BOUNDED SOLUTIONS FOR FUZZY INTEGRAL EQUATIONS

We study conditions under which the solutions of a fuzzy integral equation are bounded.


Introduction.
The concept of set-valued functions and their calculus [2] were found useful in some problems in economics [3], as well as in control theory [17]. Later on, the notion of H-differentiability was introduced by Puri and Ralesku in order to extend the differential of set-valued functions to that of fuzzy functions [29]. This in turn led Seikkala [30] to introduce the notion of fuzzy derivative, which is a generalization of the Hukuhara derivative and the fuzzy integral, which is the same as that proposed by Dubois and Prade [7,8]. A natural consequence of the above was the study of fuzzy differential and integral equations, see [5,9,10,11,12,18,19,20,21,23,24,25,26,28,29,30,31,32,33].
Fixed point theorems for fuzzy mappings, an important tool for showing existence and uniqueness of solutions to fuzzy differential and integral equations, have recently been proved by various authors, see [1,4,13,14,15,16,22,27]. In particular, in [22] Lakshmikantham and Vatsala proved the existence of fixed points to fuzzy mappings, using theory of fuzzy differential equations. Finally, stability criteria for the solutions of fuzzy differential systems are given in [21].
In this paper, we examine conditions under which all the solutions of the fuzzy integral equation and the special case are bounded. These fuzzy integral equations are proved useful when studying observability of fuzzy dynamical control systems, see [6].

Preliminaries.
By P k (R n ), we denote the family of all nonempty compact convex subsets of R n .
For A, B ∈ P k (R n ), the Hausdorff metric is defined by (2.1) A fuzzy set in R n is a function with domain R n and values in [0, 1], that is an element of [0, 1] R n (see [35,34]).
Let u ∈ [0, 1] R n , the a-level set is By E n , we denote the family of all fuzzy sets u ∈ [0, 1] R n (see [18,29,35]), for which: where d is the Hausdorff metric for nonempty compact convex subsets of R n (see [18]).

Main results
Notation 3.1. By0 ∈ E n , we denote the fuzzy set for which0 (3.1) , then all solutions of the fuzzy integral equation
and thus e at 1 r < A+ Br < r , (3.11) which is a contradiction. Thus, x(t) is bounded.
Remark 3.5. Now, since the initial value problem where f : T × E n → E n is continuous, it is equivalent to the integral equation If for the map f : T × E n → E n the conditions of Theorem 3.3 or 3.4 hold true, then all the solutions of the initial value problem (3.12) are bounded.

Conclusion.
In this paper, using a Gronwall type inequality, we give conditions under which the fuzzy integral equations (3.2) and (3.7) possess only bounded solutions. Consequently, this implies that the Cauchy problem (3.12) possesses only bounded solutions as well. It appears that, these fuzzy equations are useful when one studies the observability of fuzzy dynamical control systems. We also think that, our results can be of use in studying stability of fuzzy differential equations and fuzzy differential systems.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: