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

Authors: Marian Matłoka
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

We first define the concept of(ΦE1,ΦE2)-convex fuzzy processes. Second, we present some basic properties of such processes.


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## Body

## 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 ofΦE-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

LetXi denote a subset of the ni-dimensional Euclidean space Rni. Assume that Φi is a map satisfying the following assumption:(i)
Φi:Xi×Xi×[0,1]→Rni,(ii)
Φi(x,y,0)=y, Φ1(x,x,λ)=x, ∀x,y∈Xi, λ∈[0,1].For any subsetsA,B, of Xi let us define
(1)Φi(A,B,λ)={Φi(x,y,λ):x∈A,y∈B}.Definition 1 (see [1]).
A setXi is Φi-convex if Φi(x1,x2,λ)∈Xi for all x1,x2∈Xi, λ∈[0,1].Remark 2.
The intersection ofΦi-convex sets is still Φi-convex.Remark 3.
LetXi be a convex subset of Rni, and Φi(x1,x2,λ)=λx1+(1-λ)x2. Then a convex set is Φi-convex.Remark 4.
Ifη:Rni×Rni→Rni, Xi is a preinvex set with respect to η (see [11]), Φi(x1,x2,λ)=x2+λ·η(x1,x2), then Xi is a Φi-convex set.
Assume thatE1:Rni→Rni.Definition 5 (see [5]).
A setXi⊂Rni is said to be Ei-convex if (1-λ)E(x1)+λE(x2)∈Xi, for each x1,x2∈Xi and λ∈[0,1].Definition 6.
A setXi⊂Rni is said to be ΦEi-convex if Φi(Ei(x1),Ei(x2),λ)∈Xi, for each x1,x2∈Xi and λ∈[0,1].Remark 7.
IfΦi(x1,x2,λ)=λx1+(1-λ)x2 and Ei is the identity map then a convex set is ΦEi-convex.Definition 8.
LetXi be a ΦEi-convex set. A function f:Xi→R is said to be ΦEi-quasiconcave on Xi if for any x1,x2∈Xi and λ∈[0,1](2)f(Φi(Ei(x1),Ei(x2),λ))≥min(f(x1),f(x2)).
Let C:Rni→[0,1] denote a fuzzy set in Rni.Definition 9.
A fuzzy setC is called ΦEi-convex if and only if
(3)C(Φi(Ei(x1),Ei(x2),λ))≥min(C(x1),C(x2)),
for all x1,x2∈Rni.Definition 10.
Anα-cut of a fuzzy set C is defined as follows:
(4)Cα={{x:C(x)≥α}forα∈(0,1]{x:C(x)>α}forα=0.Proposition 11.
IfCis ΦEi-convex fuzzy set then Cα is ΦEi-convex (crisp) set.Proof.
We have to prove that ifx1,x2∈Cα then for any λ∈[0,1], Φi(Ei(x1),Ei(x2),λ)∈Cα. So, taking into account the above definitions, we observe that if x1,x2∈Cα then C(x1)≥α and C(x2)≥α and min(C(x1),C(x2))≥α. So, C(Φi(Ei(x1),Ei(x2),λ))≥min(C(x1),C(x2))≥α. This means that Φi(Ei(x1),Ei(x2),λ)∈Cα.


