Abstract
The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, \t\t\t{u(4)(t)=f(t,u(t),u′(t),u″(t),u‴(t)),t∈[0,1],u(0)=u′(0)=0,u″(1)=0,u‴(1)=g(u(1)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \\quad t\\in [0,1], \\\\ u(0)=u'(0)=0, \\\\ u''(1)=0,\\qquad u'''(1)=g(u(1)), \\end{cases} $$\\end{document} where f:[0,1]times mathbb{R}^{4}rightarrow mathbb{R}, g: mathbb{R}rightarrow mathbb{R} are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.
Highlights
In this paper, we investigate a fully fourth-order differential equation with nonlinear boundary condition⎧ ⎪⎪⎨u(4)(t) = f (t, u(t), u (t), u (t), u (t)), t ∈ [0, 1],⎪⎪⎩uu(0(1) )==u0(,0) (1) = g(u(1)), (1.1)where f : [0, 1] × R4 → R, g : R → R are continuous functions
The two-point boundary value problems (BVPs) of fourth-order differential equations are the mathematical models for describing the states of elastic beams
In [2, 7, 8], the solvability conclusions were obtained by some fixed point theorems and monotone iterative method under the conditions that the nonlinear function f only contains the firstorder derivative term
Summary
We investigate a fully fourth-order differential equation with nonlinear boundary condition. The existence theory of solutions for the cantilever beam equation with nonlinear boundary condition (1.3) was studied under the condition that the nonlinear term f does not involve the derivative terms of deformation function u. In [2, 7, 8], the solvability conclusions were obtained by some fixed point theorems and monotone iterative method under the conditions that the nonlinear function f only contains the firstorder derivative term. Because of the influence of fully derivative terms in the nonlinear function f and the nonlinearity of the boundary condition, the solvability for BVP (1.1) has not been studied extensively. As far as we know, there are fewer results on the equations of fully cantilever beam with nonlinear boundary conditions by using the method involving lower and upper solutions. We introduce the definitions of lower and upper solutions and provide some preliminary results, which are useful in the proof
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
