Heriberto Román-Flores

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).

A Chebyshev type inequality for fuzzy integrals

Abstract In this paper, we prove a Chebyshev type inequality for fuzzy integrals. More precisely, we show that: ⨍ 0 1 fg d μ ⩾ ⨍ 0 1 f d μ ⨍ 0 1 g d μ , where μ is the Lebesgue measure on R and f , g : [ 0 , 1 ] → [ 0 , ∞ ) are two continuous and strictly monotone functions, both increasing or both decreasing. Also, some examples and applications are presented.

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.

A note on transitivity in set-valued discrete systems

Abstract Let (X,d) be a metric space and f:X→X is a continuous function. If we consider the space ( K (X),H) of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d and f : K (X)→ K (X) , f (A)={f(a)/a∈A} , then the aim of this work is to show that f transitive implies f transitive. Also, we give an example showing that f transitive does not implies f transitive.

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.

Embedding of level-continuous fuzzy sets on Banach spaces☆

In this paper we present an extension of the Minkowski embedding theorem, showing the existence of an isometric embedding between the class Fc(X) of compact-convex and level-continuous fuzzy sets on a real separable Banach space X and C([0,1]×B(X∗)), the Banach space of real continuous functions defined on the cartesian product between [0,1] and the unit ball B(X∗) in the dual space X∗. Also, by using this embedding, we give some applications to the characterization of relatively compact subsets of Fc(X). In particular, an Ascoli–Arzela type theorem is proved and applied to solving the Cauchy problem x(t)=f(t,x(t)), x(t0)=x0 on Fc(X).

A note on Zadeh's extensions☆

Let f:Rn→Rn and f:F(Rn)→F(R)n be Zadehs extension of f to the space of fuzzy compact sets F(Rn). The aim of this paper is to show that if f is continuous, then f:(F(Rn),D)→(F(Rn),D) is also continuous, D being the supremum over Hausdorff distances between their corresponding level sets.

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.