## 3. Main Results

In this section, we present the definition and some properties of the(ΦE1,ΦE2)-convex fuzzy processes.LetXi denote ΦEi-convex set and F(Xi) the set of all non-void fuzzy sets in Xi(i=1,2,3).Definition 12.
A mappingA 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∈X2(5)A(Φ1(E1(x1),E1(x2),λ))(y)≥supy1,y2∈X2:y=Φ2(E2(y1),E2(y2),λ)min(A(x1)(y1),A(x2)(y2)).Example 13.
LetX1=R, X2=R-(-1/2,1/2) and let E1(x)=|x|, E2(y)=y, Φ1(x1,x2,λ)=λx1+(1-λ)x2, and
(6)Φ2(y1,y2,λ)={λy1+(1-λ)y2ify1y2>0(1+λ)y2-λy1ify1y2<0.
Consider A:X1→F(X2) defined by
(7)A(X)∶=χ〈f(x),∞),
where f(x)=k|x|, k>0, and χC denotes the characteristics function of C.
Then a mappingA is (ΦE1,ΦE2)-convex fuzzy process.
Now, let us considerA:X1→F(X2) defined by
(8)A(x)(y)={yf(x)ify∈〈0,f(x)),1ify∈〈f(x),∞),0ify∈<(-∞,0),
for all x≠0 and
(9)A(x)(y)={1ify∈〈0,∞),0ify∈(-∞,0),
for x=0, where f(x)=k|x|, k>0.
The above mapping is(ΦE1,ΦE2)-convex fuzzy mapping too.Theorem 14.
IfA 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 ify1,y2∈X2, λ∈[0,1] then for any x∈X1(10)A(x)(Φ2(E2(y1),E2(y2),λ))≥min(A(x)(y1),A(x)(y2)).
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
(11)A(x)(Φ2(E2(y1),E2(y2),λ))=A(Φ1(x,x,λ))(Φ2(E2(y),E2(y2),λ))≥supy′,y′′∈X2:Φ2(E2(y1),E2(y2),λ)=Φ2(E2(y′),E2(y′′),λ)min(A(x)(y′),A(x)(y′′))≥min(A(x)(y1),A(x)(y2)).Definition 15.
The graph of a(ΦE1,ΦE2)-convex fuzzy process A from X1 to F(X2), denoted GA, is a fuzzy set in X1×X2 such that for any (x,y)∈X1×X2(12)GA(x,y)=A(x)(y).In the analogous way as in the Definition9, we can define the (ΦE1,ΦE2)-convex fuzzy subset B of X1×X2, that is,
(13)B(Φ1(E1(x1),E1(x2),λ),Φ2(E2(y1),E2(y2),λ))≥min(B(x1,y1),B(x2,y2)),
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](14)GA(Φ1(E1(x1),E1(x2),λ),Φ2(E2(y1),E2(y2),λ))=A(Φ1(E1(x1),E1(x2),λ))(Φ2(E2(y1),E2(y2),λ))≥supy′,y′′∈X2:Φ2(E2(y1),E2(y2),λ)=Φ2(E2(y′),E2(y′′),λ)min(A(x1)(y′),A(x2)(y′′))≥min(A(x1)(y1),A(x2)(y2))=min(GA(x1,y1),GA(x2,y2)).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
(15)(B∘A)(x)(z)=supy∈X2min(A(x)(y),B(y)(z))
for any (x,z)∈X1×X3.Theorem 18.
IfA 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.
Letx1,x2∈X1 and λ∈[0,1]. Then for any z∈X3 we have
