A Note on (Φ𝐸₁,Φ𝐸₂)-Convex Fuzzy Processes

Abstract

We first define the concept of (Φ𝐸₁,Φ𝐸₂)-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 (Φ₁,Φ₂)-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 (Φ₁,Φ₂)-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 𝐸-convex sets and 𝐸-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 Φ𝐸-convex set, Φ𝐸-convex fuzzy set, and (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process. First, we will present the preliminary definitions and next the main properties of (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process.

2. Preliminaries

Let 𝑋_𝑖 denote a subset of the 𝑛_𝑖-dimensional Euclidean space 𝑅^𝑛𝑖. Assume that Φ_𝑖 is a map satisfying the following assumption:(i)Φ_𝑖∶𝑋_𝑖×𝑋_𝑖×[0,1]→𝑅^𝑛𝑖,(ii)Φ_𝑖(𝑥,𝑦,0)=𝑦, Φ₁(𝑥,𝑥,𝜆)=𝑥, ∀𝑥,𝑦∈𝑋_𝑖, 𝜆∈[0,1].

For any subsets 𝐴,𝐵, of 𝑋_𝑖 let us define Φ_𝑖(𝐴,𝐵,𝜆)=Φ_𝑖𝑥,𝑦,𝜆∶𝑥∈𝐴,𝑦∈𝐵. (1)

Definition 1 (see [1]). A set 𝑋_𝑖 is Φ_𝑖-convex if Φ_𝑖(𝑥₁,𝑥₂,𝜆)∈𝑋_𝑖 for all 𝑥₁,𝑥₂∈𝑋_𝑖, 𝜆∈[0,1].

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

Remark 3. Let 𝑋_𝑖 be a convex subset of 𝑅^𝑛𝑖, and Φ_𝑖(𝑥₁,𝑥₂,𝜆)=𝜆𝑥₁+(1−𝜆)𝑥₂. Then a convex set is Φ_𝑖-convex.

Remark 4. If𝜂∶𝑅^𝑛𝑖×𝑅^𝑛𝑖→𝑅^𝑛𝑖, 𝑋_𝑖 is a preinvex set with respect to 𝜂 (see [11]), Φ_𝑖(𝑥₁,𝑥₂,𝜆)=𝑥₂+𝜆⋅𝜂(𝑥₁,𝑥₂), then 𝑋_𝑖 is a Φ_𝑖-convex set.
Assume that 𝐸₁∶𝑅^𝑛𝑖→𝑅^𝑛𝑖.

Definition 5 (see [5]). A set 𝑋_𝑖⊂𝑅^𝑛𝑖 is said to be 𝐸_𝑖-convex if (1−𝜆)𝐸(𝑥₁)+𝜆𝐸(𝑥₂)∈𝑋_𝑖, for each 𝑥₁,𝑥₂∈𝑋_𝑖 and 𝜆∈[0,1].

Definition 6. A set 𝑋_𝑖⊂𝑅^𝑛𝑖 is said to be Φ𝐸_𝑖-convex if Φ_𝑖(𝐸_𝑖(𝑥₁),𝐸_𝑖(𝑥₂),𝜆)∈𝑋_𝑖, for each 𝑥₁,𝑥₂∈𝑋_𝑖 and 𝜆∈[0,1].

Remark 7. If Φ_𝑖(𝑥₁,𝑥₂,𝜆)=𝜆𝑥₁+(1−𝜆)𝑥₂ and 𝐸_𝑖 is the identity map then a convex set is Φ𝐸_𝑖-convex.

Definition 8. Let 𝑋_𝑖 be a Φ𝐸_𝑖-convex set. A function 𝑓∶𝑋_𝑖→𝑅 is said to be Φ𝐸_𝑖-quasiconcave on 𝑋_𝑖 if for any 𝑥₁,𝑥₂∈𝑋_𝑖 and 𝜆∈[0,1]𝑓Φ_𝑖𝐸_𝑖𝑥₁,𝐸_𝑖𝑥₂,𝜆≥min𝑓𝑥₁,𝑓𝑥₂. (2) Let 𝐶∶𝑅^𝑛𝑖→[0,1] denote a fuzzy set in 𝑅^𝑛𝑖.

