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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]

Let M n (F) be the algebra of n × n matrices over an arbitrary field F. We consider linear maps Φ : M n (F) → M r (F) preserving matrices annihilated by a fixed polynomial f (x) = (x − a 1) ⋯ (x − a m) with m ≥ 2 distinct zeroes a 1 , a 2 , ... , a m ∈ F ; namely, f (Φ (A)) = 0 whenever f (A) = 0. Suppose that f (0) = 0 , and the zero set Z (f) = { a 1 , ... , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r (F) , invertible diagonal matrices D 1 ∈ M p (F) and D 2 ∈ M q (F) , where s = r − n p − n q ≥ 0. The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f (x) in the sense that λ Z (f) ⊆ Z (f). In general, assume that Z (f) − a 1 is not an additive group. If Φ (I n) commutes with Φ (A) for all A ∈ M n (F) , or if f (x) has a unique zero multiplier λ = 1 , then Φ assumes the form (†). The above assertions follow from the special case when f (x) = x (x − 1) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n (F) → M r (F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n (F) = M 2 (Z 2). [ABSTRACT FROM AUTHOR]