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This paper is concerned with the system (NEESSVMEP) obtained by mixing a nonlinear evolutionary equation and a strong set-valued mixed equilibrium problem (SSVMEP). In our setting, the set of constraints is not necessarily compact and the problem is driven by a not necessarily monotone set-valued mapping. We show that the solution set for (SSVMEP) is nonempty, closed, convex and bounded. We then establish the upper semicontinuity and the measurability properties of the involved functions in the nonlinear evolutionary equation. Utilizing these results, we prove that the solution set for (NEESSVMEP) is nonempty and compact. [ABSTRACT FROM AUTHOR]