Theorem 3.1.
For
,
, if there is
such that
, then
is an essential weakly efficient solution of (VMP). Hence, the component that contains the point
is an essential component.
Proof.
Suppose that
,
, and there exists
such that
. For any open neighborhood
of
in
, there is an open neighborhood
of
in
such that
, where
denotes the closure of 
Since
, and
, then
Take
such that
Then for any
with
, one has
Then
by (3.2) and (3.4), one has
therefore
Then,
, and hence
, which implies
. By Definition 2.6,
is an essential weakly efficient solution to (VMP). Hence, the component that contains the point
is an essential component.
Corollary 3.2 (see [2]).
When
, for
, if
is a singleton, then
is an essential optimum solution.
Lemma 3.3.
Let
be a nonempty compact convex subset of a Banach space, the function
continuous and strongly quasiconvex, then
is a singleton.
Proof.
Suppose
, and
. By Definition 2.11,
which is a contradiction, then
is a singleton.
By Theorem 3.1 and Lemma 3.3, we have the following Theorem 3.4.
Theorem 3.4.
Let
be a nonempty compact convex subset of a Banach space, for
, if there exists
such that
is strongly quasiconvex, then
has an essential weakly efficient solution, consequently,
has an essential component.
Theorem 3.5.
For
, if
has only one component
, then
is an essential component.
Proof.
By Lemma 2.4, the mapping
is upper semicontinuous at
, hence, for any open set
with
, there exists
such that for all
with
, one has
, and hence,
. By Definition 2.5,
is an essential component.
Remark 3.6.
When
, by [3], the optimum solution set
(where
) has an essential component if and only if
is connected. But, when
, it is not true.
Example 3.7.
Let
, define
is disconnected; however,
is an essential weakly efficient solution,
is an essential component, and
is an essential component.
Example 3.8.
Let
, define
Then
,
. By Theorem 3.1,
is an essential weakly efficient solution. But,
and
are not essential weakly efficient solutions. In fact, for
, take
,
, for all
, take
as the following:
Then
, and
But
. In fact, for all
, if
,
, take
, we have
If
, take
, we have
Thus
; therefore,
is not an essential weakly efficient solution. Similarly,
is not an essential weakly efficient solution.