On the nonlinear system of fourth-order beam equations with integral boundary conditions

• Published in: AIMS mathematics Vol. 6; no. 10; pp. 11467 - 11481
• Main Authors: Khanfer, Ammar, Bougoffa, Lazhar
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: existence theorem; fourth-order differential equations; nonlocal boundary conditions; schauder's fixed point theorem
• ISSN: 2473-6988; 2473-6988
• 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 <disp-formula> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> subject to the coupled integral boundary conditions: <disp-formula> <tex-math id="FE2"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> 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.