Smooth solution of a boundary-value problem (Q2740280)

This paper deals with the following boundary-value problem NEWLINE\[NEWLINE \begin{aligned} &u_{tt}-u_{xx}=f(x,t,u),\quad x\in (0,\pi), t\in \mathbb{R}, \\ &u(0,t)=\mu(t),\quad u(\pi,t)=0,\quad t\in \mathbb{R},\\ &u(x,t+T)=u(x,t),\quad x\in [0,\pi].\end{aligned} \tag{1}NEWLINE\]NEWLINE The solution is sought in the space \(\widetilde A_3\) which consists of functions \(g(x,t)\) satisfying the condition \(g(x,t+2\pi /(2s-1))=-g(x,t)\) where \(s\in \mathbb{N}\). It is assumed that the function \(\mu(t)\) is \(T\)-periodic and \(f(t,x,g(t,x))\in \widetilde A_3\) for all \(g\in \widetilde A_3\). The existence theorem is proved by means of a contraction principle using the fact that there exists a Green function for the linearized problem in \(\widetilde A_3\). Quite similar results were obtained earlier in the papers of \textit{Yu. O. Mitropolsky} and \textit{N. G. Khoma} [Ukr. Math. J. 47, 1563-1570 (1995; Zbl 0962.35015)], \textit{Yu. O. Mitropolsky, G. P. Khoma}, and \textit{N. G. Khoma} [Ukr. Math. J. 50, No.~6, 929-933 (1998; Zbl 0937.35106)].