A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bologna, 1921 & 1923) --- I have not found an English translation, but an English summary by Tonelli is also in an AMS bulletin. It is noteworthy that Tonelli himself, in this summary but also in other writings I have found from him on the internet, does not mention Hilbert at all. Apparently he considered his contributions to be quite independent.
Carlo Beenakker
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