On differential equations with interactive fuzzy parameter via t-norms

Abstract

In this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers. The interactivity is formalized with the help of upper semicontinuous t-norms. To obtain solutions to fuzzy differential equations we use two different approaches. The first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the Zadeh extension principle. We show that the solutions obtained by the two methods are equal. Interactivity among the parameters and variable of the problem are studies with respect two methods. We show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t-norm (the minimum, the t-norm of the product, the t-norm of Lukasiewicz and t-norm of the drastic product) to model the interaction between parameters. Finally, to illustrate the concepts introduced in the paper, we study the Malthusian model with interactive parameters.

Introduction

Our goal is to study the initial value problem (IVP)

$$\{\begin{array}{ccc}\hfill x^{\prime }(t)& \hfill =\hfill & f(x(t),w)\hfill \\ \hfill x(t_0)& \hfill =\hfill & x_0,\hfill \end{array}$$

in which w is a parameter of the equation and $x_0$ the initial condition. This is a problem already well studied, if $x_0$ and w are real numbers. However, if these parameters are uncertain, there are at least two theoretical frameworks used: stochastic theory [1], [2], [3] and fuzzy sets theory [4], [5]. Here we will deal with the second case, which is designed in the literature as Environmental Fuzziness (See [6], Chapter 10), as an example of “Environmental Stochasticity” studied in Turelli [3].

We can associate Problem (1) to the initial value Problem (2) in which the parameter becomes part of the initial condition.

$$\{\begin{array}{ccc}\hfill (x^{\prime }(t),w^{\prime }(t))& \hfill =\hfill & (f(x(t),w),0)\hfill \\ \hfill (x(t_0),w(t_0))& \hfill =\hfill & (x_0,w).\hfill \end{array}$$

This study focuses on parameters $x_0$ and w in the Equation (2) that are interactive fuzzy sets [7], [8], [9] and studied from two viewpoints: fuzzy differential inclusions [10], [11], [12], [13] and also by fuzzification of the deterministic solution of Problem [4], [5]. As we will see in Sections 3 and 4, the fuzzy solutions of (2) are not obtained from the derivative of fuzzy functions, like is found, for example, in [6], [14], [15], [16], [17], [18], [19], [20]. In fact, the fuzzy functions treated in this article, Section 3, are a fuzzy bunch of functions [21].