Volume 2011, Article ID 914624,21pages doi:10.1155/2011/914624
Research Article
Critical Point Theorems and Ekeland Type
Variational Principle with Applications
Lai-Jiu Lin,
1Sung-Yu Wang,
1and Qamrul Hasan Ansari
21Department of Mathematics, National Changhua University of Education, Changhua 50058, China 2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, Taiwan
Correspondence should be addressed to Lai-Jiu Lin,[email protected]
Received 28 September 2010; Accepted 9 December 2010 Academic Editor: S. Al-Homidan
Copyrightq 2011 Lai-Jiu Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the notion of λ-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang2007. We establish some critical point theorems in the setting of λ-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al.1983 and the results given by Khanh and Quy 2010 to
λ-spaces and cone metric spaces. As applications of our results, we characterize the completeness
of λ-space cone metric spaces and quasimetric spaces are special cases of λ-space and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.
1. Introduction
In the last three decades, the famous Ekeland’s variational principlein short, EVP 1 see
also, 2,3 emerged as one of the most important tools and results in nonlinear analysis
due to its wide applications in optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, and so forth; see, for example,2–8 and references therein.
It has been extended and generalized in different directions and in different settings. See, for example, 4, 5, 7–22 and references therein. The vectorial version of EVP in short,
VEVP is considered and studied in 5,22,23 and references therein. Aubin and Frankowska
4 presented the equilibrium version of EVP in short, EEVP in the setting of complete
metric spaces. Such version of EVP is further studied in 9, 11, 21 with applications to
an equilibrium problem which is a unified model of several problems, namely, variational inequalities, complementarity problems, fixed point problem, optimization problem, Nash equilibrium problem, saddle point problem, and so forth; see, for example, 24, 25 and
spaces with a Q-function which generalizes the notions of τ-function 17 and a w-distance
8. They proved some equivalences of EEVP with a fixed point theorem of Caristi-Kirk
type for multivalued maps 12, Takahashi’s minimization theorem 8, and some other
related results. As applications, they derived the existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. They also extended the Nadler’s fixed point theorem 26 for multivalued maps to a Q-function in the setting of complete
quasimetric spaces. As a consequence, they proved the Banach contraction theorem for a Q-function in the setting of complete quasimetric spaces. Ansari10 extended EEVP for vector
valued functions in the setting of complete quasimetric spaces with a W-distance. By using this result, he derived the existence of solutions of a vector equilibrium problem25, that is,
equilibrium problem for vector valued functions. He established some equivalent results to EEVP for vector valued functions and also established Caristi-Kirk fixed point theorem for multivalued maps12 in a more general setting.
In 1983, Dancs et al.27 established a critical point theorem in the setting of complete
metric spaces and proved EVP by using their result. Lin et al.18,19 used the critical point
theorem of Dancs et al.27 to establish EEVP for multivalued bifunctions. They also studied
intersection theorems, variational inclusion problems, and some other related problems by using the critical point theorem of Dancs et al. 27. Khanh and Quy 15 generalized the
critical point theorem of Dancs et al.27 in the setting of metric spaces with τ-functions 17.
Khanh and Quy15,16 used their result to established EEVP for vector valued functions.
In this paper, we introduce the concepts of λ-function and λ-space which are much weaker than those of cone metric and cone metric space defined by Huang and X. Zhang 28, respectively. We establish some critical point theorems in the setting of λ-spaces and, in
particular, in the setting of complete cone metric spaces. Our results generalize those results given in15,27. As applications of our results, we characterize the completeness of cone
metric spaces and give VEVP and EEVP for multivalued maps in the setting of cone metric spaces. Moreover, we improve and generalize many Ekeland type variational principles and critical point theorems.
2. Preliminaries
Let E be a topological vector space with origin 0. For a given nontrivial, pointed, closed and convex cone P ⊂ E, we define on E a partial ordering ≤ resp., < with respect to P by x ≤ y resp., x < y if y − x ∈ P resp., y − x ∈ P \ {0}, and we denote by x y if y − x ∈ int P interior of P. When E is a normed space, the nontrivial, pointed, closed, and convex cone P is said to be normal cone if there exists K > 0 such that for all x, y ∈ E,
0 ≤ x ≤ y implies x ≤ Ky. 2.1
The least positive number K satisfying 2.1 is called the normal constant of P.
