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"established successively" is perfectly correct and usable in written English
It is used to describe something that has been brought about over a period of time, step-by-step. For example, "After many years of hard work, the business had been established successively."
Exact(3)
The studios Godfrey established successively in the 1960s and 1970s became focal points for wannabe creatives, including the young Terry Gilliam, who was refused work on the grounds that he was too good.
Since 2014, some reduced-order FE extrapolating methods based on the POD method for partial differential equations have been established successively by Luo's team (see, e.g., [24 26]) in order to avert the valueless repeated computations.
Since 2014, some reduced-order extrapolating finite difference (FD) schemes based on the POD technique for PDEs have been established successively by Luo's team (see, e.g., [29 32]) in order to avert the valueless repeated computations.
Similar(57)
Antony successively established his headquarters at Ephesus (Selçuk), Athens, and Patras (Pátrai) and marshalled his principal fleet in the gulf of Ambracia (northwestern Greece).
Upon hyphopodium formation, this structure is established by each successively infected cell and guides fungal hyphae towards the inner cortex [ 21].
In 1125, they defeated the Liao, and continued on to defeat the Northern Song, successively establishing capitals in modern-day Beijing and Kaifeng.
Two offset surfaces of the design surface, the scallop surface and the tool center surface, are employed to successively establish scallop curves on the scallop surface and cutter location tool paths for the design surface.
Between 1865 and 1870, the ratification of the Thirteenth, Fourteenth, and Fifteenth Amendments successively abolished slavery, established black citizenship and equal protection of laws, and extended the vote to African-American men, resulting in the election of more than six hundred black local, state, and federal officeholders, many of whom were themselves former slaves.
Because cigarette smoke contains halogenated and polycyclic aromatic hydrocarbons (Lofroth and Zebuhr 1992; Muto and Takizawa 1989), we examined whether air pollution caused by cigarette smoke can be detected and monitored directly and successively using the established mice.
The uptick rule (rule 10a-1) went into effect in 1938 as a means of "preventing short-sellers from accelerating a declining market by exhausting all remaining bids at one price level, causing successively lower prices to be established by long-sellers".
Successively, Sehgal and Bharucha-Reid [5, 6] established fixed point theorems in probabilistic metric spaces (for short, PM-spaces).
More suggestions(15)
established thereafter
established separately
established respectively
established sequentially
established gradually
established repeatedly
established afterwards
established straight
tested successively
established successfully
established consecutively
established progressively
drew successively
established after
established consecutive
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com