On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational

Abstract

The semilinear wave equation u t t − u x x + f ( u ) = g ( t ) {u_{tt}} - {u_{xx}} + f(u) = g(t) with u ( 0 , t ) = u ( π , t ) = 0 u(0,t) = u(\pi ,t) = 0 , u u is T T -periodic in t t , is considered for some situations in which T T is not a rational multiple of π \pi . Various existence results depending on the range of f ′ f’ are given, which contrast sharply with the case where T T is a rational multiple of π \pi .