Existence of positive solutions for the cantilever beam equations with fully nonlinear terms

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• 作者：Yongxiang Li ; ^{liyxnwnu@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fully fourth-order boundary value problem ; Cantilever beam equation ; Positive solution ; Cone ; Fixed point index
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：27
• 期：Complete
• 页码：221-237
• 全文大小：667 K

文摘

In this paper we discuss the existence of positive solutions of the fully fourth-order boundary value problem which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where $f:[0,1]\times {{\mathbb{R}}_{+}}^3\times {\mathbb{R}}_{-}\to {\mathbb{R}}_{+}$ is continuous. Some inequality conditions on f$f$ guaranteeing the existence of positive solutions are presented. Our conditions allow that $f(t,x_0,x_1,x_2,x_3)$ is superlinear or sublinear growth on $x_0,x_1,x_2,x_3$. For the superlinear case, a Nagumo-type condition is presented to restrict the growth of f$f$ on x₂$x_2$ and x₃$x_3$. Our discussion is based on the fixed point index theory in cones.