Existence of Periodic Solutions of Semilinear Parabolic Equations and the Method of Upper and Lower Solutions
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The existence of periodic solutions of semilinear parabolic equations has been investigated by several authors [1-10,15-18] by different methods such as the method of Poincare operator and the theory of monotone operators. In a recent paper  Amann also obtains multiplicity results. Recently, an attempt was made successfully in [11,12,16] to combine the two basic techniques, namely the method of upper and lower solutions and the method of Lyapunov-Schmidt to investigate existence of periodic solutions of first and second order equations. In this paper we continue this fruitful approach to study existence of periodic solutions of semilinear parabolic equations with homogeneous Neumann boundary conditions. The results obtained are exhaustive in the sense that they not only contain the known results, but also deal with other possible situations. Although our approach is extendable to general case, we have considered a simple parabolic equation in one space variable so as to bring out the ideas involved clearly.