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volume 5, issue 4, article 110, 2004.

Received 10 October, 2004;

accepted 11 November, 2004.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

ON GENERALIZED PREINVEX FUNCTIONS AND MONOTONICITIES

MUHAMMAD ASLAM NOOR

Etisalat College of Engineering P.O. Box 980, Sharjah United Arab Emirates.

EMail:noor@ece.ac.ae

c

2000Victoria University ISSN (electronic): 1443-5756 215-04

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On Generalized Preinvex Functions and Monotonicities

Muhammad Aslam Noor

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J. Ineq. Pure and Appl. Math. 5(4) Art. 110, 2004

http://jipam.vu.edu.au

Abstract

In this paper we consider some classes ofα-preinvex andα-invex functions.

We study some properties of these classes ofα-preinvex (α-invex) functions.

In particular, we establish the equivalence among theα-preinvex functions,α- invex functions and αη-monotonicity of their differential under some suitable conditions.

2000 Mathematics Subject Classification:26D07, 26D10, 39B62.

Key words: Preinvex functions,η-monotone operators, Invex functions, Pseudo invex functions.

1. Introduction

In recent years, several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson [1]. Hanson’s initial result inspired a great deal of subsequent work which has greatly expanded the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences. Weir and Mond [13], Jeyakumar and Mond [3] and Noor [5,7]

have studied the basic properties of the preinvex functions and their role in optimization and mathematical programming problems. It is well-known that the preinvex functions and invex sets may not be convex functions and convex sets.

In recent years, these concepts and results have been investigated extensively in [6,7,8,11,12]. It is noted that some of the results obtained in [8] are incor- rect and misleading. The main purpose of this paper to suggest some appropri- ate and suitable modifications. We also consider some classes of preinvex and invex functions, which are called α-preinvex and α-invex functions. Several new concepts ofαη-monotonicity are introduced. We establish the relationship between these classes and derive some new results. As special cases, one can obtain some new and correct versions of known results. Results obtained in this paper present a refinement and improvement of previously known results.

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2. Preliminaries

Let K be a nonempty closed set in H. We denote byh·,·i andk · k the inner product and norm respectively. Let F : K → H andη(·,·) : K ×K → Rbe continuous functions. Letα;K×K −→R\{0}be a bifunction.

First of all, we recall the following well known results and concepts.

Definition 2.1. Let u ∈ K. Then the set K is said to be α-invex at u with respect toη(·,·)andα(·,·), if , for allu, v ∈K, t ∈[0,1],

u+tα(v, u)η(v, u)∈K.

K is said to be anα-invex set with respect toη andα, ifK isα-invex at each u∈K.Theα-invex setK is also calledαη-connected set. Note that the convex set withα(v, u) = 1andη(v, u) = v−uis an invex set, but the converse is not true. For example, the setK =R\ −¹₂,¹₂

is an invex set with respect toηand α(v, u) = 1,where

η(v, u) =

( v−u, for v >0, u >0 or v <0, u <0 u−v, for v <0, u >0 or v <0, u <0.

It is clear thatK is not a convex set.

Remark 2.1. (i) Ifα(v, u) = 1,then the setKis called the invex (η-connected) set, see [6,7,12,13].

(ii) If η(v, u) = v −u and 0 < α(v, u) < 1, then the set K is called the star-shaped.

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(iii) Ifα(v, u) = 1andη(v, u) =v−u,then the setK is called the convex set.

From now onwardK is a nonempty closedα-invex set inHwith respect to α(·,·)andη(·,·), unless otherwise specified.

Definition 2.2. The function F on the α-invex set K is said to be α-preinvex with respect toη, if

F(u+tα(v, u)η(v, u))≤(1−t)F(u) +tF(v), ∀u, v ∈K, t∈[0,1].

The functionF is said to beα-preconcave if and only if−F isα-preinvex. Note that every convex function is a preinvex function, but the converse is not true.

For example, the function F(u) = −|u| is not a convex function, but it is a preinvex function with respect toηandα(v, u) = 1,where

η(v, u) =

( v−u, if v ≤0, u≤0 and v ≥0, u≥0 u−v, otherwise.

Definition 2.3. The functionF on theα-invex setK is called quasiα-preinvex with respect toαandη,if

F(u+tα(v, u)η(v, u))≤max{F(u), F(v)}, ∀u, v∈K, t ∈[0,1].

Definition 2.4. The function F on the α-invex set K is said to be logarithmic α-preinvex with respect toαandη,if

F(u+tα(v, u)η(v, u))≤(F(u))^1−t(F(v))^t, u, v ∈K, t ∈[0,1], whereF(·)>0.

