Engel condition of composition of generalized derivations in prime rings

Let [Formula: see text] be a prime ring, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] the extended centroid of [Formula: see text], where [Formula: see text] is the Utumi quotient ring of [Formula: see text]. Suppose that [Formula: see text] are two nonzero generalized derivations of [Formula: see text]. Recently, [Dhara et al., Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48(1) (2020) 154–167] studied the case [Formula: see text] for all [Formula: see text], where [Formula: see text] is a derivation (which is independent of [Formula: see text]) and [Formula: see text] fixed positive integers and obtained the structure of [Formula: see text] and the ring [Formula: see text]. [Liu, An Engel condition with two generalized derivations on Lie ideals of prime rings, J. Algebra and Its Applications, 22(5) (2023) 2350115] studied a similar identity replacing derivation [Formula: see text] with another generalized derivation [Formula: see text]. Recently, [Pandey, Pair of generalized derivations on Lie ideals in prime rings, Asian-European J. Math. 16(1) (2022) 2350002] studied an identity considering composition of mapping, that is [Formula: see text] for all [Formula: see text]. In this paper, we consider the situation containing composition of mappings as well as Engel condition, that is, [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed integers and then obtain the nature of mappings [Formula: see text], [Formula: see text] and the ring [Formula: see text].