The author is concerned with the existence and uniqueness of solutions to the problem \[ u_{tt}-u_{xx}=\epsilon f(x,t,u,u_ t,u_ x)+\epsilon \int^{h(x,t)}_{0}\phi (x,t,s,u(x,s),u_ t(x,s),u_ x(x,s))ds, \] u(0,t)\(=u(\pi,t)=0\), where \(\epsilon\) is a parameter, \(h: \{\) \(0\leq x\leq \pi\), \(t\in R\}\to R\), while f(x,t,u,v,w) and \(\phi\) (x,t,s,u,v,w) are defined for \(0\leq x\leq \pi\), \(t\in R\), \(s\in R\), \(| u| \leq a\), \(| v| \leq a\), \(| w| \leq a\), and satisfy appropriate smoothness conditions.