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From the above definitions, we have

F(u+tα(v, u)η(v, u))≤(F(u))^1−t(F(v))^t

≤(1−t)F(u) +tF(v)

≤max{F(u), F(v)}

<max{F(u), F(v)}.

Fort= 1,Definitions2.2and2.4reduce to:

Condition A.

F(u+α(v, u)η(v, u))≤F(v), ∀u, v ∈K,

which plays an important part in studying the properties of the α-preinvex (α- invex) functions. Some properties of theα-preinvex functions have been studied in [7,11].

Forα(v, u) = 1, Condition A reduces to the following for preinvex functions.

Condition B.

F(u+η(v, u))≤F(v), ∀u, v ∈K.

For the applications of ConditionB, see [7,11,12].

Definition 2.5. The function F on the α-invex set K is said to be pseudo α- preinvex with respect toαandη,if there exists a strictly positive functionb(·,·) such that

F(v)≤F(u)

=⇒F(u+tα(v, u)η(v, u))≤F(u) +t(t−1)b(u, v), u, v ∈K t∈[0,1].

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Lemma 2.1. If the function F is α-preinvex function with respect toα andη, thenF is pseudoα-preinvex function with respect toαandη.

Proof. Without loss of generality, we assume thatF(v) < F(u), ∀u, v ∈K.

For everyt∈[0,1],we have

F(u+tα(v, u)η(v, u))≤(1−t)F(u) +tF(v)

< F(u) +t(t−1){F(u)−F(v)}

=F(u) +t(t−1)b(v, u), whereb(v, u) =F(v)−F(u)>0.

Thus it follows that the function F is pseudo α-preinvex function with respect toαandη,the required result.

Definition 2.6. The differentiable functionF on theα-invex setK is said to be anα-invex function with respect toα(·,·)andη(·,·), if

F(v)−F(u)≥ hα(v, u)F⁰(u), η(v, u)i, ∀u, v ∈K,

where F⁰(u) is the differential of F at u. The concepts of the α-invex and α-preinvex functions have played a very important role in the development of convex programming, see [2,3].

Definition 2.7. An operatorT :K −→His said to be:

(i). stronglyαη-monotone, iff, there exists a constantα >0such that hα(v, u)T u, η(v, u)i+hα(u, v)T v, η(u, v)i

≤ −α{kη(v, u)k²+kη(u, v)k²}, ∀u, v ∈K.

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(ii). αη-monotone, iff,

hα(v, u)T u, η(v, u)i+hα(u, v)T v, η(u, v)i ≤0, ∀u, v ∈K.

(iii). stronglyαη-pseudomonotone, iff, there exists a constantν > 0such that hα(v, u)T u, η(v, u)i+νkη(v, u)k² ≥0

=⇒ −hα(u, v)T v, η(u, v)i ≥0,∀u, v ∈K.

(iv). strictlyη-monotone, iff,

hα(v, u)T u, η(v, u)i+hα(u, v)T v, η(u, v)i<0, ∀u, v ∈K.

(v). αη-pseudomonotone, iff,

hα(v, u)T u, η(v, u)i ≥0 =⇒ hα(u, v)T v, η(u, v)i ≤0, ∀u, v ∈K.

(vi). quasiαη-monotone, iff,

hα(v, u)T u, η(v, u)i>0 =⇒ hα(u, v)T v, η(u, v)i ≤0, ∀u, v ∈K.

(vii). strictlyη-pseudomonotone, iff,

hα(v, u)T u, η(v, u)i ≥0 =⇒ hα(u, v)T v, η(u, v)i<0, ∀u, v ∈K.

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Note for α(v, u) = 1, ∀u, v ∈ K, the α-invex setK becomes an invex set. In this case, Definition 2.7 is exactly the same as in [7, 11]. In addition, if α(v, u) = 1 and η(v, u) = v −u,then the α-invex setK is the convex set K and consequently Definition2.7reduces to the one in [9] for the convex set K.This clearly shows that Definition2.7is more general than and includes the ones in [7,9,12] as special cases.

Definition 2.8. A differentiable function F on anα-invex setK is said to be strongly pseudoαη-invex function, iff, there exists a constantµ > 0such that hα(v, u)F⁰(u), η(v, u)i+µkη(u, v)k² ≥0 =⇒F(v)−F(u)≥0, ∀u, v ∈K.

