- Research Article
- Open access
- Published:
Generalized
Invariant Monotonicity and Generalized
Invexity of Nondifferentiable Functions
Journal of Inequalities and Applications volume 2009, Article number: 393940 (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 [4–6] 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ4_HTML.gif)
(2) is said to be strictly
invariant monotone on
with respect to
and
if for any
with
and any
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ5_HTML.gif)
Remark 3.2.
-
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ6_HTML.gif)
(2)the function is said to be strictly
invex with respect to
and
on
if for any
with
and any
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ7_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ12_HTML.gif)
then, by Definition 4.3, is
pseudoinvex function with respect to
and
on
. On the other hand, if the following is OK:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ13_HTML.gif)
then, by Definition 4.4, is strictly
pseudoinvex function with respect to
and
on
.
Thus, we must conclude that the following implication holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ17_HTML.gif)
Take .
When , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ18_HTML.gif)
Then, Clearly, by (4.7),
is strictly
quasi-invex with respect to
and
on
, but
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ19_HTML.gif)
Thus, is not strictly
invariant quasimonotone.
On the other hand, when ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ20_HTML.gif)
but
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ25_HTML.gif)
By quasi-invexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ26_HTML.gif)
Note that quasi-invexity of
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ27_HTML.gif)
Therefore, is a
invariant quasimonotone mapping with respect to the same
and
on
.
Condition C (see [12]).
Let Then, for any
and for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ29_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ30_HTML.gif)
but
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ31_HTML.gif)
By hypothesis 3 and (4.23), there exist ,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ32_HTML.gif)
It follows from Condition C, (4.25), and hypothesis 4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ33_HTML.gif)
Since is a
invariant quasimonotone mapping with respect to
and
on
, (4.26) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ34_HTML.gif)
From Condition C, hypothesis 4, and (4.27), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ37_HTML.gif)
(2) is said to be strictly
invariant pseudomonotone on
with respect to
and
if for any
with
and any
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ39_HTML.gif)
(2)the function is said to be strictly
pseudoinvex with respect to
and
on
if for any
with
and any
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ40_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ41_HTML.gif)
We need to show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ42_HTML.gif)
Suppose, on the contrary, , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ43_HTML.gif)
Since is a
pseudoinvex function with respect to
and
on
, (5.7) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ44_HTML.gif)
From hypothesis 3 and (5.8), such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ45_HTML.gif)
Hence, pseudoinvexity of
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ46_HTML.gif)
From Condition C, hypothesis 4, and (5.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ48_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ49_HTML.gif)
We need to show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ50_HTML.gif)
Assume, on the contrary,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ51_HTML.gif)
By hypothesis 3, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ52_HTML.gif)
From Condition C, hypothesis 4, and (5.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ53_HTML.gif)
Since is an invariant pseudomonotone mapping with respect to
and
, (5.17) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ54_HTML.gif)
From Condition C, hypothesis 4, and (5.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ55_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ56_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ59_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ61_HTML.gif)
Let . By the Mean Value Theorem, there exist
and
where
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ62_HTML.gif)
Hence, by Condition C, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ63_HTML.gif)
and there exists , where
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ64_HTML.gif)
Thus, by Condition C, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ65_HTML.gif)
On the other hand, it follows from hypothesis 4, Condition C, and (6.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ66_HTML.gif)
By strongly invariant pseudomonotonicity of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ67_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ68_HTML.gif)
Then, by (6.5), (6.9), and hypothesis 4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ69_HTML.gif)
Furthermore, by (6.7), (6.10), and hypothesis 4, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ70_HTML.gif)
Adding inequalities (6.11) and (6.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ71_HTML.gif)
By hypothesis 3, it is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ72_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ73_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393940/MediaObjects/13660_2008_Article_1949_Equ74_HTML.gif)
Under certain conditions, we obtain the relationships between generalized invexity in Diagram (7.2) and the corresponding generalized
invariant monotonicity in Diagram (7.1).
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Acknowledgment
This research was partially supported by the National Natural Science Foundation of China (no.10771228 and no.10831009).
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Liu, C., Yang, X. Generalized Invariant Monotonicity and Generalized
Invexity of Nondifferentiable Functions.
J Inequal Appl 2009, 393940 (2009). https://doi.org/10.1155/2009/393940
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DOI: https://doi.org/10.1155/2009/393940