Monotone iterative solutions for a coupled system of p-Laplacian differential equations involving the Riemann–Liouville fractional derivative

Applying the monotone iterative technique and the method of upper and lower solutions, we investigate the existence of extremal solutions for a nonlinear system of p-Laplacian differential equations with nonlocal coupled integral boundary conditions. We present a numerical example to illustrate the main result.

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

<abstract><p>The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u&amp; = &amp;\lambda f(x, u, v, u'', v''), \ 0&lt;x&lt;1, \\ v^{(4)}+B(x)v&amp; = &amp;\mu g(x, u, v, u'', v''), \ 0&lt;x&lt;1 \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>subject to the coupled integral boundary conditions:</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)&amp; = &amp; \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)&amp; = &amp; \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda &gt; 0, \mu &gt; 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.</p></abstract>

The Uniqueness Theorem of the Solution for a Class of Differential Systems with Coupled Integral Boundary Conditions

We discuss the uniqueness of the solution to a class of differential systems with coupled integral boundary conditions under a Lipschitz condition. Our main method is the linear operator theory and the solvability for a system of inequalities. Finally, an example is given to demonstrate the validity of our main results.