Definition 2.9. A differentiable function F on theα-invex set K is said to be strongly quasiα-invex, if there exists a constantµ >0such that

F(v)≤F(u) =⇒ hα(v, u)F⁰(u), η(v, u) +µkη(v, u)k² ≤0, ∀u, v ∈K.

Definition 2.10. The function F on theα-invex setK is said to be pseudoα- invex, if

hα(v, u)F⁰(u), η(v, u)i ≥0, =⇒ F(v)≥F(u), ∀u, v ∈K.

Note that if α(v, u) = 1,then the α-invex setK is exactly the invex setK and consequently Definitions 2.8 – 2.10 are exactly the same same as in [7].

In particular, if η(v, u) = −η(v, u),∀u, v ∈ K, that is, the function η(·,·)is skew-symmetric, then Definitions2.7–2.10reduce to the ones in [8,11]. This shows that the concepts introduced in this paper represent an improvement of the previously known ones. All the concepts defined above play an important and fundamental part in mathematical programming and optimization problems.

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We also need the following assumption regarding the functionsη(·,·), and α(·,·).

Condition C. Let η(·,·) : K ×K −→ H and α(·,·) : K ×K −→ R\{0}

satisfy the assumptions

η(u, u+tα(v, u)η(v, u)) = −tη(v, u)

η(v, u+tα(v, u)η(v, u)) = (1−t)η(v, u), ∀u, v ∈K, t∈[0,1].

Clearly for t = 0, we haveη(u, v) = 0, if and only ifu = v,∀u, v ∈ K. One can easily show [11] that

η(u+tα(v, u)η(v, u), u) =tη(v, u), ∀u, v ∈K.

Note that for α(v, u) = 1, Condition C collapses to the following condition, which is due to Mohan and Neogy [4].

Condition D. Letη(·,·) :K×K −→Hsatisfy the assumptions η(u, u+tη(v, u)) =−tη(v, u)

η(v, u+tη(v, u)) = (1−t)η(v, u), ∀u, v ∈K, t∈[0,1]

For the applications of ConditionD, see [7,11,12] and the references therein.

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3. Main Results

In this section, we study some basic properties of α-preinvex functions on the α-invex setK.

Theorem 3.1. Let F be a differentiable function on the α-invex setK and let ConditionChold. Then the functionF is aα-preinvex function if and only ifF is aα-invex function.

Proof. Let F be a α-preinvex function on the α-invex set K. Then, ∀u, v ∈ K, t ∈[0,1], u+tα(v, u)η(v, u)∈Kand

F(u+tα(v, u)η(v, u))≤(1−t)F(u) +tF(v), ∀u, v ∈K, which can be written as

F(v)−F(u)≥ F(u+tα(v, u)η(v, u))−F(u)

t .

Lettingt−→0in the above inequality, we have

F(v)−F(u)≥ hα(v, u)F⁰(u), η(v, u)i, which implies thatF is aα-invex function.

Conversely, let F be a α-invex function on the α-invex function K. Then

∀u, v ∈ K, t ∈ [0,1], v_t =u+tα(v, u)η(v, u) ∈K and using ConditionC, we have

F(v)−F(u+tα(v, u)η(v, u))

≥ hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u+tα(v, u)η(v, u))i

= (1−t)hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i.

(3.1)

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In a similar way, we have

F(u)−F(u+tα(v, u)η(v, u))

≥ hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(u, u+tα(v, u)η(v, u))i

=−thα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u))i.

(3.2)

Multiplying (3.1) bytand (3.2) by(1−t)and adding the resultant, we have F(u+tα(v, u)η(v, u))≤(1−t)F(u) +tF(v).

showing thatF is aα-preinvex function.

Ifα(v, u) = 1,then Theorem 3.1 reduces to the following result, which is mainly due to Mohan and Neogy [4] for the preinvex and invex functions on the invex set.

Theorem 3.2. Let F be a differentiable function on the invex set K and let ConditionDhold. Then the functionF is a preinvex function if and only ifF is an invex function.

Theorem 3.3. LetF be differntiable function on the invex setK.IfF isα-invex (α-preinvex) function, then its differentialF⁰(u)isαη-monotone.

Proof. LetF be aα-invex function on theα-invex setK.Then (3.3) F(v)−F(u)≥ hα(v, u)F⁰(u), η(v, u)i, ∀u, v ∈K.

Changing the role ofuandvin (3.3), we have

(3.4) F(u)−F(v)≥ hα(u, v)F⁰(v), η(u, v)i, ∀u, v ∈K.

