Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 290845, 4 pages

doi:10.1155/2012/290845

Research Article

A Note on -Convex Fuzzy Processes

Marian Matłoka

Department of Applied Mathematics, Poznań University of Economics, Al. Niepodległości 10, 61-875 Poznań, Poland

Received 18 June 2012; Accepted 2 September 2012

Academic Editor: Salvatore Sessa

Copyright © 2012 Marian Matłoka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We first define the concept of -convex fuzzy processes. Second, we present some basic properties of such processes.

1. Introduction

In recent years, many generalizations of convexity have appeared in the literature aiming at applications to duality theory and optimality conditions. In 1997, Pini and Singh [1] introduced -convex functions and studied some of their properties. They showed that some of the well-known classes of generalized convex functions (e.g., B-vex functions [2], geodesic convex functions [3], and invex functions [4]) form subclasses of the class of -convex functions. In 1999, Youness [5] showed that many results for convex sets and convex functions actually hold for a wider class of sets and functions, called -convex sets and -convex functions.

Convex processes were studied first by Rockafellar [6] who was interested in extending properties of linear transformations to a large class of maps preserving convexity and which arise naturally in economic theory.

The extension of this notion to the fuzzy framework was done by Matłoka [7, 8] and was investigated by Syau et al. [9] and by Chalco-Cano et al. [10].

In this work, we extend the notion of -convex set, -convex fuzzy set, and -convex fuzzy process. First, we will present the preliminary definitions and next the main properties of -convex fuzzy process.

2. Preliminaries

Let denote a subset of the -dimensional Euclidean space . Assume that is a map satisfying the following assumption:

(i),(ii), , , .

For any subsets , of let us define

Definition 1 (see [1]). A set is -convex if for all , .

Remark 2. The intersection of -convex sets is still -convex.

Remark 3. Let be a convex subset of , and . Then a convex set is -convex.

Remark 4. If, is a preinvex set with respect to (see [11]), , then is a -convex set.

Assume that .

Definition 5 (see [5]). A set is said to be -convex if , for each and .

Definition 6. A set is said to be -convex if , for each and .

Remark 7. If and is the identity map then a convex set is -convex.

Definition 8. Let be a -convex set. A function is said to be -quasiconcave on if for any and

Let denote a fuzzy set in .

Definition 9. A fuzzy set is called -convex if and only if

for all .

Definition 10. An -cut of a fuzzy set is defined as follows:

Proposition 11. If is -convex fuzzy set then is -convex (crisp) set.

Proof. We have to prove that if then for any , . So, taking into account the above definitions, we observe that if then and and . So, . This means that .

3. Main Results

In this section, we present the definition and some properties of the -convex fuzzy processes.

Let denote -convex set and the set of all non-void fuzzy sets in .

Definition 12. A mapping from to is called -convex fuzzy process if and only if for any and and

Example 13. Let , and let , , , and

Consider defined by

where , , and denotes the characteristics function of .

Then a mapping is -convex fuzzy process.

Now, let us consider defined by

for all and

for , where , .

The above mapping is -convex fuzzy mapping too.

Theorem 14. If is a -convex fuzzy process from to and is an identity mapping then for any is a -convex fuzzy set in .

Proof. We will prove that if , then for any

Let us note that for any .

So, using the definition of -convex fuzzy processes, we have

Definition 15. The graph of a -convex fuzzy process from to , denoted , is a fuzzy set in such that for any

In the analogous way as in the Definition 9, we can define the -convex fuzzy subset of , that is,

for , , .

Theorem 16. The graph of a -convex fuzzy process from to is a -convex fuzzy subset of .

Proof. Taking into account the definitions of the graph and -convex fuzzy process, we observe that for any and

Definition 17. A composition of -convex fuzzy process from to and -convex fuzzy process from to is mapping from to such that

for any .

Theorem 18. If is -convex fuzzy process from to and is -convex fuzzy process from to , then is a -convex fuzzy process from to .

Proof. Let and . Then for any we have

So, is a -convex fuzzy process.

For any set , we put

for any .

Theorem 19. If is -convex fuzzy process from to and is a -convex subset of and is an identity mapping then is -convex subset of .

Proof. Let and . Then we have

So, is a -convex fuzzy subset of .

Theorem 20. If is -convex fuzzy process from to then for any

for any , .

Proof. According to the definition of -cut, we have

Moreover,

This means that if then , that is,

Now, for any and any -quasiconcave function , let us define a function

Theorem 21. If is a -convex fuzzy process from to , then the function is -quasiconcave.

Proof. Let , . Then we have

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