Some properties of semi-E -preinvex maps in Banach spaces

Nonlinear Analysis: Real World Applications 12 (2011) 1243–1249

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Some properties of semi-E-preinvex maps in Banach spaces Zhiming Luo a,∗ , Jinbao jian b a

Department of Information, Hunan University Commerce, Changsha, Hunan 410205, PR China

b

College of Mathematics and Information Science, Guangxi University, Nannin, Guangxi, 530004, PR China

article

info

Article history: Received 6 August 2009 Accepted 24 September 2010 Keywords: E-convex maps E-preinvex maps Semi-E-convex maps Semi-E-preinvex maps

abstract In this paper, we extend the class of E-convex, semi-E-convex, E-preinvex functions in n-dimensional spaces introduced by Youness (1999) [10], respectively by Syau and Lee (2005) [14], Chen (2002) [12], Fulga and Preda (2009) [15] to semi-E-preinvex maps in Banach spaces. Some properties of these classes are studied. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Convex functions have important properties that make them an ideal tool for solving practical problems. Attempts have been made to generalize convexity by preserving some of the properties of convex functions. Mangasarian [1] studied the concepts of quasiconvexity and pseudoconvexity. Functions belonging to these classes and to their associate subclasses satisfy certain local–global minimum properties. Further extensions of these concepts to semilocally quasiconvex and semilocally pseudoconvex functions were introduced by Kaul and Kaur [2,3]. Ortega and Rheinboldt [4] extended the families of quasiconvex and strongly quasiconvex functions by defining other generalized convex functions. Avriel and Zang [5] introduced and studied the class of arcwise connected functions. Hanson [6] introduced invex functions which satisfy the property that every stationary point is a global minimum. Another generalization of convex functions, namely B-vex functions which satisfy many of the basic properties of convex functions, was introduced by Bector and Singh [7]. For the nonlinear multiobjective programming Stancu–Minasian [8] derived some optimality and duality results when the functions involved are generalized semilocally B-preinvex, Antczak [9] introduced the concept of (p, r )-invexity and obtained some duality results in the differentiable case. Recently, the concepts of E-convex sets and E-convex functions were introduced by Youness [10]. In this paper, motivated by earlier research works, see [11–14,1,15], we introduce the class of semi-E-preinvex maps in Banach spaces. Also, properties and characterizations of these classes are given. 2. Preliminaries Let X and Y be two real Banach spaces, K ⊂ X and C ⊂ Y be two nonempty sets. Recall that K is a convex set, if

(1 − λ)x + λy ∈ K , ∗

∀x, y ∈ K , 0 ⩽ λ ⩽ 1.

Corresponding author. E-mail address: [email protected] (Z. Luo).

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.09.019

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And C is said to be a closed, convex and pointed cone with apex at the origin iff C is closed and the following conditions hold:

(i) λC ⊂ C ,

∀λ > 0;

(ii) C + C ⊂ C ;

(iii) C ∩ (−C ) = 0.

With a cone C , we can define relations ‘‘⩽C ’’ and ‘‘ C ’’ as follows: x ⩽C y ⇐⇒ y − x ∈ C and x C y ⇐⇒ y − x ∈ C . Let X be a Banach space, we give the definitions of invex sets, E-convex sets, E-invex sets, and the concepts of E-convex maps, semi-E-convex maps, preinvex maps and E-preinvex maps in X as the following while X = Rn these definitions and the concepts were given by earlier researches (see [11,12,15] respectively). Definition 1. A set K ⊆ X is said to be E-convex iff there is a map E : X → X such that

(1 − λ)E (x) + λE (y) ∈ K ,

∀x, y ∈ K , 0 ⩽ λ ⩽ 1.

Definition 2. A map f : X → Y is said to be E-convex on a set K ⊆ X iff there is a map E : X → X such that K is an E-convex set and f (λE (x) + (1 − λ)E (y)) ⩽C λf (E (x)) + (1 − λ)f (E (y)) for each x, y ∈ K , and 0 ⩽ λ ⩽ 1. Definition 3. A map f : X → Y is said to be semi-E-convex on a set K ⊆ X iff there is a map E : X → X such that K is an E-convex set and f (λE (x) + (1 − λ)E (y)) ⩽C λf (x) + (1 − λ)f (y) for each x, y ∈ K , and 0 ⩽ λ ⩽ 1. Definition 4. A set K ⊆ X is said to be invex with respect to a given mapping η : X × X → X if y + λη(x, y) ∈ K ,

∀x, y ∈ K , 0 ⩽ λ ⩽ 1.

