Global Periodic Solutions of the Nonlinear Wave Equation

where u and 4~ denote scalar functions on R 2, and g is a scalar function on R. We shall assume g and ff continuous, and we shall determine sufficient conditions in the form of inequalities to guarantee that problem (1) has continuous solutions. Thus, the solutions which we consider will be weak solutions in the sense of distributions. In terms of functional analysis and the alternative approach [cf. 3, 6], problem (1) is difficult since the kernel of the underlying operator ~ u = utt U r r , with u doubly T-periodic, is infinite dimensional. CESARt [1], HALE [5], and more recentlY CESARI and KANNAN [4] have considered problem (1) and the analogous problem