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Adding (3.3) and (3.4), we have

hα(v, u)F⁰(u), η(v, u)i+hα(u, v)F⁰(v), η(u, v)i ≤0, which shows thatF⁰isαη-monotone.

We now prove the converse of Theorem3.3for the caseα(v, u) = α(u, v), that is, the functionα(v, u) is a symmetric function. However, in general, the converse of Theorem3.3is an open problem.

Theorem 3.4. Let Conditions Aand C hold and the functionα(v, u) be sym- metric. If the differential F⁰(u) of a function F(u) is αη-monotone, then the functionF(u)isα-invex (α-preinvex) function.

Proof. LetF⁰(u)beαη-monotone, that is,

hα(u, v)F⁰(v), η(u, v)i+hα(v, u)F⁰(u), η(v, u)i ≤0, ∀u, v ∈K, which implies that

(3.5) hF⁰(v), η(u, v)i ≤ −hF⁰(u), η(v, u)i, sinceα(v, u)is a positive symmetric function.

SinceKis aα-invex set,∀u, v ∈K, t∈[0,1], v_t=u+tα(v, u)η(v, u)∈K.

Takingv ≡v_t,in (3.5) and using ConditionC, we have

−thF⁰(u+tα(v, u)η(v, u)), η(v, u)i ≤ −thF⁰(u), η(v, u)i, which implies that

(3.6) hF⁰(u+tα(v, u)η(v, u)), η(v, u)i ≥ hF⁰(u), η(v, u)i.

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Let

g(t) = F(u+tα(v, u)η(v, u)), ∀u, v ∈K, t∈[0,1].

Then, using (3.6), we have

g⁰(t) = hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i

≥ hα(v, u)F⁰(u), η(v, u)i.

Integrating the above relation between0and1,we have g(1)−g(0)≥ hα(v, u)F⁰(u), η(v, u)i, that is,

F(u+α(v, u)η(v, u))−F(u)≥ hα(v, u)F⁰(u), η(v, u)i, which implies, using ConditionA,

F(v)−F(u)≥ hα(v, u)F⁰(u), η(v, u)i,

which shows that the functionF(u)is aα-invex (α-preinvex) function, the required result.

Forα(v, u) = 1,theα-invex setK becomes the invex set and consequently from Theorem3.3and Theorem3.4, we have the following result for preinvex and invex functions.

Theorem 3.5. Let Conditions B and Dhold and let K be an invex set. Then the differential F⁰(u)of a functionF(u)isη-monotone if and only ifF(u)is a preinvex(invex) function on the invex setK.

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We now give a necessary condition for stronglyαη-pseudo-invex functions, which is also a generalization and refinement of a result proved in [8,11].

Theorem 3.6. Let the differentialF⁰(u)of a functionF(u)on theα-invex setK be stronglyαη-pseudomonotone. If ConditionsAandChold, thenF is strongly pseudoαη-invex function.

Proof. LetF⁰(u)be stronglyαη-pseudomonotone. Then

hα(v, u)F⁰(u), η(v, u)i+µkη(v, u)k² ≥0, ∀u, v ∈K, implies that

(3.7) −hα(u, v)F⁰(v), η(u, v)i ≥0, ∀u, v ∈K.

SinceKis anα-invex set,∀u, v ∈K, t∈[0,1], v_t=u+tα(v, u)η(v, u)∈K.

Takingv =v_tin (3.7) and using ConditionC, we have

hα(v_t, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i ≥0, ∀u, v ∈K, which implies that

(3.8) hF⁰(u+tα(v, u)η(v, u)), η(v, u)i ≥0, ∀u, v ∈K.

Let

g(t) = F(u+tα(v, u)η(v, u)), ∀u, v ∈K, t∈[0,1].

Then, using (3.8), we have

g⁰(u) = hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i ≥0.

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Integrating the above relation between0and1,we have g(1)−g(0) ≥0,

that is,

F(u+α(v, u)η(v, u))−F(u)≥0, which implies, using ConditionA, that

F(v)−F(u)≥0,

showing that the functionF(u)is strongly pseudoαη-invex function.

As special cases of Theorem3.6, we have the following:

Theorem 3.7. Let the differentialF⁰(u)of a functionF(u)on theα-invex setK beαη-pseudomonotone. If ConditionsAandChold, thenF is pseudoαη-invex function.

Theorem 3.8. Let the differentialF⁰(u)of a functionF(u)on theα-invex setK be stronglyη-pseudomonotone. If ConditionsAandChold, thenF is a strongly pseudoη-invex function.