Throughout the paper, unless otherwise specified, we assume that X is a nonempty set, E is a real topological vector space with origin 0 ordered by a nontrivial, pointed, closed and convex cone P and λ : X × X → E is a vector valued function. We denote by 2X the family of all subsets of X. If E is a normed space, for each s ∈ E, r ∈Ê 0, ∞, we use the
following notations:
i Bs, r {y ∈ E : s − y < r}, ii B∗s, r {y ∈ E : s − y < r} \ {s}.
Definition 2.1see 28. Let X be a nonempty set, and let E be a topological vector space
ordered by a nontrivial, pointed, and closed convex cone P . Let d : X × X → E be a vector valued function such that the following conditions hold:
d1 0 ≤ dx, y for all x, y ∈ X and dx, y 0 if and only if x y, d2 dx, y dy, x for all x, y ∈ X,
d3 dx, y ≤ dx, z dz, y for all x, y, z ∈ X.
Then d is called a cone metric on X, and the set X with a cone metric d is called a cone metric
space and it is denoted byX, d.
Definition 2.2. A vector valued function λ : X × X → E is said to be a λ-function if for all x, y ∈ X,
λ1 λx, y ≥ 0,
λ2 x / y ⇒ λx, y / 0.
A nonempty set X with a λ-function is called a λ-space, and it is denoted by X, λ.
Clearly, every cone metric space is a λ-space, but the converse is not true; see below,
Example 2.4.
Definition 2.3. i A sequence {xn}n∈in a λ-space X, λ is said to be the following:
a λ-Cauchy sequence resp., quasi-λ-Cauchy sequence if for every c ∈ E with c 0, there exists a positive integer N such that λxn, xm c resp., λxn, xm < c for
all n, m ≥ N.
b λ-convergent resp., quasi-λ-convergent if there exists x ∈ X such that for every
c ∈ E with c 0, there exists a positive integer N such that λxn, x c resp.,
λxn, x < c for all n ≥ N. In this case, we say that {xn} λ-converges resp.,
quasi-λ-converges to x in X, λ, and we denote it by xn−→ x resp., xλ n q−λ
−−−→ x. The point
x ∈ X is called a λ-limit point resp., quasi-λ-limit point of the sequence {xn}.
ii A λ-space X, λ is said to be λ-complete resp., quasi-λ-complete if every λ-Cauchy sequence resp., quasi-λ-Cauchy sequence is a λ-convergent resp., quasi-λ-convergent sequence.
iii A subset D of a λ-space X, λ is said to be the following:
a λ-closed resp., quasi-λ-closed in X, λ if for every x ∈ X with a sequence {xn} ⊂ D
such that{xn}λ-converges resp., quasi-λ-converges to x in X, λ, then x ∈ D; the
λ-closure of a set D in X, λ is the intersection of all λ-closed sets containing D.
b λ-open resp., quasi-λ-open in X, λ if Dc X \ D the complement of D in X is
λ-closed resp., quasi-λ-closed.
If E Ê and P Ê, then the definitions of quasi-λ-Cauchy sequence and quasi-λ-convergent
sequence are equal to the ones of λ-Cauchy sequence and λ-convergent sequence, respectively. Example 2.4. Let X E Êand P Ê. Define λ : X × X → E by
λx, y ⎧ ⎨ ⎩ x − y if x, y ∈É rational numbers, 1 otherwise. 2.2
Then, λ is a λ-function and X, λ is a λ-space. But X, λ is not a cone metric space. Furthermore, the subsetÉis λ-closed in X, λ.
Proof. If x ∈ X with a net {xn}n∈ ⊂
É such that xn
λ
−→ x, then x ∈ É. Indeed, suppose that
x /∈É, then λxn, x 1 for all n ∈Æ. This contradicts the fact that xn
λ
−→ x.
Lemma 2.5. If xn−→ x, then for every subsequence {xλ nk} of {xn}, one has xnk −→ x. Similar resultsλ
for λ-Cauchy sequences, quasi-λ-convergent sequences and quasi-λ-Cauchy sequence also hold. Proof. By definition, for any c 0, there exists a positive integer N such that λxn, x c
for all n ≥ N. For every subsequence {xnk} of {xn}, we have λxn, x c for all nk ≥ N and
hence{xnk} also λ-converges to x.
