On the existence of positive solutions for the bending elastic beam equations

doi:10.1016/j.amc.2006.11.144

Applied Mathematics and Computation

Volume 189, Issue 1, 1 June 2007, Pages 821-827

https://doi.org/10.1016/j.amc.2006.11.144 Get rights and content

Abstract

This paper discusses the existence of positive solutions of the fourth-order boundary value problem $\begin{matrix} u^{(4)} = f (t, u, u^{″}), 0 < t < 1, \\ u (0) = u (1) = u^{″} (0) = u^{″} (1) = 0, \end{matrix}$ which models a statically bending elastic beam whose two ends are simply supported, where $f : [0, 1] \times R^{+} \times R^{-} \to R^{+}$ is continuous. The essential conditions on f guaranteeing the existence of positive solution are presented. The discussion is based on the fixed point index theory in cones.

Section snippets

Introduction and main results

The deformations of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by fourth-order ordinary differential equation boundary value problem (BVP) $\{\begin{matrix} u^{(4)} (t) = f (t, u (t), u^{″} (t)), 0 < t < 1, \\ u (0) = u (1) = u^{″} (0) = u^{″} (1) = 0, \end{matrix}$ where $f : [0, 1] \times R \times R \to R$ is continuous [3], [4], and the u″ in f is the bending moment term which represents bending effect. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors, see [1], [2], [3], [4],

Preliminaries

We denote the maximum norm of C(I) by ∥u∥. Let C⁺(I) be the cone of all nonnegative functions in C(I). We choose an equivalent norm in Banach space C²(I) by $‖ u ‖_{2} = ‖ u ‖ + ‖ u^{″} ‖ = \max_{t \in I} | u (t) | + \max_{t \in I} | u^{″} (t) | .$ Given $h \in C (I)$ , we consider the linear boundary value problem (LBVP) $\{\begin{matrix} u^{(4)} = h (t), t \in I, \\ u (0) = u (1) = u^{″} (0) = u^{″} (1) = 0 . \end{matrix}$ Let G(t, s) be the Green’s function to the linear boundary value problem $- u^{″} = 0, u (0) = u (1) = 0,$ which is explicitly expressed by $G (t, s) = \{\begin{matrix} t (1 - s), 0 ⩽ t ⩽ s ⩽ 1, \\ s (1 - t), 0 ⩽ s ⩽ t ⩽ 1 . \end{matrix}$ It is easy to see that G(t, s) has

Proof of the main results

Proof of Theorem 1

We show that the mapping A defined by (10) has a non-zero fixed point by counting of the fixed point index in cone K in C²(I).

Choose $r \in (0, r_{0})$ , where r₀ is the constant in assumption (F1). We prove that $λ Au \neq u$ for $u \in \partial K_{r}$ and $0 < λ ⩽ 1$ . In fact, if there exist $u_{0} \in \partial K_{r}$ and $0 < λ_{0} ⩽ 1$ such that $λ_{0} {Au}_{0} = u_{0}$ , then by definition of A, u₀(t) satisfies differential equation $u_{0}^{(4)} (t) = λ_{0} f (t, u_{0} (t), u_{0}^{″} (t)), 0 ⩽ t ⩽ 1,$ and boundary condition $u_{0} (0) = u_{0} (1) = u_{0}^{″} (0) = u_{0}^{″} (1) = 0 .$ Since $0 ⩽ u_{0} (t), - u_{0}^{″} (t) ⩽ ∥ u_{0} ∥_{2} = r < r_{0}$ , from assumption (F1) and

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This paper discusses the solvability of the fourth-order boundary value problem
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^☆: Research supported by NNSF of China (10271095) and the NSF of Gansu Province (ZS031-A25-003-Z).

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On the existence of positive solutions for the bending elastic beam equations☆

Abstract

Section snippets

Introduction and main results

Preliminaries

Proof of the main results

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

On fourth-order boundary value problems arising in beam analysis

Diff. Integral Equat.

Existence and uniqueness theorems for a bending of an elastic beam equation

Appl. Anal.

Fourth-order two-point boundary value problems

Proc. Am. Math. Soc.