On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions

Ammar Khanfer; Lazhar Bougoffa; Ammar Khanfer; Lazhar Bougoffa

doi:10.3934/math.2021575

AIMS Mathematics

2021, Volume 6, Issue 9: 9899-9910. doi: 10.3934/math.2021575

Previous Article Next Article

Research article

On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions

Ammar Khanfer ^{1
,
,},
Lazhar Bougoffa ^{2
,
,}

1.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
2.
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia

Received: 02 May 2021 Accepted: 23 June 2021 Published: 01 July 2021
MSC : 34B15, 34B18

The purpose of this paper is to establish an existence and uniqueness theorem for the nonlocal fourth-order nonlinear beam differential equations with a parameter

$ \begin{equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0<x<1 \end{equation*} $

subject to the integral boundary conditions:

$ \begin{equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, \end{equation*} $

where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $
- fourth-order beam equation,
- nonlocal boundary conditions,
- existence and uniqueness theorem,
- Schauder's fixed point theorem
Citation: Ammar Khanfer, Lazhar Bougoffa. On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions[J]. AIMS Mathematics, 2021, 6(9): 9899-9910. doi: 10.3934/math.2021575

Related Papers:

Abstract

The purpose of this paper is to establish an existence and uniqueness theorem for the nonlocal fourth-order nonlinear beam differential equations with a parameter

$ \begin{equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0<x<1 \end{equation*} $

subject to the integral boundary conditions:

$ \begin{equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, \end{equation*} $

where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $

References

[1]	D. G. Zill, M. R. Cullen, Differential Equations with Boundary-value Problems, 2 Eds., Brooks, Cole, 2008.
[2]	R. A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc., 77 (1979), 329–335. doi: 10.1090/S0002-9939-1979-0545591-4
[3]	A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415–426. doi: 10.1016/S0022-247X(86)80006-3
[4]	Y. S. Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc., 104 (1988), 175–180. doi: 10.1090/S0002-9939-1988-0958062-3
[5]	Z. Bai, H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357–368. doi: 10.1016/S0022-247X(02)00071-9
[6]	R. Ma, Multiple positive solutions for a semipositone fourth-order boundary value problem, Hiroshima Math. J., 33 (2003), 217–227.
[7]	Q. Zhang, S. Chena, J. Lv, Upper and lower solution method for fourth-order four-pointboundary value problems, J. Comput. Appl. Math., 196 (2006), 387–393. doi: 10.1016/j.cam.2005.09.007
[8]	Z. Wei, C. Pang, The method of lower and upper solutions for fourth order singular m-point boundary value problems, J. Math. Anal. Appl., 322 (2006), 675–692. doi: 10.1016/j.jmaa.2005.09.064
[9]	G. Infante, P. Pietramala, A cantilever equation with nonlinear boundary conditions, Electron. J. Qual. Theory Differ. Equ., 15 (2009), 1–14.
[10]	H. Ma, Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear Anal., 68 (2008), 645– 651. doi: 10.1016/j.na.2006.11.026
[11]	X. Han, H. Gao, J. Xu, Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter, Fixed Point Theory Appl., 2011 (2011), 1–11.
[12]	A. Cabada, S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261–5267.
[13]	R. Jiang, C. Zhai, Positive solutions for a system of fourth-order differential equations with integral boundary conditions and two parameters, Nonlinear Anal-Model, 23 (2018), 401–422. doi: 10.15388/NA.2018.3.7
[14]	Z. Bai, Positive solutions of some nonlocal fourth-order boundary value problem, Appl. Math. Comput., 215 (2010), 4191–4197.
[15]	X. Lv, L. Wang, M. Pei, Monotone positive solution of a fourth-order BVP with integral boundary conditions, Bound. Value Probl., 2015 (2015), 172. doi: 10.1186/s13661-015-0441-2
[16]	J. M. Gere, B. J. Goodno, Mechanics of Materials, Cengage Learning, 8th edition SI, 2012.
[17]	L. Bougoffa, A. Khanfer, Existence and uniqueness theorems of second-order equations with integral boundary conditions, Bull. Korean Math. Soc., 55 (2018), 899–911.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)