Michał Boczek

On Chebyshev type inequalities for generalized Sugeno integrals

Abstract We give the necessary and sufficient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions. For several choices of operators, we characterize the classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satisfied.

On the Jensen type inequality for generalized Sugeno integral

We prove necessary and sufficient conditions for the validity of Jensen type inequalities for generalized Sugeno integral. Our proofs make no appeal to the continuity of neither the fuzzy measure nor the operators. For several choices of operators, we characterize the classes of functions for which the corresponding inequalities are satisfied.

On S-homogeneity property of seminormed fuzzy integral

We give an?answer to Problem 9.3 stated by Mesiar and Stupňanova 8. We show that the class of semicopulas solving this problem contains any associative semicopula S such that for each a ? 0, 1 the function x ? S(a, x) is continuous and increasing on a countable number of intervals.

On conditions under which some generalized Sugeno integrals coincide: A solution to Dubois' problem

Abstract In the paper, we characterize the class of all pairs of fuzzy implications and dual fuzzy implications such that the q-integral and q-cointegral coincide. This gives a solution to the problem posed by Didier Dubois during the conference FSTA 2016, the 13th International Conference on Fuzzy Set Theory and Applications. As a corollary, we obtain the necessary and sufficient conditions under which the seminormed fuzzy integral and the semiconormed fuzzy integral are equal for all measurable functions.

On comonotone commuting and weak subadditivity properties of seminormed fuzzy integrals

Abstract In the paper, we characterize the class of all the binary operators ∘-commuting with the generalized Sugeno integral generated by a strict t -norm. For the Sugeno integral, this problem was solved by Ouyang and Mesiar. We also present the necessary and sufficient conditions for the weak subadditivity of the generalized Sugeno integral.

Steffensen type inequalities for fuzzy integrals

We provide several Steffensen type inequalities for the Sugeno integral. The inequalities are of the form - ? a b f g d µ ≤ A - ? a a + λ f d µ + B or - ? a b f g d µ ≤ A - ? b - λ b f d µ + B , where A, B are constants, µ is a fuzzy measure on R , g: a, b] ? 0, 1], f : a , b ] ? R + is nonincreasing or nondecreasing and λ = ( b - a ) ? - ? a b g d µ . We show that some of our sufficient conditions are also necessary.