TRACING THE LEGACY AND EXTENSION OF NAGUMO’S THEOREM TO FRACTIONAL CALCULUS

Abstract This article delves into the significant trace of Nagumo’s theorem, tracing its inception in the mid-twentieth century and its subsequent evolution into Fractional Calculus. Mitio Nagumo (1905–1995), a prominent Japanese mathematician, is renowned for his extensive contributions to various mathematical domains, particularly for formulating this theorem. A special attention is given to the generalization of Nagumo’s theorem within the framework of Fractional Calculus, demonstrating its contemporary significance and applicability. Moreover, the article extends Nagumo’s uniqueness theorem for a class of fractional initial value problems with Hilfer derivative. The results are constructed by generalizing the classical mean value theorem of differential calculus, using the Hilfer derivative instead of the classical first-order derivative.