Nonlinear Oscillations, Vol. 9, No. 2, 2006 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH W λ 0 -PSEUDOMONOTONE MAPS P.O. Kas’yanov, 1 V.S. Mel’nik, 2 and S. Toscano 3 UDC 517.9 We consider differential-operator equations with W λ 0 -pseudomonotone operators. The problem of study- ing periodic solutions by the Faedo–Galerkin method is considered. Important a priori estimates are ob- tained. A topological description of resolvent operators is given. 1. Introduction One of the most efficient approaches to the investigation of nonlinear problems defined by partial differential equations with boundary values consists of their transformation into equations in Banach spaces governed by nonlinear operators. In order to study these equations, modern methods of nonlinear analysis have been used [1–3]. In [4], by using a special basis, the Cauchy problem for a class of equations with operators of Volterra type was studied. An important periodic problem for equations with monotone differential operators of Volterra type was studied in [1]. Periodic solutions for pseudomonotone operators were considered in [2]. In the present paper, we introduce a new construction of bases to prove the existence of periodic solutions of differential-operator equations by using the Faedo–Galerkin method for W λ 0 -pseudomonotone operators. From the viewpoint of applications, we substantially extend the class of operators considered by other authors (see [4–6]). 2. Statement of the Problem Let (V 1 , ‖·‖ V 1 ) and (V 2 , ‖·‖ V 2 ) be some reflexive separable Banach spaces continuously imbedded in a Hilbert space (H, (·, ·)) such that V := V 1 ∩ V 2 is dense in the spaces V 1 ,V 2 , and H. (2.1) After the identification H ∼ = H ∗ , we get V 1 ⊂ H ⊂ V ∗ 1 , V 2 ⊂ H ⊂ V ∗ 2 (2.2) with continuous and dense imbeddings [1]; here, (V ∗ i , ‖·‖ V i ) is the space topologically conjugate to V i with respect to the canonical bilinear form 〈·, ·〉 V i : V ∗ i × V i → R, i =1, 2, which coincides on H with the inner product (·, ·) on H. Let us consider the functional spaces X i = L r i (S ; H ) ∩ L p i (S ; V i ), where S = [0,T ], 0 <T < +∞, and 1 <p i ≤ r i < +∞, i =1, 2. The spaces X i are Banach spaces with the norms ‖y‖ X i = ‖y‖ Lp i (S;V i ) + ‖y‖ Lr i (S;H) . Moreover, X i is a reflexive space. 1 Shevchenko Kiev University, Kiev. 2 Institute of Applied Systems Analysis, Ukrainian Academy of Sciences, Kiev. 3 University of Salerno, Fisciano (Salerno), Italy. Published in Neliniini Kolyvannya, Vol. 9, No. 2, pp. 187–212, April–June, 2006. Original article submitted March 31, 2006. 1536–0059/06/0902–0181 c  2006 Springer Science+Business Media, Inc. 181