Definition 5. A map f : X → Y is said to be preinvex on a set K ⊆ X with respect to a given mapping η : X × X → X if f (y + λη(x, y)) ⩽C λf (x) + (1 − λ)f (y) for each x, y ∈ K , and 0 ⩽ λ ⩽ 1. Definition 6. A set K ⊆ X is said to be E-invex with respect to a given mapping η : X × X → X if E (y) + λη(E (x), E (y)) ∈ K ,

∀x, y ∈ K , 0 ⩽ λ ⩽ 1.

Definition 7. Let K ⊆ X be an E-invex set with respect to a given mapping η : X × X → X . A function f : X → Y is said to be E-preinvex on K with respect to η if ∀x, y ∈ K , 0 ⩽ λ ⩽ 1, f (E (y) + λη(E (x), E (y))) ⩽C λf (E (x)) + (1 − λ)f (E (y)). Next, in Banach spaces, we introduce the concept of semi-E-preinvex maps on a semi-E-invex set. Throughout this paper, E : X → X and η : X × X → X are two fixed mappings. Definition 8. A map f : X → Y is said to be semi-E-preinvex on a set K ⊆ X with respect to a given mapping η : X × X → X iff there is a map E : X → X such that K is an E-preinvex set and f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y) for each x, y ∈ K and 0 ⩽ λ ⩽ 1. 3. Semi-E-preinvex maps and main results Now, we discuss the properties of semi-E-convex maps. In the following, we consider the problem Min f (x),

s.t. x ∈ K = {x ∈ X : gi (x) ≤Di 0, i = 1, 2, . . . , m}

(P )

where f : X → Y and gi : X → Zi , i = 1, 2, . . . , m, are maps, K is E-preinvex, C ⊂ X and Di ⊂ Zi are closed, convex and pointed cone with apex at the origin, (Y , ⩽C ) and (Zi , ⩽Di ) are ordered Banach spaces ordered by C and Di respectively.

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Proposition 1. If map f : X → Y is semi-E-preinvex on an E-convex set K ⊆ X , then f (E (x)) ⩽C f (x) for each x ∈ K . Proof. Since f is semi-E-preinvex on an E-invex set K E (y) + λη(E (x), E (y)) ∈ K and

⊆ X , then for any x, y ∈ K and 0 ⩽ λ ⩽ 1, we have

f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y). Thus for λ = 0, f (E (y)) ⩽C f (y) is derived.



Remark 1. An E-preinvex map on E-invex is not necessary a semi-E-preinvex map. Example 1. Let A = [−2, −1] ∪ [1, 2]. It is easy to prove that A is invex with respect to η : R × R → R defined by x − y, if x ≥ 0, y ⩾ 0, or x ≤ 0, y ⩽ 0; η(x, y) = −1 − y, if x > 0, y ⩽ 0, or x ⩾ 0, y < 0; 1 − y, if x < 0, y ⩾ 0, or x ⩽ 0, y > 0.



Also, we can prove that A is E-invex with respect to E : R → R defined by E (x) =



x2 ,

√

if |x| ⩽ √ 2; −1, if |x| > 2.

1 We notice that A is E-invex but is not E-convex; indeed, if we take x = 11 , y = 32 and λ = 10 then Definition 2 is not 10 satisfied. √ √ Let f : R → R be defined as f (x) = x2 , f is an E-preinvex function on A. Since f (E ( 2)) = 4 > f ( 2) = 2, then from Proposition 1, it follows that f is not a semi-E-preinvex function on set E.

Proposition 2. If the maps fi : X → Y are all semi-E-preinvex on an E-invex set K ⊂ X , i = 1, 2, . . . , k, with the same map E, ∑k then the function h(x) = i=1 ai fi (x) is semi-E-preinvex on K for ai ⩾ 0, i = 1, 2, . . . , k. Proposition 3. If map f : X → Y is semi-E-preinvex on an E-invex set K ⊂ X , then for any α ∈ Y , the level set Kα = {x|x ∈ K , f (x) ⩽C α} is E-invex. Proof. For any x, y ∈ Kα and 0 ⩽ λ ⩽ 1, then, f (x) ⩽C α, f (y) ⩽C α . Since f is semi-E-invex on an E-invex set K ⊆ X , then, E (y) + λη(E (x), E (y)) ∈ K , and f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y) ⩽ α i.e., E (y) + λη(E (x), E (y)) ∈ Kα , hence, Kα is E-invex.