Definition 9. A fuzzy set 𝐶 is called Φ𝐸_𝑖-convex if and only if 𝐶Φ_𝑖𝐸_𝑖𝑥₁,𝐸_𝑖𝑥₂,𝜆≥min𝐶𝑥₁,𝐶𝑥₂, (3) for all 𝑥₁,𝑥₂∈𝑅^𝑛𝑖.

Definition 10. An 𝛼-cut of a fuzzy set 𝐶 is defined as follows: 𝐶^𝛼={𝑥∶𝐶(𝑥)≥𝛼}for(]{(𝑥)}𝛼∈0,1𝑥∶𝐶>𝛼for𝛼=0. (4)

Proposition 11. If 𝐶is Φ𝐸_𝑖-convex fuzzy set then 𝐶^𝛼 is Φ𝐸_𝑖-convex (crisp) set.

Proof. We have to prove that if 𝑥₁,𝑥₂∈𝐶^𝛼 then for any 𝜆∈[0,1], Φ_𝑖(𝐸_𝑖(𝑥₁),𝐸_𝑖(𝑥₂),𝜆)∈𝐶^𝛼. So, taking into account the above definitions, we observe that if 𝑥₁,𝑥₂∈𝐶^𝛼 then 𝐶(𝑥₁)≥𝛼 and 𝐶(𝑥₂)≥𝛼 and min(𝐶(𝑥₁),𝐶(𝑥₂))≥𝛼. So, 𝐶(Φ_𝑖(𝐸_𝑖(𝑥₁),𝐸_𝑖(𝑥₂),𝜆))≥min(𝐶(𝑥₁),𝐶(𝑥₂))≥𝛼. This means that Φ_𝑖(𝐸_𝑖(𝑥₁),𝐸_𝑖(𝑥₂),𝜆)∈𝐶^𝛼.

3. Main Results

In this section, we present the definition and some properties of the (Φ𝐸₁,Φ𝐸₂)-convex fuzzy processes.

Let 𝑋_𝑖 denote Φ𝐸_𝑖-convex set and 𝐹(𝑋_𝑖) the set of all non-void fuzzy sets in 𝑋_𝑖(𝑖=1,2,3).

Definition 12. A mapping 𝐴 from 𝑋₁ to 𝐹(𝑋₂) is called (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process if and only if for any 𝑥₁,𝑥₂∈𝑋₁ and 𝜆∈[0,1] and 𝑦∈𝑋₂𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆𝑦≥sup_{𝑦1,𝑦2∈𝑋2∶𝑦=Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)}min𝐴𝑥₁𝑦₁,𝐴𝑥₂𝑦₂. (5)

Example 13. Let 𝑋₁=𝑅, 𝑋₂=𝑅−(−1/2,1/2) and let 𝐸₁(𝑥)=|𝑥|, 𝐸₂(𝑦)=𝑦, Φ₁(𝑥₁,𝑥₂,𝜆)=𝜆𝑥₁+(1−𝜆)𝑥₂, and Φ₂𝑦₁,𝑦₂,𝜆=𝜆𝑦₁+(1−𝜆)𝑦₂if𝑦₁𝑦₂()𝑦>01+𝜆₂−𝜆𝑦₁if𝑦₁𝑦₂<0. (6) Consider 𝐴∶𝑋₁→𝐹(𝑋₂) defined by 𝐴(𝑋)∶=𝜒_{⟨𝑓(𝑥),∞)}, (7) where 𝑓(𝑥)=𝑘|𝑥|, 𝑘>0, and 𝜒_𝐶 denotes the characteristics function of 𝐶.
Then a mapping 𝐴 is (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process.
Now, let us consider 𝐴∶𝑋₁→𝐹(𝑋₂) defined by 𝐴(𝑥)𝑦=𝑦𝑓(𝑥)if(𝑥),1𝑦∈0,𝑓if𝑓(𝑥),0𝑦∈,∞if𝑦∈<(−∞,0), (8) for all 𝑥≠0 and 𝐴(𝑥)𝑦=1if⟨),0𝑦∈0,∞if𝑦∈(−∞,0), (9) for 𝑥=0, where 𝑓(𝑥)=𝑘|𝑥|, 𝑘>0.
The above mapping is (Φ𝐸₁,Φ𝐸₂)-convex fuzzy mapping too.

