Open Access

Generalized Invariant Monotonicity and Generalized Invexity of Nondifferentiable Functions

Journal of Inequalities and Applications20092009:393940

https://doi.org/10.1155/2009/393940

Received: 11 December 2008

Accepted: 9 February 2009

Published: 18 February 2009

Abstract

New concepts of generalized invex functions for non-differentiable functions and generalized invariant monotone operators for set-valued mappings are introduced. The relationships between generalized invexity of functions and generalized invariant monotonicity of the corresponding Clarke's subdifferentials are studied. Some of our results are extension and improvement of some results obtained in (Jabarootion and Zafarani (2006); Behera et al. (2008)).

1. Introduction

Convexity plays a central role in mathematical economics, engineering, management sciences, and optimization. In recent years several extensions and generalizations have been developed for classical convexity. An important generalization of convex functions is invex functions introduced by Hanson (1981) [1]. He has shown that the Kuhn-Tuker conditions are sufficient for optimality of nonlinear programming problems under invexity conditions. Kaul and Kaur (1985) [2] presented the conpects of pseudoinvex and quasi-invex functions and investigated their applications in nonlinear programming. A concept closely related to the convexity of function is the monotonicity of function; it is worth noting that monotonicity plays a very important role in the study of the existence and sensitivity analysis of solutions for variational inequality problems. An important breakthrough was given by Karamardian and Schaible (1990) [3]. They have proved that the generalized convexity of the function is equivalent to the generalized monotonicity of its gradient function . Motivated by the work of Karamardian and Schaible (1990), there has been increasing interest in the study of monotonicity and generalized monotonicity and their relationships to convexity and generalized convexity. Ruiz-Garzón et al. (2003) [4] introduced strongly invex and strongly pseudoinvex functions in and gave the sufficient conditions for (strictly, strongly) invex monotonicity, strictly pseudoinvex monotonicity, and quasi-invex monotonicity. Moreover, in [4] the necessary conditions for strictly pseudoinvex monotonicity and pseudoinvex monotonicity were obtained. The necessary conditions for strongly pseudoinvex monotonicity were given by Yang et al. (2005) [5]. The results on generalized invexity and generalized invex monotonicity obtained in [46] are studied in -dimensional Euclidean space. Several generalizations in real Banach space have been developed for generalized invexity and generalized invex monotonicity. Recently, Fan et al. (2003) [7] have studied the relationships between (strict, strong) convexity, pseudoconvexity, and quasiconvexity of functions and (strict, strong) monotonicity, pseudomonotonicity, and quasimonotonicity of its Clarke's generalized subdifferential mapping, respectively. Jabarootian and Zafarani (2006) [8] generalized convexity to invexity and obtained the relationships between several kinds of generalized invexity of functions and generalized invariant monotonicity of its Clarke's generalized subdifferential mapping.

Behera et al. (2008) [9] introduce the concepts of generalized invariant monotone operators and generalized invex functions and discuss the relationships between generalized invariant monotonicity and generalized invexity. Some examples are presented by Behera et al. to illustrate the proper generalizations for the corresponding concepts of generalized invariant monotone. However, it is noted that the definition of strictly quasi-invex is not defined precisely in [9], and Theorem 3.2 of [9] contains some errors. The purpose of this paper is to point out these errors and to suggest appropriate modifications. In real Banach space, we define new concepts of generalized invexity for non-differentiable functions and generalized invariant monotonicity for set-valued mappings which are extension and improvement of the corresponding definitions of [8, 9]. In [9], some sufficient conditions for generalized invariant pseudomonotonicity and generalized invariant quasimonotonicity were given. We will give some necessary conditions. We also introduce the concept of strongly invariant pseudomonotone and give its necessary conditions. Some results that obtained in this paper are the improvement of the corresponding results of [8, 9].

2. Preliminaries

In this paper, let be a real Banach space endowed with a norm and its dual space with a norm . We denote by , , , and the family of all nonempty subsets of , the dual pair between and , the line segment for , and the interior of , respectively. Let be a nonempty open subset of , a set-valued mapping, a vector-valued function, and a non-differentiable real-valued function. The following concepts and results are taken from [10].

Definition 2.1.

Let be locally Lipschitz continuous at a given point and any vector in . The Clarke's generalized directional derivative of at in the direction , denoted by , is defined by
(2.1)

Definition 2.2.

Let be locally Lipschitz continuous at a given point and let be any vector in . The Clarke's generalized subdifferential of at , denoted by , is defined as follows:
(2.2)

Lemma 2.3 (Mean Value Theorem).

