Fréchet and Gateaux gH-differentiability for interval valued functions of multiple variables

Abstract

In this work we extend concepts of differentiability for Interval-Valued functions (IV-functions) of multiple variables, based on generalized Hukuhara gH-difference. In this context, we introduce a new concept of gH-linearity and characterize the class of gH-linear IV-functions, which are fundamental for a general approach to Fréchet-type and Gateaux-type gH-differentiability of the first order. We moreover consider vector IV-functions and outline the definition of the gH-Jacobian; by representing intervals and IV-functions in midpoint-radius notation, we establish properties and relations between pointwise, Fréchet and Gateaux gH-differentiability. Finally, higher-order differentiability and gH-Hessian matrix are considered. These intuitive concepts are mathematically and computationally easy to work with.