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By using some new developments in the theory of equilibrium problems, we study the existence of anti-periodic solutions for nonlinear evolution equations associated with time-dependent pseudomonotone and quasimonotone operators in the topological sense. More precisely, we establish new existence results for mixed equilibrium problems associated with pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are therefore applied to study the existence of anti-periodic solutions for nonlinear evolution equations in the setting of reflexive 123 J Optim Theory Appl Banach spaces. This new approach leads us to improve and unify most of the recent results obtained in this direction.