Periodic solutions of some vector retarded functional differential equations

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has no nontrivial T-periodic solution (see the books of Krasnosel’skii [6], Reissig, Sansone, and Conti [14], and Roseau [15]). This result has been extended to the case of the vector retarded functional differential equation by Fennel1 [2]. When the requirement upon the linear part is not satisfied, supplementary conditions upon f are needed to ensure the existence of T-periodic solutions, as follows immediately from the Fredholm alternative [4] for the special case off independent of x. Such auxiliary conditions appear in papers of Lazer [7] (for a second order scalar differential equation), Ezeilo [l], Sedsiwy [16], Villari [19], and the author [8] (f or a third order scalar differential equation), Sedsiwy [17] and Reissig [ll, 121 (for the n-th order scalar case) and the same authors [18, 131 f or the corresponding vector case. Also, Fennel1 [2] has extended Lazer’s result and method to the retarded case. Those results are proved by various methods (Brouwer, Schauder, or Leray-Schauder fixed point theorems) which make the arguments rather tedious. On the other hand, all the equations considered in those papers are