Remark 2. The converse of Proposition 4 is not true. We give an example of a function f (x), whose level set Kα = {x|x ∈ M , f (x) ⩽ α} is E-invex. Example 2. In Example 1, the level sets

 A, √α ⩾ 4; √ Kα = [− α, 1] ∪ [1, α], φ, α < 1

1 ⩽ α < 4;

are always E-invex with respect to E. Since

 f

E (1) +

1 4



η(E (1.4), E (1)) = f (1.24) >

1 4

f (1.4) +

3 4

f (1) = 1.24

then the function f (x) is not semi-E-preinvex on the E-invex set A. Proposition 4. Suppose map f : X → Y is E-preinvex on an E-invex set K ⊂ X . Then f is semi-E-invex on set K iff f (E (x)) ⩽C f (x) for each x ∈ K . Proof. Suppose the map f : X → Y is E-preinvex on an E-invex set K and f (E (x)) ⩽C f (x) for each x ∈ K , then for any x, y ∈ K and 0 ⩽ λ ⩽ 1, we have f (E (y) + λη(E (X ), E (Y ))) ⩽C λf (E (x)) + (1 − λ)f (E (y)) ⩽C λf (x) + (1 − λ)f (y).

Hence, f is semi-preinvex on the E-invex set K . The other hand is derived from Proposition 1.  Definition 9. The mapping f : X → Y is said to be quasi-semi-E-preinvex on a set K ⊆ X , if at least one of the following holds: f (E (y) + λη(E (x), E (y))) ⩽C f (x)

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and f (E (y) + λη(E (x), E (y))) ⩽C f (y) for each x, y ∈ K , λ ∈ [0, 1] such that E (y) + λη(E (x), E (y)) ∈ K . Proposition 5. Let K ⊆ X is an E-invex set. If the map f : K → Y is quasi-semi-E-preinvex, then the level set Kα = {x|x ∈ K , f (x) ⩽C α} is E-invex for each α ∈ X . Proof. If f is quasi-semi-E-preinvex on an E-invex set K . Thus, for any x, y ∈ Kα , and λ ∈ [0, 1], we have E (y) + λη(E (x), E (y)) ∈ K , f (x) ⩽C α, f (y) ⩽C α and at least one of the following holds: f (E (y) + λη(E (x), E (y))) ⩽C f (x) and f (E (y) + λη(E (x), E (y))) ⩽C f (y). It follows that E (y) + λη(E (x), E (y)) ∈ Kα and the set Kα is E-invex.



Proposition 6. Let gi : X → Zi is quasi-semi-E-preinvex on X , i = 1, 2, . . . , k. Then the set K = {x ∈ X | gi (x) ⩽Di 0, i = 1, 2, . . . , k} is E-invex. Proof. From Proposition 5, it follows that the set Ki = {x ∈ X | gi (x) ⩽Di 0} is E-invex, i = 1, 2, . . . , k, which implies that the set K =

k

i =1

Ki is E-invex.



Proposition 7. Let f : X → Y . Then a semi-E-preinvex map f on an E-invex set K ⊆ X is also quasi-semi-preinvex on K . Given a map E : X → X , let the mapping E × I : X × Y → X × Y be (E × I )(x, t ) = (E (x), t ), for (x, t ) ∈ X × Y , where I is identity. It is easy to show that K ⊆ X is E-invex, if and only if K × Y is E × I-invex. Now, we define an epigraph of f as follows: epi(f ) = {(x, α) : x ∈ K , α ∈ Y , f (x) ⩽C α}, the following Proposition 9 gives a characterization of a semi-E-preinvex function in terms of its epi(f ). Proposition 8. Assume K is E-invex, then f is a semi-E-preinvex function on K if and only if epi(f ) is E × I-invex on K × Y . Proof. Assume that f is semi-E-preinvex on K . Let (x, α), (y, β) ∈ epi(f ), λ ∈ [0, 1], it then follows that E (y) + λη(E (x), E (y)) ∈ K , and f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y) ⩽C λα + (1 − λ)β.

Thus

(E (y) + λη(E (x), E (y)), λα + (1 − λ)β) ∈ epi(f ), which implies that epi(f ) is E × I-invex on K × Y with respect to a given mapping η × η0 , where η0 : Y × Y → Y , (α, β) → α − β. Conversely, let epi(f ) is E × I-invex on K × Y . Let x, y ∈ K , λ ∈ [0, 1], then (x, f (x)) ∈ epi(f ), and (y, f (y)) ∈ epi(f ) is E × I-invex on K × Y , we have

(E (y) + λη(E (x), E (y)), f (y) + λη0 (f (x), f (y))) = (E (y) + λη(E (x), E (y)), λf (x) + (1 − x)f (y)) ∈ epi(f ), which implies that f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − x)f (y), this shows that f is a semi-E-preinvex function on K .



