Fourth-order \(p\)-Laplacian nonlinear systems via the vector version of Krasnosel'skiĭ's fixed point theorem
The paper deals with the fourth-order nonlinear differential system with \(p\)-Laplacian \[ \begin{cases}(\varphi_p(u_1''))''=\lambda_1a_1(t)f_1(u_1,u_2),\;t\in (0,1),\\ (\varphi_p(u_2''))''=\lambda_2a_2(t)f_2(u_1,u_2),\;t\in (0,1),\\ u_1(0)=u_1(1)=u_1''(0)=u_1''(1)=0,\\ u_2(0)=u_2(1)=u_2''(0)=u_2''(1)=0,\end{cases}\tag{1} \] where \(f_i\in C([0,+\infty)^2,[0,+\infty))\), \(a_i\in C([0,1],[0,+\infty))\), \(\lambda_i\) are positive parameters for \(i=1,2\) and \(\varphi_p(s)=s|s|^{p-2}\), \(p>1\). By using Krasnoselskii's fixed point theorem and a vector version of it, the authors present sufficient conditions for the existence of positive solutions of \((1)\).
Krasnosel'skij's cone fixed point theorem
scientific article; zbMATH DE number 5659719
scientific article