Electron. J. Diff. Eqns., Vol. 2009(2009), No. 33, pp. 1-8.

Existence and uniqueness of solutions for fourth-order boundary-value problems in Banach spaces

Yujun Cui, Yumei Zou

Abstract:
This article concerns the fourth-order boundary-value problem
$$\displaylines{ x^{(4)}(t)=f(t,x(t),x''(t)),\quad t\in (0,1),\cr x(0)=x(1)=x''(0)=x''(1)=\theta, }$$
in a Banach space. We present some spectral conditions, on the nonlinearity $f(t,u,v)$, to guarantee the existence and uniqueness of solutions. Our method is different from the one used in the references, even the above problem in a scalar space.

Submitted March 14, 2008. Published February 19, 2009.
Math Subject Classifications: 47H10.
Key Words: Spectral conditions; Monch fixed-point theorem.

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Yujun Cui
Institute of Mathematics
Shandong University of Science and Technology
Qingdao 266510, China
email: cyj720201@163.com, sdcyj72@163.com
Yumei Zou
Institute of Mathematics
Shandong University of Science and Technology
Qingdao 266510, China
email: sdzouym@126.com

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