The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters
$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0<x<1, \\ v^{(4)}+B(x)v& = &\mu g(x, u, v, u'', v''), \ 0<x<1 \end{array} \right. \end{eqnarray*} $
subject to the coupled integral boundary conditions:
$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)& = & \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)& = & \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $
where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda > 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.
Citation: Ammar Khanfer, Lazhar Bougoffa. On the nonlinear system of fourth-order beam equations with integral boundary conditions[J]. AIMS Mathematics, 2021, 6(10): 11467-11481. doi: 10.3934/math.2021664
The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters
$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0<x<1, \\ v^{(4)}+B(x)v& = &\mu g(x, u, v, u'', v''), \ 0<x<1 \end{array} \right. \end{eqnarray*} $
subject to the coupled integral boundary conditions:
$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)& = & \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)& = & \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $
where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda > 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.
| [1] | F. L. Zhu, L. S. Liu, Y. H. Wu, Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems, Appl. Math. Comput., 216 (2010), 448–457. |
| [2] | Q. Y. Wang, L. Yang, Positive solutions for a nonlinear system of fourth-order ordinary differential equations, Electron. J. Differ. Eq., 2020 (2020), 1–15. |
| [3] | R. T. Jiang, C. B. 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. |
| [4] | A. Khanfer, L. Bougoffa, On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions, AIMS Mathematics, 6 (2021), 9899–9910. |
| [5] | Z. B. Bai, H. Y. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357–368. |
| [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. H. Chena, J. H. Lv, Upper and lower solution method for fourth-order four-pointboundary value problems, J. Comput. Appl. Math., 196 (2006), 387–393. |
| [8] | Z. L. Wei, C. 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. |
| [9] | G. Infante, P. Pietramala, A cantilever equation with nonlinear boundary conditions, Electron. J. Qual. Theor., 15 (2009), 1–14. |
| [10] | H. L. Ma, Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear Anal.-Theor., 68 (2008), 645–651. |
| [11] | X. L. Han, H. L. Gao, J. Xu, Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter, Fixed Point Theory A., 2011 (2011), 604046. |
| [12] | J. B. Diaz, F. T. Metcalf, Variations of Wirtinger's inequality, New York: Academic Press, 1976. |
Ammar Khanfer, Lazhar Bougoffa. On the nonlinear system of fourth-order beam equations with integral boundary conditions[J]. AIMS Mathematics, 2021, 6(10): 11467-11481. doi: 10.3934/math.2021664
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