(16)(B∘A)(Φ1(E1(x1),E1(x2),λ))(z)=supy∈X2min(A(Φ1(E1(x1),E1(x2),λ))(y),B(y)(z))=supy1,y2∈X2:y=Φ2(E2(y1),E2(y2),λ)min(A(Φ1(E1(x1),E1(x2),λ))(Φ2(E2(y1),E2(y2),λ)),B(Φ2(E2(y1),E2(y2),λ))(z))≥supy1,y2∈X2:y=Φ2(E2(y1),E2(y2),λ)min(A(x1)(y1),A(x2)(y2),supz1,z2∈X3:z=Φ3(E3(z1),E3(z2),λ)min(B(y1)(z1),B(y2)(z2)))≥supz1,z2∈X3:z=Φ3(E3(z1),E3(z2),λ)supy1,y2∈X2min(A(x1)(y1),A(x2)(y2),B(y1)(z1),B(y2)(z2))=supz1,z2∈X3:z=Φ3(E3(z1),E3(z2),λ)min(supy1∈X2min(A(x1)(y1),B(y1)(z1)),supy2∈X2min(A(x2)(y2),B(y2)(z2)))=supz1,z2∈X3:z=Φ3(E3(z1),E3(z2),λ)min((B∘A)(x1)(z1),(B∘A)(x2)(z2)).
So, B∘A is a (ΦE1,ΦE3)-convex fuzzy process.
For any setC⊂X1, we put
(17)A(C)(y)=supx∈CA(x)(y)
for any y∈X2.Theorem 19.
IfA 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.
Lety1,y2∈X2 and λ∈[0,1]. Then we have
(18)A(C)(Φ2(E2(y1),E2(y2),λ))=supx∈CA(x)(Φ2(E2(y1),E2(y2),λ))=supx∈CA(Φ1(x,x,λ))(Φ2(E2(y1),E2(y2),λ))≥supx∈Csupy′,y′′∈X2:Φ2(E2(y′),E2(y′′),λ)=Φ2(E2(y1),E2(y2),λ)min(A(x)(y′),A(x)(y′′))=supx∈Cmin(A(x)(y1),A(x)(y2))=min(supx∈CA(x)(y1),supx∈CA(x)(y2))=min(A(C)(y1),A(C)(y2)).
So, A(C) is a ΦE2-convex fuzzy subset of X2.Theorem 20.
IfA is (ΦE1,ΦE2)-convex fuzzy process from X1 to F(X2) then for any α∈[0,1](19)[A(Φ1(E1(x1),E1(x2),λ))]α⊃Φ2(E2([A(x1)]α),E2([A(x2)]α),λ)
for any x1,x2∈X1, λ∈[0,1].Proof.
According to the definition ofα-cut, we have
(20)[A(Φ1(E1(x1),E1(x2),λ))]α={y∈X2:A(Φ1(E1(x1),E1(x2),λ))(y)≥α},E2([A(x1)]α)={E2(y1)∈X2:A(x1)(y1)≥α},E2([A(x2)]α)={E2(y2)∈X2:A(x2)(y2)≥α}.
Moreover,
(21)Φ2(E2([A(x1)]α),E2([A(x2)]α),λ)={y=Φ2(E2(y1),E2(y2),λ):y1∈[A(x1)]α,y2∈[A(x2)]α},A(Φ1(E1(x1),E1(x2),λ))(y)≥supy1,y2∈X2:y=Φ2(E2(y1),E2(y2),λ)min(A(x1)(y1),A(x2)(y2)).
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,
(22)[A(Φ1(E1(x1),E1(x2),λ))]α⊃Φ2(E2([A(x1)]α),E2([A(x2)]α),λ).
Now, for anyα∈[0,1] and any ΦE2-quasiconcave function g:X2→R, let us define a function
(23)qgα(x)=maxy∈[A(x)]αg(y)forx∈X1.Theorem 21.
IfA is a (ΦE1,ΦE2)-convex fuzzy process from X1 to F(X2), then the function qgα is ΦE1-quasiconcave.Proof.
Letx1,x2∈X1, λ∈[0,1]. Then we have
(24)qgα(Φ1(E1(x1),E1(x2),λ))=maxy∈[A(Φ1(E1(x1),E1(x2),λ))]αg(y)≥maxy∈Φ2(E2([A(x1)]α),E2([A(x2)]α),λ)g(y)=max{g(y):y=Φ2(E2(y1),E2(y2),λ),y1∈[A(x1)]α,y2∈[A(x2)]α}=max{g(Φ2(E2(y1),E2(y2),λ):y1∈[A(x1)]α,y2∈[A(x2)]α}≥max{min(g(y1),g(y2)):y1∈[A(x1)]α,y2∈[A(x2)]α}≥min(maxy1∈[A(x1)]αg(y1),maxy2∈[A(x2)]αg(y2))=min(qgα(x1),qgα(x2)).



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*Source: 290845-2012-10-03.xml*