Periodic solutions of a quasilinear parabolic differential equation

AbstractThis paper treats the quasilinear, parabolic boundary value problem uxx − ut = −ƒ(x, t, u)u(0, t) = ϑ1(t); u(l, t) = ϑ2(t) on an infinite strip {(x, t) ¦ 0 < x < l, −∞ < t < ∞} with the functions ƒ(x, t, u), ϑ1(t), ϑ2(t) being periodic in t. The major theorem of the paper gives sufficient conditions on ƒ(x, t, u) for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on ƒ(x, t, u) and indicate a method for determining the initial estimate at which the iteration may begin