Let and be point in and suppose that is Lipschitz near each point of a nonempty closed convex set containing the line segment . Then there exists a point such that
(2.3)

3. Invariant Monotonicity and Invexity

Let be nonempty subset of and let and be two vector-valued functions from to and

Definition 3.1.

Let be nonempty subset of and let be a set-valued mapping:

(1) is said to be invariant monotone on with respect to and if for any and any , , one has
(3.1)
(2) is said to be strictly invariant monotone on with respect to and if for any with and any , , one has
(3.2)
Remark 3.2.
  1. (1)

    When , it is [9, Definition 2.3].

     

(2)When , every (strictly) invariant monotone function is a (strictly) invariant monotone function defined by Jabarootian and Zafarani [8], but the converse is not true. Examples 2.1 and 2.2 of [9] are two counterexamples, where is defined as .

(3)When , , Definition 3.1(1) is the Definition 3.2(3) in [8], that is, strongly invariant monotone.

Next we introduce invex functions under non-differentiable condition.

Definition 3.3.

Let be nonempty subset of , and let be locally Lipschitz continuous on . Then,

(1)the function is said to be invex with respect to and on if for any and any , one has
(3.3)
(2)the function is said to be strictly invex with respect to and on if for any with and any , one has
(3.4)

The following lemma under non-differentiable condition is similar to Lemmas 2.3 and 2.4 under Fréchet differentiable condition in [9].

Lemma 3.4.

Let be locally Lipschitz on . If is (strictly) invex with respect to and on , then is (strictly) invariant monotone with respect to the same and on .

4. Invariant Quasimonotonicity and Quasi-Invexity

In this section, we will point out some errors in [9].

Definition 4.1 (see [11]).

A set is said to be invex with respect to if there exists an such that, for any and ,
(4.1)

Definition 4.2 (see [9, Definition 3.2]).

A Fréchet differentiable function is said to be quasi-invex with respect to and on if for any with , we have
(4.2)

If strict inequality holds, then it is said to be strictly quasi-invex, where is the Fréchet differential of .

Definition 4.3 (see [9, Definition 4.2]).

A Fréchet differentiable function is said to be pseudoinvex with respect to and on if for any with , we have
(4.3)

Definition 4.4 (see [9, Definition 5.2]).

A Fréchet differentiable function is said to be strictly pseudoinvex with respect to and on if for any with , we have
(4.4)

Remark 4.5.

In Definition 4.2, the definition of strictly quasi-invex is not defined precisely, as can be seen below. If it holds that
(4.5)
then, by Definition 4.3, is pseudoinvex function with respect to and on . On the other hand, if the following is OK:
(4.6)

then, by Definition 4.4, is strictly pseudoinvex function with respect to and on .

Thus, we must conclude that the following implication holds
(4.7)

that is, the "strict inequality" for strictly quasi-invex functions in Definition 4.2 means (4.7).

Definition 4.6 (see [9, Definition 3.1]).

A function is said to be invariant quasimonotone (strictly) with respect to and on if for any with , we have
(4.8)

Definition 4.7 (see [9, Definition 4.1]).

A function is said to be invariant pseudomonotone with respect to and on if for any ,we have
(4.9)

The following Theorem is due to Behera et al. in [9].

Theorem 4 (see [9, Theorem 3.2]).

Let be an invex set with respect to , and let be Fréchet differentiable on . If is strictly quasi-invex with respect to and on , then is a strictly invariant quasimonotone function with respect to the same and on .

Remark 4.8.

By Remark 4.5, Theorem A is not true as can be seen from the following example.

Example 4.9.

Let , , and be functions defined by
(4.10)

Take .

When , we have
(4.11)
Then, Clearly, by (4.7), is strictly quasi-invex with respect to and on , but
(4.12)

Thus, is not strictly invariant quasimonotone.

On the other hand, when ,
(4.13)
but
(4.14)

Thus, is not invariant pseudomonotone.

Furthermore, is invariant quasimonotone. Hence, Example 4.9 also illustrates that invariant quasimonotone is not necessarily strictly invariant quasimonotone or invariant pseudomonotone.

In [9], sufficient conditions for invariant quasimonotone for Fréchet differentiable functions were given. However, necessary conditions have been missing. In what follows, we will give the necessary conditions for invariant quasimonotonity and study the relationship between invariant quasimonotonicity and quasi-invexity.

Definition 4.10.

Let be a nonempty subset of and is said to be invariant quasimonotone on with respect to and if for any and any , , one has
(4.15)

Remark 4.11.