Definition 10. The map f : X → Y is said to be a pseudo-semi-preinvex E-invex set K ⊆ X , if there exists a strictly positive function b : X × X → Y such that f (x)
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Proof. Suppose f (x) ⩽c f (y) and f is semi-E-preinvex on an E-invex set K , then for all x, y ∈ K and λ ∈ (0, 1), we have f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y) = f (y) + λ(f (x) − f (y)) ⩽C f (y) + λ(1 − λ)(f (x) − f (y)) = f (y) + λ(λ − 1)(f (y) − f (x))

= f (y) + λ(λ − 1)b(x, y), where b(x, y) = f (y) − f (x) >C 0, the required result.



Proposition 10. Let f : X → Y be a quasi-semi-E-preinvex function on E-invex set K , if φ : Y → Y is an order-preserving transformation, then the composite function φ ◦ f is a quasi-semi-E-preinvex on K . Proof. Since f : X → Y is a quasi-semi-E-preinvex function on E-invex set K , then at least one of the following holds: f (E (y) + λη(E (x), E (y))) ⩽C f (x) and f (E (y) + λη(E (x), E (y))) ⩽C f (y) for each x, y ∈ K , λ ∈ [0, 1] such that E (y) + λη(E (x), E (y)) ∈ K . Thus at least one of the following holds:

φ ◦ f (E (y) + λη(E (x), E (y))) ⩽C φ f (y) = φ ◦ f (x) and

φ ◦ f (E (y) + λη(E (x), E (y))) ⩽C φ f (x) = φ ◦ f (y). From which it follows that the composite function φ ◦ f is a quasi-semi-E-preinvex on K .



Definition 11. The feasible solution x¯ of the following problem (PE ) is said to be a weakly efficient solution, if there is a non-existent x ∈ K such that f (¯x) − f (x) ∈ intC . Theorem 1. Assume K is an E-invex set, and f (E (x)) ⩽C f (x), for each x ∈ K , x¯ is a weakly efficient solution of the following problem: min(f ◦ E )(x),

s.t.

x ∈ K.

(PE )

Then, E (¯x) is a weakly efficient solution of problem (P ). Proof. Let x¯ be a weakly efficient solution of problem (PE ), then there is a neighborhood of x¯ , U such that ∀y ∈ K ∩ U, the following does not hold f (¯x) − f (y) ∈ intC . If E (¯x) is not a weakly efficient solution of problem (P ), then there is y ∈ K such that f (E (¯x)) − f (x) ∈ intC . Since f (¯x − f (E (¯x))) ∈ C and f (¯x) − f (y) ∈ intC , then f (¯x) − f (y) = [f (¯x) − f (E (¯x))] − [f (E (¯x)) − f (y)] ∈ C + intC ⊂ intC which contradicts that x¯ be a weakly efficient solution of problem (PE ). Hence, E (¯x) is a weakly efficient solution of problem (P ).  Theorem 2. Assume function f : X → Y is semi-E-preinvex on an E-invex set K ⊆ X and x¯ is a weakly efficient solution of the following problem: min(f ◦ E )(x),

s.t.

x ∈ K.

(PE )

Then, E (¯x) is a solution of the problem (P ). Proof. Follows from Theorem 1.



Theorem 3. If x0 = E (z 0 ) ∈ E (K ) is a local weakly efficient solution of the problem (P ) on an E-invex set K , and f : X → Y is E-preinvex on the set K , and f (E (x)) ⩽C f (x) for each x ∈ K , then x0 is a global weakly efficient solution of problem (P ) on K . Proof. Let x0 = E (z 0 ) ∈ E (K ) is not a global weakly efficient solution of the problem (P ) on K , then, there is y ∈ K such that f (x0 ) − f (y) = f (E (z 0 )) − f (y) ∈ intC . Since function f : X → Y is E-preinvex and f (x) − f (E (x)) ∈ C for each x ∈ K , it implies that f (x0 + λη(E (x0 ), E (y))) = f (E (z 0 ) + λη(E (z 0 ), E (y))) ⩽C λf (E (y)) + (1 − λ)f (E (z 0 )) = λf (y) + (1 − λ)f (x0 ).