Remark 2.6. a If X, d is a cone metric space, and if we replace λ by d inDefinition 2.3i
andii, we obtain the definitions of a d-Cauchy sequence and d-convergent sequence in a cone metric space and the definition of a d-complete cone metric space in 28, respectively.
If there is no danger of confusion, then we will not use the letter d before these definitions. b As it is proved in 28, every convergent sequence is a Cauchy sequence in a cone
metric space. But this assertion is not true for a λ-convergent sequence in X, λ. For example, let X E Êand P Ê. Define λ : X × X → E by
λx, y ⎧ ⎨ ⎩ x − y if x 0 or y 0, 1 otherwise. 2.3
Then,{1/n}n∈is a λ-convergent sequence with λ-limit 0, but it is not a λ-Cauchy sequence.
c With the help of λ-open resp., open sets and λ-closed resp., quasi-λ-closed sets, we can easily endow X with topology which would be weaker than the topology generated by quasimetric spaces and cone metric spaces.
Proof. i It is obvious that the empty set ∅ and X are λ-closed sets.
ii Let A and B be λ-closed sets, and let {xn} ⊂ A ∪ B be a sequence such that xn−→ xλ
for some x ∈ X Note that this x may not be unique. For each such x with xn −→ x, withoutλ
loss of generality, we can extract a subsequence{xnk} ⊂ A of {xn}. Since {xnk}λ-converges to
x in X, λ and A is λ-closed, then x ∈ A ⊂ A ∪ B, and therefore the set A ∪ B is λ-closed.
iii Let {Ai}i∈Ibe any family of λ-closed sets, and let {yn} ⊂
i∈IAibe any sequence
which λ-converges to y in X, λ. Then, for each i ∈ I, {yn} ⊂ Aiand yn −→ y ∈ Aλ isince Aiis
λ-closed. Therefore, y ∈ i∈IAi, and hence i∈IAiis a λ-closed set. Therefore, λ-space X, λ
is a topological space with the topology consists of all λ-open sets.
The proof for the case of quasi-λ-closed sets or quasi-λ-open sets lies on the same lines of the above proof.
Definition 2.7. Let A be a nonempty subset of a λ-space X, λ, and let {An} be a sequence of
nonempty subsets inX, λ. We adopt the following notations.
i δA < c for some c ∈ E with c ≥ 0 if λx, y < c for all x, y ∈ A.
ii ρA sup{λx, y : x, y ∈ A} if E is a normed vector space with an ordered cone
iii δAn −→ 0 of the first type if for every c ∈ E with c > 0, there exists a positiveλ
integer N such that δAn < c for all n ≥ N.
iv δAn−→ 0 of the second type if for every c ∈ E with c 0, there exists a positiveλ
integer N such that δAn < c for all n ≥ N.
v δAn−→ 0 of the first type w.r.t. {yλ n} ⊆ X if for every c ∈ E with c > 0, there exists
a positive integer N such that for each n ≥ N, we have λyn, u < c for all u ∈ An.
vi δAn −→ 0 of the second type w.r.t. {yλ n} ⊆ X if for every c ∈ E with c 0, there
exists a positive integer N such that for each n ≥ N, we have λyn, u < c for all
u ∈ An.
Now, we recall the definitions of τ-functions and weak τ-functions.
Definition 2.8see 17. Let X, d be a metric space. A function p : X × X → Ê is said to be
a τ-function if the following conditions are satisfied. τ1 For all x, y, z ∈ X, px, z ≤ px, y py, z. τ2 px, · isÊ-lower semicontinuous, for each x ∈ X.
τ3 For any sequences {xn} and {yn} in X with lim supn → ∞{pxn, xm : m > n} 0 and
limn → ∞pxn, yn 0, one has limn → ∞dxn, yn 0.
τ4 For x, y, z ∈ X, px, y 0, and px, z 0 imply y z.
Definition 2.9. LetX, d be a quasimetric space i.e., symmetricity is not required. A function p : X × X → Ê is said to be a weak τ-function if the conditions τ1, τ3, and τ4 hold.
Remark 2.10. The definition of weak τ-functions on a metric space is given in 15.
Definition 2.11see 27. Let F : X → 2X be a multivalued map. A point x ∈ X is said to be
a critical point of F if and only if Fx {x}.
