Yurilev Chalco-Cano

Fuzzy differential equations and the extension principle

We study the Cauchy problem for differential equations, considering its parameters and/or initial conditions given by fuzzy sets. These fuzzy differential equations are approached in two different ways: (a) by using a family of differential inclusions; and (b) the Zadeh extension principle for the solution of the model. We conclude that the solutions of the Cauchy problem obtained by both are the same. We also provide some illustrative examples.

Comparation between some approaches to solve fuzzy differential equations

In this paper, we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via Zadehs extension principle. We get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented. We also give some properties and we show the relationships with others interpretation. Finally, we propose a numerical procedure for obtaining the fuzzy solution.

A Jensen type inequality for fuzzy integrals

In this paper, we show a Jensen type inequality for the Sugeno integral. We also discuss some conditions assuring the satisfaction of opposite inequality (reverse Jensen inequality).

Generalized derivative and π-derivative for set-valued functions

In this paper we study the generalized derivative and the @p-derivative for interval-valued functions. We show the connections between these derivatives. Some illustrative examples and applications to interval differential equations and fuzzy functions are presented.

Calculus for interval-valued functions using generalized Hukuhara derivative and applications

This paper is devoted to studying differential calculus for interval-valued functions by using the generalized Hukuhara differentiability, which is the most general concept of differentiability for interval-valued functions. Conditions, examples and counterexamples for limit, continuity, integrability and differentiability are given. Special emphasis is set to the class F(t)=C.g(t), where C is an interval and g is a real function of a real variable. Here, the emphasis is placed on the fact that F and g do not necessarily share their properties, underlying the extra care that must be taken into account when dealing with interval-valued functions. Two applications of the obtained results are presented. The first one determines a Delta method for interval valued random elements. In the second application a new procedure to obtain solutions to an interval differential equation is introduced. Our results are relevant to fuzzy set theory because the usual fuzzy arithmetic, extension functions and (mathematical) analysis are done on @a-cuts, which are intervals.

Goodness-of-fit tests based on empirical characteristic functions

A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.

SUGENO INTEGRAL AND GEOMETRIC INEQUALITIES

In this work, we prove a Prekopa-Leindler type inequality for the Sugeno integral. More precisely, if 0 < λ < 1 and h((1-λ)x+λy) ≥ f(x)1-λg(y)λ, ∀ x,y ∈ ℝn, where h, f and g are nonnegative μ-measurable functions on ℝn, then , for any concave fuzzy measure μ. Also, we derive a general Brunn-Minkowski inequality (standard form) for any homogeneous quasiconcave fuzzy measure μ on ℝn.

H-continuity of fuzzy measures and set defuzzification

The purpose of this work is to analyze the continuity of fuzzy measures with respect to the Hausdorff metric on K(R^n), the class of all nonempty compact subsets of R^n. As an application, we study the continuity of set defuzzification processes on R^n which are defined by Aumann integral methods.

A convolution type inequality for fuzzy integrals

In this paper we show a Bushell–Okrasinski type inequality for fuzzy integrals. More precisely, we show that: s⨍01(1-t)s-1g(t)sdt⩾⨍01g(t)dts, where g:[0,1]→[0,∞) is a continuous and strictly decreasing function and s ⩾ 2.

Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative

This paper addresses the optimization problems with interval-valued objective function. For this we consider two types of order relation on the interval space. For each order relation, we obtain KKT conditions using of the concept of generalized Hukuhara derivative (