liu

Abstract:
In this paper, we use the lower and upper solution method to obtain an existence theorem for the fourth-order boundary-value problem
$$\displaylines{ u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad 0<t<1,\cr u(0)=u'(1)=u''(0)=0,\quad u'''(1)=g\big(\int^1_0u''(t)d\theta(t)\big), }$$
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.

Submitted October 19, 2007. Published April 10, 2008.
Math Subject Classifications: 34B15.
Key Words: Fourth-order boundary-value problem; upper and lower solution; Riemann-Stieltjies integral; Nagumo-type condition.

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Yang Liu
Department of Mathematics, Yanbian University
Yanji, Jilin 133000, China.
Department of Mathematics, Huaiyin Teachers College
Huaian, Jiangsu 223300, China
email: liuyang19830206@yahoo.com.cn

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