The relaxation of monotonicity requirements is a trend in the theory of aggregation functions. In the recent literature, we can find several relaxed forms of monotonicity, such as weak, directional, ...cone, ordered directional and strengthened directional monotonicity. All these forms of monotonicity are global properties in the sense that they are imposed for all the points in the domain of a function. In this work, we introduce a local notion of monotonicity called pointwise directional monotonicity, or directional monotonicity at a point. Based on this concept, we characterize all the previously defined notions of monotonicity and, in the final part of the paper, we present some geometric aspects of the global weaker forms of monotonicity, stressing their relations and singularities.
The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties ...and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on
R
+
and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts in contrast to Köthe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950,
2000
). The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.
Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces
L
Φ
(
μ
)
...endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces
E
Φ
(
μ
)
with the F-norm induced from
L
Φ
(
μ
)
. The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.
We consider mixture functions, which are a type of weighted averages for which the corresponding weights are calculated by means of appropriate continuous functions of their inputs. In general, these ...mixture function need not be monotone increasing. For this reason we study sufficient conditions to ensure standard, weak and directional monotonicity for specific types of weighting functions. We also analyze directional monotonicity when differentiability is assumed.
In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional ...monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Łukasiewicz implication.
The uniqueness obtained in 1 may not right if f(s)/sk is only decreasing for s>0. We correct this error by requiring f(s)/sk is “strictly” decreasing for s>0.
The paper introduces a new class of functions from 0,1n to 0,n called d-Choquet integrals. These functions are a generalization of the “standard” Choquet integral obtained by replacing the difference ...in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all “standard” Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied.
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