Nontrivial periodic solutions of a nonlinear beam equation

We consider the existence of a nontrivial solution of the following equation:\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \begin{array}{*{20}c} {} & {u_{tt} + u_{xxxx} + g(u) = 0} & {(x,t) \in Q = (0,\pi ) \times (0,2\pi )} \\ \end{array} \\ \begin{array}{*{20}c} {(0)} & {u(0,t) = u(\pi,t) = u_{xx} (\pi,t) = 0,t \in (0,2\pi )} & {} \\ \end{array} \\ \begin{array}{*{20}c} {} & {u(x,0) = (x,2\pi ),} & x \\ \end{array} \in (0,\pi ) \\ \end{array} $$\end{document} where g is a nondecreasing function defined on R 1 , satisfies g (O) = O, and some other additional conditions. Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]‐[8]:\documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = 0} & {(x,t) \in 0} \\ \end{array} \\ \begin{array}{*{20}c} {u(0,t) = u(\pi,t) = 0} & {t\varepsilon (0,2\pi )} \\ \end{array} \\ \begin{array}{*{20}c} {u(x,0) = u(x,2\pi )} & {x\varepsilon (0,\pi )} \\ \end{array} \\ \end{array} $$\end{document} .