Theorem 14. If 𝐴 is a (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₂) and 𝐸₁ is an identity mapping then for any 𝑥∈𝑋₁𝐴(𝑥) is a Φ𝐸₂-convex fuzzy set in 𝑋₂.

Proof. We will prove that if 𝑦₁,𝑦₂∈𝑋₂, 𝜆∈[0,1] then for any 𝑥∈𝑋₁𝐴(𝑥)Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆≥min𝐴(𝑥)𝑦₁,𝐴(𝑥)𝑦₂. (10) Let us note that for any ∈𝑋₁𝐸₁(𝑥)=𝑥=Φ₁(𝑥,𝑥,𝜆).
So, using the definition of (Φ𝐸₁,Φ𝐸₂)-convex fuzzy processes, we have 𝐴(𝑥)Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂Φ,𝜆=𝐴₁(𝑥,𝑥,𝜆)Φ₂𝐸₂𝑦,𝐸₂𝑦₂,𝜆≥sup_{𝑦′,𝑦′′∈𝑋2∶Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)=Φ2(𝐸2(𝑦′),𝐸2(𝑦′′),𝜆)}min𝐴(𝑥)𝑦^′,𝐴(𝑥)𝑦^′′≥min𝐴(𝑥)𝑦₁,𝐴(𝑥)𝑦₂. (11)

Definition 15. The graph of a (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process 𝐴 from 𝑋₁ to 𝐹(𝑋₂), denoted 𝐺_𝐴, is a fuzzy set in 𝑋₁×𝑋₂ such that for any (𝑥,𝑦)∈𝑋₁×𝑋₂𝐺_𝐴𝑥,𝑦=𝐴(𝑥)𝑦. (12)

In the analogous way as in the Definition 9, we can define the (Φ𝐸₁,Φ𝐸₂)-convex fuzzy subset 𝐵 of 𝑋₁×𝑋₂, that is, 𝐵Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆,Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂𝐵𝑥,𝜆≥min₁,𝑦₁,𝐵𝑥₂,𝑦₂, (13) for 𝑥₁,𝑥₂∈𝑋₁, 𝑦₁,𝑦₂∈𝑋₂, 𝜆∈[0,1].

Theorem 16. The graph of a (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process 𝐴 from 𝑋₁ to 𝐹(𝑋₂) is a (Φ𝐸₁,Φ𝐸₂)-convex fuzzy subset of 𝑋₁×𝑋₂.

Proof. Taking into account the definitions of the graph and (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process, we observe that for any (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑋₁×𝑋₂ and 𝜆∈[0,1]𝐺_𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆,Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂Φ,𝜆=𝐴₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆≥sup_{𝑦′,𝑦′′∈𝑋2∶Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)=Φ2(𝐸2(𝑦′),𝐸2(𝑦′′),𝜆)}min𝐴𝑥₁𝑦^′,𝐴𝑥₂𝑦^′′≥min𝐴𝑥₁𝑦₁,𝐴𝑥₂𝑦₂=min𝐺_𝐴𝑥₁,𝑦₁,𝐺_𝐴𝑥₂,𝑦₂. (14)

Definition 17. A composition of (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process 𝐴 from 𝑋₁ to 𝐹(𝑋₂) and (Φ𝐸₂,Φ𝐸₃)-convex fuzzy process 𝐵 from 𝑋₂ to 𝐹(𝑋₃) is mapping 𝐵∘𝐴 from 𝑋₁ to 𝐹(𝑋₃) such that (𝐵∘𝐴)(𝑥)(𝑧)=sup_𝑦∈𝑋2min𝐴(𝑥)𝑦,𝐵𝑦(𝑧) (15) for any (𝑥,𝑧)∈𝑋₁×𝑋₃.

