Tatjana Grbić

Pseudo-Riemann-Stieltjes integral

An extension of the classical Riemann-Stieltjes integral to the field of pseudo-analysis is being investigated through this paper. Core of the construction presented here consists of generalized pseudo-operations given by monotone generating function.

Inequalities of Jensen and Chebyshev Type for Interval-Valued Measures Based on Pseudo-integrals

Since interval-valued measures have applications in number of practical areas, this paper is focused on two approaches to this problem as well as on the corresponding generalizations of the Jensen and the Chebyshev integral inequalities. The first approach is based on an interval-valued measure defined by the pseudo-integral of interval-valued function, while the second approach considers an interval-valued measure obtained through pseudo-integrals of real-valued functions.

Pseudo-integral of interval-valued functions and some limit properties

Pseudo-integral of the interval-valued functions based on the pseudo-operations has been introduced. Some basic properties, as well as, some limit properties of the pseudo-integral of interval-valued functions have been investigated.

A note on absolute continuity for the interval-valued measures based on pseudo-integral of interval-valued function

Interval-valued functions and integration of interval-valued functions are important mathematical tools that play a crucial role in several practical areas, e.g., mathematical economics. This paper is focused on interval-valued measures based on pseudo-integration of interval-valued function. Some important characteristics of this type of measures, such as absolute continuity of type I and type VI, are investigated.

The pseudo-linear superposition principle for nonlinear partial differential equations and representation of their solution by the pseudo-integral

In this paper, for a pseudo-linear partial differential equation of the nth order the pseudo-superposition principle is proved, which means that a pseudo-linear combination of solutions of the equation is again a solution of this equation. Especially, this principle is proved for some equations of the second order imposing weaker conditions. In addition, theorems giving the representation of the solution of Cauchy problem by pseudo-integral are obtained.

Interval-valued Chebyshev, Hölder and Minkowski inequalities based on g-integrals

A natural generalization of (classical) measures are monotone set valued functions, the so called non-additive measures. Further generalization of measures are interval-valued measures and interval-valued non-additive measures. Since interval-valued ⊕-measures, as a special case of interval-valued non-additive measures, have been extensively applied in the mathematical representation of the various aspects of uncertainty, the present paper offers a generalization of Chebyshev, Hölder and Minkowski types inequalities obtained by g-integrals for non-negative real-valued functions with respect to an interval-valued ⊕-measures.

Inequalities of Hölder and Minkowski type for pseudo-integrals with respect to interval-valued ⊕-measures

Abstract In the present paper, the Holder and Minkowski type of inequality for the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure is proven. Two cases of semirings are considered. In the first case, pseudo-operations (pseudo-addition and pseudo-multiplication) are set by a strictly monotone continuous function. In the second case, the pseudo-addition is the idempotent operation sup, and pseudo-multiplication is specified by a strictly monotone continuous function, as in the first case. Trivial and nontrivial examples of interval-valued pseudo-additive measures are provided, as well as Holder and Minkowski type of inequalities with respect to those measures.

Weak convergence of sequences of distorted probabilities

In this paper, the portmanteau theorem which provides the equivalent conditions of the weak convergence of a sequence of probability measures is extended on the sequence of distorted probabilities.

Chebyshev type inequalities for pseudo-integrals of set-valued functions

Chebyshev type inequalities for pseudo-integrals of set-valued functions for two classes of semirings are shown. One of them is the case when pseudo-integral of set-valued function is based on a g-semiring, where g is an increasing and continuous function, and the other case is when semiring is ([a, b], max, ⨀), where ⨀ is generated by an increasing and continuous generator g.

Large deviation principle with generated pseudo measures

Contemporary large deviation theory uses various approaches. One of possible recent approaches is based on the idempotent sup-measure and related integrals based on operations sup and the usual product. In this paper, the large deviation convergence of a sequence of generated pseudo-measures to sup-decomposable measure is introduced. The main result of the paper gives a necessary condition for introduced convergence.