When , every invariant quasimonotone function is an invariant quasimonotone function defined by Jabarootian and Zafarani [8], but the converse is not true. See [9, Example 3.1], for a counterexample, where
(4.16)

Definition 4.12.

Let be a nonempty subset of , and let be locally Lipschitz continuous on . Then, the function is said to be quasi-invex with respect to and on if for any and any , one has
(4.17)

The following theorem under non-differentiable condition is similar to Theorem 3.1 of Behera et al. in [9].

Theorem 4.13.

Let be locally Lipschitz continuous on . If is quasi-invex with respect to and on , then is a invariant quasimonotone mapping with respect to the same and on .

Proof.

Suppose is quasi-invex with respect to and on . Let , , and be such that
(4.18)
By quasi-invexity of , we have
(4.19)
Note that quasi-invexity of implies
(4.20)

Therefore, is a invariant quasimonotone mapping with respect to the same and on .

Condition C (see [12]).

Let Then, for any and for any ,
(4.21)

Definition 4.14 (see [4]).

A function is said to be a skew function if for all

Now we will give the sufficient conditions for quasi-invexity.

Theorem 4.15.

Let be an open invex set with respect to , and let be locally Lipschitz continuous on . Suppose that

(1) is a invariant quasimonotone mapping with respect to and on ;

(2) satisfies Condition C;

(3)for each and , there exist and , such that
(4.22)

(4)

Then, is a quasi-invex function with respect to the same and on .

Proof.

Suppose that is not a quasi-invex function with respect to and on . Then, there exist and , such that
(4.23)
but
(4.24)
By hypothesis 3 and (4.23), there exist , such that
(4.25)
It follows from Condition C, (4.25), and hypothesis 4 that
(4.26)
Since is a invariant quasimonotone mapping with respect to and on , (4.26) implies
(4.27)
From Condition C, hypothesis 4, and (4.27), we obtain
(4.28)

which contradicts (4.24). Hence, is a quasi-invex function with respect to the same and on .

Similar to proof for Theorem 4.15, we can obtain the following theorem.

Theorem 4.16.

Let be an open convex subset of , and let be locally Lipschitz continuous on . Suppose that

(1) is a invariant quasimonotone mapping with respect to and on ;

(2) and are affine in the first argument and skew;

(3)for each and , there exist , , where , such that
(4.29)

Then, is a quasi-invex function with respect to the same and on .

5. Invariant Pseudomonotonicity and Pseudoinvexity

Definition 5.1.

Let be a nonempty subset of , and let be a set-valued mapping. Then,

(1) is said to be invariant pseudomonotone on with respect to and if for any and any , , one has
(5.1)
(2) is said to be strictly invariant pseudomonotone on with respect to and if for any with and any , , one has
(5.2)

Remark 5.2.

When , every (strictly) invariant pseudomonotone function is (strictly) invariant pseudomonotone function [8] on with respect to the same , but the converse is not ture. Examples 4.1 and 5.1 of [9] are two counterexamples, where is defined as .

Definition 5.3.

Let be a nonempty subset of , and let be locally Lipschitz continuous on . Then,

(1)the function is said to be pseudoinvex with respect to and on if for any and any , one has
(5.3)
(2)the function is said to be strictly pseudoinvex with respect to and on if for any with and any , one has
(5.4)

In this section we will give sufficient conditions and necessary conditions for invariant pseudomonotonicity.

Theorem 5.4.

Let be an open invex set with respect to , and let be locally Lipschitz continuous on . Suppose that

(1) is a pseudoinvex function with respect to and on ;

(2) satisfies Condition C;

(3)for any and , there exists such that ;

(4)

Then, is a invariant pseudomonotone mapping with respect to the same and on .

Proof.

Suppose that is a pseudoinvex function with respect to and on . Let . If ,
(5.5)
We need to show
(5.6)
Suppose, on the contrary, , such that
(5.7)
Since is a pseudoinvex function with respect to and on , (5.7) implies
(5.8)
From hypothesis 3 and (5.8), such that
(5.9)
Hence, pseudoinvexity of implies
(5.10)
From Condition C, hypothesis 4, and (5.10), we have
(5.11)

which contradicts (5.5). Hence, is a invariant pseudomonotone mapping with respect to the same and on .

Corollary 5.5.

Let be an open invex set with respect to , and let be locally Lipschitz continuous on , and let satisfy Condition C. If, for any

(1) such that ;

(2) is a pseudoinvex function with respect to on ,

then, is an invariant pseudomonotone mapping with respect to the same on .

Theorem 5.6.