Theorem 18. If 𝐴 is (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₂) and 𝐵 is (Φ𝐸₂,Φ𝐸₃)-convex fuzzy process from 𝑋₂ to 𝐹(𝑋₃), then 𝐵∘𝐴 is a (Φ𝐸₁,Φ𝐸₃)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₃).

Proof. Let 𝑥₁,𝑥₂∈𝑋₁ and 𝜆∈[0,1]. Then for any 𝑧∈𝑋₃ we have (𝐵∘𝐴)Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆(𝑧)=sup_𝑦∈𝑋2min𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆𝑦,𝐵𝑦(𝑧)=sup_{𝑦1,𝑦2∈𝑋2∶𝑦=Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)}min𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂Φ,𝜆₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝐵Φ,𝜆₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆(𝑧)≥sup_{𝑦1,𝑦2∈𝑋2∶𝑦=Φ2𝐸2𝑦1,𝐸2𝑦2,𝜆}min𝐴𝑥₁𝑦₁,𝐴𝑥₂𝑦₂,sup_{𝑧1,𝑧2∈𝑋3∶𝑧=Φ3𝐸3𝑧1,𝐸3𝑧2,𝜆}min𝐵𝑦₁𝑧₁,𝐵𝑦₂𝑧₂≥sup_{𝑧1,𝑧2∈𝑋3∶𝑧=Φ3𝐸3𝑧1,𝐸3𝑧2,𝜆}sup_{𝑦1,𝑦2∈𝑋2}min𝐴𝑥₁𝑦₁,𝐴𝑥₂𝑦₂,𝐵𝑦₁𝑧₁,𝐵𝑦₂𝑧₂=sup_{𝑧1,𝑧2∈𝑋3∶𝑧=Φ3𝐸3𝑧1,𝐸3𝑧2,𝜆}minsup_{𝑦1∈𝑋2}min𝐴𝑥₁𝑦₁,𝐵𝑦₁𝑧₁,sup_{𝑦2∈𝑋2}min𝐴𝑥₂𝑦₂,𝐵𝑦₂𝑧₂=sup_{𝑧1,𝑧2∈𝑋3∶𝑧=Φ3𝐸3𝑧1,𝐸3𝑧2,𝜆}min(𝐵∘𝐴)𝑥₁𝑧₁,(𝐵∘𝐴)𝑥₂𝑧₂. (16) So, 𝐵∘𝐴 is a (Φ𝐸₁,Φ𝐸₃)-convex fuzzy process.
For any set 𝐶⊂𝑋₁, we put 𝐴(𝐶)𝑦=sup_𝑥∈𝐶𝐴(𝑥)𝑦 (17) for any 𝑦∈𝑋₂.

Theorem 19. If 𝐴 is (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₂) and 𝐶 is a Φ𝐸₁-convex subset of 𝑋₁ and 𝐸₁ is an identity mapping then 𝐴(𝐶) is Φ𝐸₂-convex subset of 𝑋₂.

Proof. Let 𝑦₁,𝑦₂∈𝑋₂ and 𝜆∈[0,1]. Then we have 𝐴(𝐶)Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆=sup_𝑥∈𝐶𝐴(𝑥)Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆=sup_𝑥∈𝐶𝐴Φ₁(𝑥,𝑥,𝜆)Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝜆≥sup_𝑥∈𝐶sup_{𝑦′,𝑦′′∈𝑋2∶Φ2(𝐸2(𝑦′),𝐸2(𝑦′′),𝜆)=Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)}min𝐴(𝑥)𝑦^′,𝐴(𝑥)𝑦^′′=sup_𝑥∈𝐶min𝐴(𝑥)𝑦₁,𝐴(𝑥)𝑦₂=minsup_𝑥∈𝐶𝐴(𝑥)𝑦₁,sup_𝑥∈𝐶𝐴(𝑥)𝑦₂=min𝐴(𝐶)𝑦₁,𝐴(𝐶)𝑦₂. (18) So, 𝐴(𝐶) is a Φ𝐸₂-convex fuzzy subset of 𝑋₂.