Theorem 3.9. Let the differentialF⁰(u)of a functionF(u)on the invex setK be stronglyη-pseudomonotone. If ConditionsBandDhold, thenF is a strongly pseudoη-invex function.

Theorem 3.10. Let the differentialF⁰(u)of a functionF(u)on the invex setK be η-pseudomonotone. If Conditions B andD hold, then F is a pseudo invex function.

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Theorem 3.11. Let the differentialF⁰(u)of a differentiableα-preinvex function F(u) be Lipschitz continuous on theα-invex set K with a constant β > 0.If ConditionAholds, then

F(v)−F(u)≤ hα(v, u)F⁰(u), η(v, u)i+β

2kα(v, u)η(v, u)k², ∀u, v ∈K.

Proof. ∀u, v ∈K, t ∈[0,1], u+tα(v, u)η(v, u)∈K,sinceKis an α-invex set. Now we consider the function

ϕ(t) = F(u+tα(v, u)η(v, u))−F(u)−thα(v, u)F⁰(u), η(v, u)i.

from which it follows thatϕ(0) = 0and

(3.9) ϕ⁰(t) =hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i

− hα(v, u)F⁰(u), η(v, u)i.

Integrating (3.9) between0and1,we have

ϕ(1) =F(u+α(v, u)η(v, u))−F(u)− hα(v, u)F⁰(u), η(v, u)i

≤ Z 1

0

|ϕ⁰(t)|dt

= Z 1

0

|hα(v, u)F⁰(u+tα(v, u)η(v, u)), η(v, u)i − hα(v, u)F⁰(u), η(v, u)i|dt

≤β Z 1

0

tkα(v, u)η(v, u)k²dt = β

2kα(v, u)η(v, u)k²,

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which implies that

(3.10) F(u+α(v, u)η(v, u))−F(u)

≤ hα(v, u)F⁰(u), η(v, u)i+β

2kα(v, u)η(v, u)k². from which, using ConditionA, we obtain

F(v)−F(u)≤ hα(v, u)F⁰(u), η(v, u)i+β

2kα(v, u)η(v, u)k².

Remark 3.1. Forη(v, u) =v−uandα(v, u) = 1,theα-invex setK becomes a convex set and consequently Theorem 3.11 reduces to the well known result in convexity, see [14].

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References

[1] M.A. HANSON, On sufficiency of the Kuhn-Tucker conditions, J. Math.

Anal. Appl., 80 (1981), 545–550.

[2] V. JEYAKUMAR, Strong and weak invexity in mathematical program- ming, Methods Oper. Research, 55 (1985), 109–125.

[3] V. JEYAKUMAR AND B. MOND, On generalized convex mathematical programming, J. Austral. Math. Society Ser. B., 34 (1992), 43–53.

[4] S.R. MOHAN ANDS.K. NEOGY, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908.

[5] M.A. NOOR, Generalized convex functions, PamAmer. Math. J., 4(3) (1994), 73–89.

[6] M.A. NOOR, Invex equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463–475.

[7] M.A. NOOR, Properties of preinvex functions, Pre-print, Etisalat College of Engineering, Sharjah, UAE, 2004.

[8] G. RUIZ-GARZION, R. OSUNA-GOMEZ AND A. RUFIAN-LIZAN, Generalized invex monotonicity, European J. Oper. Research, 144 (2003), 501–512.

[9] S. SCHAIBLE, Generalized monotonicity: concepts and uses, in: Varia- tional Inequalities and Network Equilibrium Problems (eds. F. Giannessi and A. Maugeri ), Plenum Press, New York, (1995), 289–299.

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[10] X.Q. YANG, Generalized convex functions and vector variational inequal- ities, J. Optim. Theory and Appl., 79 (1993), 563–580.

[11] X.M. YANG, X.Q. YANG ANDK.L. TEO, Criteria for generalized invex monotonicities, European J. Oper. Research, xxx (2004).

[12] X.M. YANG, X.Q. YANG ANDK.L. TEO, Generalized invexity and gen- eralized invariant monotonicity, J. Optim. Theory Appl., 117 (2003), 607–

625.

[13] T. WEIR AND B. MOND, Preinvex functions in multiobjective optimiza- tion, J. Math. Anal. Appl., 136 (1988), 29–38.

[14] D.L. ZHU AND P. MARCOTTE, Co-coercivity and its role in the con- vergence of iterative schemes for solving variational inequalities, Siam J.

Optim., 6 (1996), 714–726.