Fre´chet differentiation of nonlinear operators between fuzzy normed spaces

By the rapid advances in linear theory of fuzzy normed spaces and fuzzy bounded linear operators it is natural idea to set and improve its nonlinear peer. We aimed in this work to realize this idea by introducing fuzzy Fre´chet derivative based on the fuzzy norm definition in Bag and Samanta [Bag T, Samanta SK. Finite dimensional fuzzy normed linear spaces. J Fuzzy Math 2003;11(3):687–705]. The definition is divided into two part as strong and weak fuzzy Fre´chet derivative so that it is compatible with strong and weak fuzzy continuity of operators. Also we restate fuzzy compact operator definition of Lael and Nouroizi [Lael F, Nouroizi K. Fuzzy compact linear operators. Chaos, Solitons & Fractals 2007;34(5):1584–89] as strongly and weakly fuzzy compact by taking into account the compatibility. We prove also that weak Fre´chet derivative of a nonlinear weakly fuzzy compact operator is also weakly fuzzy compact.