Theorem 20. If 𝐴 is (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₂) then for any 𝛼∈[0,1]𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆^𝛼⊃Φ₂𝐸₂𝐴𝑥₁^𝛼,𝐸₂𝐴𝑥₂^𝛼,𝜆 (19) for any 𝑥₁,𝑥₂∈𝑋₁, 𝜆∈[0,1].

Proof. According to the definition of 𝛼-cut, we have 𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆^𝛼=𝑦∈𝑋₂∶𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂𝑦,𝐸,𝜆≥𝛼₂𝐴𝑥₁^𝛼=𝐸₂𝑦₁∈𝑋₂∶𝐴𝑥₁𝑦₁,𝐸≥𝛼₂𝐴𝑥₂^𝛼=𝐸₂𝑦₂∈𝑋₂∶𝐴𝑥₂𝑦₂≥𝛼. (20) Moreover, Φ₂𝐸₂𝐴𝑥₁^𝛼,𝐸₂𝐴𝑥₂^𝛼=,𝜆𝑦=Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂∶𝑦,𝜆₁∈𝐴𝑥₁^𝛼,𝑦₂∈𝐴𝑥₂^𝛼,𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆𝑦≥sup_{𝑦1,𝑦2∈𝑋2∶𝑦=Φ2(𝐸2(𝑦1),𝐸2(𝑦2),𝜆)}min𝐴𝑥₁𝑦₁,𝐴𝑥₂𝑦₂. (21) This means that if 𝑦∈Φ₂(𝐸₂([𝐴(𝑥₁)]^𝛼),𝐸₂([𝐴(𝑥₂)]^𝛼),𝜆) then 𝑦∈[𝐴(Φ₁(𝐸₁(𝑥₁),𝐸₁(𝑥₂),𝜆)]^𝛼, that is, 𝐴Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆^𝛼⊃Φ₂𝐸₂𝐴𝑥₁^𝛼,𝐸₂𝐴𝑥₂^𝛼,𝜆. (22)
Now, for any 𝛼∈[0,1] and any Φ𝐸₂-quasiconcave function 𝑔∶𝑋₂→𝑅, let us define a function 𝑞^𝛼_𝑔(𝑥)=max_{𝑦∈[𝐴(𝑥)]𝛼}𝑔𝑦for𝑥∈𝑋₁. (23)

Theorem 21. If 𝐴 is a (Φ𝐸₁,Φ𝐸₂)-convex fuzzy process from 𝑋₁ to 𝐹(𝑋₂), then the function 𝑞^𝛼_𝑔 is Φ𝐸₁-quasiconcave.

Proof. Let 𝑥₁,𝑥₂∈𝑋₁, 𝜆∈[0,1]. Then we have 𝑞^𝛼_𝑔Φ₁𝐸₁𝑥₁,𝐸₁𝑥₂,𝜆=max_{𝐴Φ𝑦∈1𝐸1𝑥1,𝐸1𝑥2,𝜆𝛼}𝑔𝑦≥max_{𝑦∈Φ2(𝐸2([𝐴(𝑥1)]𝛼),𝐸2([𝐴(𝑥2)]𝛼),𝜆)}𝑔𝑦=max𝑔𝑦∶𝑦=Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂,𝑦,𝜆₁∈𝐴𝑥₁^𝛼,𝑦₂∈𝐴𝑥₂^𝛼=max𝑔(Φ₂𝐸₂𝑦₁,𝐸₂𝑦₂∶𝑦,𝜆₁∈𝐴𝑥₁^𝛼,𝑦₂∈𝐴𝑥₂^𝛼≥maxmin𝑔𝑦₁,𝑔𝑦₂∶𝑦₁∈𝐴𝑥₁^𝛼,𝑦₂∈𝐴𝑥₂^𝛼≥minmax_{𝑦1∈𝐴𝑥1𝛼}𝑔𝑦₁,max_{𝑦2∈𝐴𝑥2𝛼}𝑔𝑦₂=min𝑞^𝛼_𝑔𝑥₁,𝑞^𝛼_𝑔𝑥₂. (24)