Due to their numerous applications such as in decision making, information fusion, game theory , and data mining , Choquet integrals have recently attracted much attention. In this study, two generalization types of Choquet integrals are presented. First, a generalized Choquet type integral of a single-valued function is introduced with respect to a set-function and measure. Several of its properties, such as convergence theorems and Jensen's inequality, are proved. Second, in the spirit of the single-valued Choquet integral, a generalized Choquet type set-valued integral for a single-valued function with respect to a set-multifunction and measure is introduced using Aumann integrals as well as various properties, including convergence theorems.

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is restricted because of the special operations used in the construction of these integrals. Therefore, we provide a concept of integrals generalizing both the Choquet and the Sugeno case. For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated, which can be defined on arbitrary measurable spaces and for arbitrary monotone measures. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced, and the restriction to the unit interval, i.e., to fuzzy integrals, is considered.

A general fixed point theorem for a multi-valued probabilistic q-contraction f : S → 2S is proved, where (S, ℱ, T) is a complete Menger space and ℱ satisfies a growth condition which is connected with the countable extension of a t-norm T. As a corollary a generalization of Tardiff's result27 is obtained. A random fixed point result is proved, where a measure space related to decomposable measure is used.

Using the g-calculus, introduced earlier by E. Pap, the g-operator field ℱgis introduced analogously to the classical field of Mikusiňski operators ℱ. The obtained results enable the solving of a certain nonlinear PDE of the Burgers type in a wider framework of the generalized functions, which is a generalization of the earlier obtained result.

We shall consider non-associative and non-commutative pseudo-addition and pseudo-multiplication, i.e., generalized pseudo-operations. More precise, we shall consider special class of generalized pseudo-operations that have the following form: x⊕ y=k-1(ε k(x)+k(y)), x⊙ y=k-1(k(x)ε k(y)), where ε is arbitrary fixed positive real number and k is a positive strictly monotone function. Using previous pseudo-operations, corresponding pseudo-measure and pseudo-integral will be introduced. Pseudo-convolution based on pseudo-measure and pseudo-integral will be constructed.

To cover almost all known Lebesgue type integrals constructed by means of pseudo-operations, the (S, U)-integral based on a t-conorm S, a uninorm (or t-norm) U and an S-measure (decomposable measure) is introduced and its properties are studied. Also its relationship to aggregation operators is discussed.

Operation with functions known as pseudo-convolution and its generalization as well as theirs basic properties has been presented. Then, it has been proved that pseudo-convolution which core is pseudo-integral based on max or min decomposable measure can be obtained as limit of g-convolutions, i.e., pseudo-convolutions with pseudo-integrals based on ⊕-decomposable measures where ⊕ is generated pseudo-addition, as their cores.

When using a t-norm for combining fuzzy sets, no compensation between small and large degrees of membership takes place. On the other hand, a t-conorm provides full compensation. Since many real situations do not fall into either one category, so-called compensatory operators have been proposed in the literature [H.-J. Zimmermann and P. Zysno, Fuzzy Sets and Systems4 (1980) 37–51] which are non-associative in nature. In this paper, associative compensatory operators (whose domain is the unit square with the exception of the two points (0, 1) and (1, 0) and whose only associative extensions to the whole unit square are the aggregative operators suggested in [J. Dombi, Europ. J. Oper. Res.10 (1982) 282–293]) are studied and their representation in terms of multiplicative generators is given. It is shown that these operators are constructed with the help of strict t-norms and t-conorms, in a way which is similar to ordinal sums. Finally, the duals of such operators are shown to be again associative compensatory operators, and a characterization of self-dual operators is given.