Let be an open invex set with respect to and let be locally Lipschitz continuous on . Suppose that

(1) is a invariant pseudomonotone mapping with respect to and on ;

(2) satisfies Condition C;

(3)for any and , there exist , , such that
(5.12)

(4)

Then, is a pseudoinvex function with respect to the same and on .

Proof.

Suppose that is a invariant pseudomonotone mapping with respect to and on . Let for all be such that
(5.13)
We need to show
(5.14)
Assume, on the contrary,
(5.15)
By hypothesis 3, such that
(5.16)
From Condition C, hypothesis 4, and (5.16), we have
(5.17)
Since is an invariant pseudomonotone mapping with respect to and , (5.17) implies
(5.18)
From Condition C, hypothesis 4, and (5.18), we obtain
(5.19)

which contradicts (5.13). Hence, is a pseudoinvex function with respect to the same and on .

Corollary 5.7.

Let be an open invex set with respect to , let be locally Lipschitz continuous on , and let satisfy Condition C. For any , such that
(5.20)

If is an invariant pseudomonotone mapping with respect to on , then is a pseudoinvex function with respect to the same on .

Similar to proof for Theorems 5.4 and 5.6, we can obtain the following two theorems.

Theorem 5.8.

Let be an open convex subset of , and let be locally Lipschitz continuous on . Suppose that

(1) is a pseudoinvex function with respect to and on ;

(2) and are affine in the first argument and skew;

(3)for any and , there exists such that
(5.21)

Then, is a invariant pseudomonotone mapping with respect to the same and on .

Theorem 5.9.

Let be an open convex subset of , and let be locally Lipschitz continuous on . Suppose that

(1) is a invariant pseudomonotone mapping with respect to and on ;

(2) and are affine in the first argument and skew;

(3)for each and , there exist , , where such that
(5.22)

Then, is a pseudoinvex function with respect to the same and on .

6. Strongly Invariant Pseudomonotonicity and Strongly Pseudoinvexity

In this section, we introduce the concepts of strongly invariant pseudomonotonicity and strongly pseudoinvexity. We will give a necessary condition for strongly invariant pseudomonotonicity.

Definition 6.1.

Let be a nonempty subset of . Then, is said to be strongly invariant pseudomonotone on with respect to and if there exists a constant , such that for any and any , , one has
(6.1)

Definition 6.2.

Let be a nonempty subset of , and let be locally Lipschitz continuous on . Then, the function is said to be strongly pseudoinvex with respect to and on , if there exists a constant , such that for any and any , one has
(6.2)

Theorem 6.3.

Let be an open invex set with respect to , and let be locally Lipschitz continuous on . Suppose that

(1) is a strongly invariant pseudomonotone with respect to and on ;

(2) satisfies Condition C;

(3) , for any ;

(4)

Then, is a strongly pseudoinvex function with respect to the same and on .

Proof.

Suppose that is strongly invariant pseudomonotone with respect to and on . Let and for all be such that
(6.3)
Let . By the Mean Value Theorem, there exist and where , such that
(6.4)
Hence, by Condition C, we have
(6.5)
and there exists , where , such that
(6.6)
Thus, by Condition C, we have
(6.7)
On the other hand, it follows from hypothesis 4, Condition C, and (6.3) that
(6.8)
By strongly invariant pseudomonotonicity of , we obtain
(6.9)
(6.10)
Then, by (6.5), (6.9), and hypothesis 4, we have
(6.11)
Furthermore, by (6.7), (6.10), and hypothesis 4, we obtain
(6.12)
Adding inequalities (6.11) and (6.12), we have
(6.13)
By hypothesis 3, it is clear that
(6.14)

Thus, is a strongly pseudoinvex function with respect to the same and on .

7. Conclusions

In this paper, we introduced various concepts of generalized invariant monotonicity and established their relations with generalized invexity. Diagram (7.1) summarizes these relations, where means that the implication relation holds and means that the implication relation does not hold:
(7.1)

where IM stands for invariant monotonicity, IPM stands for invariant pseudomonotonicity, and IQM stands for invariant quasimonotonicity.

For the generalized invexity of a real-valued function, the relations in Diagram (7.2) hold:
(7.2)

Under certain conditions, we obtain the relationships between generalized invexity in Diagram (7.2) and the corresponding generalized invariant monotonicity in Diagram (7.1).

Declarations

Acknowledgment

This research was partially supported by the National Natural Science Foundation of China (no.10771228 and no.10831009).

Authors’ Affiliations

(1)
College of Mathematics Science, Inner Mongolia University
(2)
College of Mathematics and Computer Science, Chongqing Normal University

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Copyright

© C. Liu and X. Yang. 2009

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