Thus, for any small λ ∈ (0, 1), f (x0 ) − f (x0 + λη(E (x0 ), E (y))) ∈ intC , which contradicts the local weakly efficient solution of x0 for problem (P ). Hence, x0 is a global weakly efficient solution of problem (P ) on K . 

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Theorem 4. Assume map f : X → Y is strictly semi-E-preinvex on an E-invex set K ⊆ X , i.e., f (E (y) + λη(E (x), E (y)))


Theorem 6. Assume map f : X → Y is quasi-semi-E-preinvex on E-invex set K ⊆ X . Then the set of weakly efficient solutions of problem (P ) is E-invex. Proof. Let x¯ be a weakly efficient solution of problem (P ), and let α = f (¯x). Let Z be the set of weakly efficient solutions for problem (P ) as follows: Z = {x ∈ K | f (x) ⩽C α}, for any x, y ∈ Z and 0 ⩽ λ ⩽ 1, since map f : X → Y is semi-E-preinvex on an E-invex set K ⊆ X , then, f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y) ⩽C α. Thus, E (y) + λη(E (x), E (y)) ∈ Z , it follows that Z is E-invex.



Theorem 7. If map f : X → Y and gi : X → Zi are quasi-semi-E-preinvex on X , i = 1, 2, . . . , m. Then the set of weakly efficient solutions of problem (P ) is E-invex. Proof. From Proposition 7, it follows that K is E-invex. Hence by Theorem 8, the set X = {x ∈ K | f (x) = α} of weakly efficient solutions of the problem (P ) is E-invex.  Corollary. If map f : X → Y and gi : X → Zi are semi-E-preinvex on K , i = 1, 2, . . . , m, then the set of weakly efficient solutions of problem (P ) is E-invex. Theorem 8. Let map f : X → Y is quasi-semi-preinvex on E-invex set K , let φ : Y → Y be positively homogeneous orderpreserving transformation, then the composite function φ ◦ f is semi-E-preinvex on K . Proof. Since f is semi-E-preinvex on E-invex set K , we have, for all x, y ∈ K f (E (y) + λη(E (x), E (y))) ⩽C λf (x) + (1 − λ)f (y)

⇒ φ ◦ f (E (y) + λη(E (x), E (y))) ⩽C φ ◦ [λf (x) + (1 − λ)f (y)] ⩽C λφ ◦ f (x) + (1 − λ)φ ◦ f (y) from which it follows that φ ◦ f is semi-E-preinvex on K .



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Theorem 9. Assume map f : X → Y is Fréchet differentiable semi-E-preinvex on an E-invex set K ⊆ X , and u ∈ K is a fixed point of the map E, i.e., u = E (u). Then u ∈ K is the weakly efficient solution of map f on K if and only if u ∈ K satisfies the inequality f ′ (E (u)) · η(E (v), E (u)) ⩾C 0,

∀v ∈ M ,

(1)

where f ′ is the Fréchet differential of function f at E (u). Proof. Let u ∈ K is a weakly efficient solution of map f on K . Then f (E (u)) ⩽C f (E (v)),

∀v ∈ K .

Since K is E-preinvex, we have E (u) + λη(E (v), E (u)) ∈ K ,

∀u, v ∈ K .

Since f is Fréchet differential on K , we have f (E (u)) ⩽C f (E (u) + λη(E (v), E (u)))

= f (E (u)) + λf ′ (E (u)) · η(E (v), E (u)) + o(λ). Dividing the above inequality by λ and taking λ → 0, we have f ′ (E (u)) · η(E (v), E (u)) = f ′ (u) · η(E (v), u) ⩾C 0. Which is the required result (1). Conversely, let u ∈ K satisfy (1). Since f is semi-E-preinvex on E-invex set K , for all u, v ∈ K , λ ∈ [0, 1], we have E (u) + λη(E (v), E (u)) ∈ K , and f (E (u) + λη(E (v), E (u))) ⩽C (1 − λ)f (u) + λf (v) = f (E (u)) + λf (f (v) − f (u)). Which implies that f (v) − f (u) ⩾C

f (E (u) + λη(E (v), E (u)) − f (E (u)))

λ

.

Letting λ → 0, we have f (v) − f (u) ⩾C f ′ (E (u)) · η(E (v), E (u)) ⩾C 0. Using (1), which implies that f (u) ⩽C f (v), ∀v ∈ K , showing that u ∈ K is the weakly efficient solution of map f on K .



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