In a recent study, Hölder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differentialequations has been formed, by the classical Ishii–Lionsʼs method. The researchers thus extended the Hölder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, this study deals with a new class of nonlocal equations that termed as mixed integro-differentialequations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local–nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

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