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We study $\mathbb{Z}_2$-graded identities of Lie superalgebras of the type$b(t), t\ge 2$, over a field of characteristic zero. Our main result is thatthe $n$-th codimension is strictly less than $(\dim b(t))^n$ asymptotically. Asa consequence we obtain an upper bound for ordinary (non-graded) PI-exponentfor each simple Lie superalgebra $b(t), t\ge 3$.

Graded identities of some simple Lie superalgebras