The fuzzy integral for monotone functions

Abstract

In this paper, we give some optimal upper bounds for the Sugeno’s integral of monotone functions. More precisely, we show that: If g : [0, ∞) → [0, ∞) is a continuous and strictly monotone function, then the fuzzy integral value p = ⨍ 0 a g d μ , with respect to the Lebesgue measure μ, verifies the following sharp inequalities: ( a ) g ( a - p ) ⩾ p for the increasing case, and ( b ) g ( p ) ⩾ p for the decreasing case. Moreover, we show that under adequate conditions, these optimal inequalities provides a powerful tool for solving fuzzy integrals. Also, some examples and application are presented.