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Existence and uniqueness of nonlinear deflections of an infinite beam resting on a non-uniform nonlinear elastic foundation

Sung Woo Choi1 and Taek Soo Jang2*

Author Affiliations

1 Department of Mathematics, Duksung Womens's University, Seoul 132-714, Republic of Korea

2 Department of Naval Architecture and Ocean Engineering, Pusan National University, Busan 609-735, Republic of Korea

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Boundary Value Problems 2012, 2012:5  doi:10.1186/1687-2770-2012-5

Published: 17 January 2012


We consider the static deflection of an infinite beam resting on a nonlinear and non-uniform elastic foundation. The governing equation is a fourth-order nonlinear ordinary differential equation. Using the Green's function for the well-analyzed linear version of the equation, we formulate a new integral equation which is equivalent to the original nonlinear equation. We find a function space on which the corresponding nonlinear integral operator is a contraction, and prove the existence and the uniqueness of the deflection in this function space by using Banach fixed point theorem.

2010 Mathematics Subject Classification: 34A12; 34A34; 45G10; 74K10.

Infinite beam; elastic foundation; nonlinear; non-uniform; fourth-order ordinary differential equation; Banach fixed point theorem; contraction.