3. Critical Point Theorems
We present the following critical point theorem in the setting of λ-spaces.
Theorem 3.1. Let X, λ be a quasi-λ-complete space, and let F : X → 2X be a multivalued map
with nonempty quasi-λ-closed values. Assume that
i for all x, y ∈ X, y ∈ Fx implies Fy ⊆ Fx,
ii for every sequence {xn} with xn1∈ Fxn, one has δFxn−→ 0 of the first type.λ
Then, for eachx ∈ X, there exists x∗∈ Fx such that Fx∗ {x∗} and λx∗, x∗ 0.
Proof. For any fixed x ∈ X, let x1 x and take xn1 ∈ Fxn for all n ∈ Æ. By conditionsi
andii, we have Fxn1 ⊆ Fxn for all n ∈Æ and δFxn
λ
−→ 0 of the first type. So, for any
c ∈ E with c > 0, there exists a positive integer N such that λyn, zn
Therefore,{xn} is a quasi-λ-Cauchy sequence in X, λ. Since X, λ is a quasi-λ-complete,
{xn} quasi-λ-converges to some point x in X, λ Note that the above quasi-λ-limit point x
may not be unique. For each quasi-λ-limit point x∗of{x
n}, x∗ ∈ Fxn for all n ∈Æ because
Fxn is quasi-λ-closed and Fxn1 ⊆ Fxn for all n ∈Æ. Since δFxn
λ
−→ 0 of the first type, we have
n∈
Fxn {x∗}. 3.2
Indeed, if there exists y ∈ n∈Fxn with y / x
∗, let c λx∗, y > 0. Since δFx
n−→λ
0 of the first type, there exists a positive integer Nsuch that δFx
n < c/2 for all n ≥ N.
Then, c λx∗, y < c/2. This leads to a contradiction. Therefore, ∅ / Fx∗ ⊆ n∈Fxn
{x∗}, and hence Fx∗ {x∗}. Therefore, for each quasi-λ-limit point x∗ of {x
n}, we have
Fx∗ n∈Fxn {x
∗} and hence the net {xn} has a unique quasi-λ-limit point, say x with Fx {x}. Further, since δFxn−→ 0 of the first type, λx, x 0.λ
Example 3.2. Let X E Êand P Ê. Define λ : X × X → E by
λx, y max |x|,y . 3.3
ThenX, λ is a λ-space but it is neither a cone metric space nor a quasimetric space. If we define a multivalued map F on X by
Fx ⎧ ⎪ ⎨ ⎪ ⎩ −|x| 2 , |x| 2 if x / 0, {0} if x 0. 3.4
Then, all the conditions ofTheorem 3.1are satisfied, and hence there exists a critical point of
F in X, λ, but neither 27, Theorem 3.1 nor 15, Lemma 3.4 is applicable in this example.
In fact, F0 {0} and λ0, 0 0.
Remark 3.3. When E Ê, P ÊandX, λ is a metric space,Theorem 3.1reduces to Theorem
3.1 in27.
For a transitive relationÊi.e., xÊy and yÊz imply xÊz in a topological space Y , we
say that
iÊ is lower closed if for any Ê-monotone i.e., · · ·ÊxnÊ· · ·Êx2Êx1 convergent
sequence xn → x one has xÊxnfor all n ∈Æ,
ii a subset A ⊆ Y is Ê-complete if any Cauchy sequence in A if the definition of
Cauchy sequence is given which isÊ-monotone converges to a point of A.
Remark 3.4. If the relationÊonX, λ inTheorem 3.1is given by
xÊy ⇐⇒ x ∈ F
then the assumptions “F has quasi-λ-closed values” and “quasi-λ-completeness of X, λ” in
Theorem 3.1in fact, in all the results of this paper can be replaced by the assumptions “Êis
lower closed” and “Fx isÊ-complete for all x ∈ X”, respectively.
Proof. Construct a net{xα} in the same manner as inTheorem 3.1. From conditionsi and
ii ofTheorem 3.1, we have xn1Êxnfor all n ∈Æ and δFxn
λ
−→ 0 of the first type. So, for any c ∈ E with c > 0, there exists a positive integer N such that
λyn, zn
< c ∀yn, zn∈ Fxn whenever n ≥ N. 3.6
Therefore, {xn} is a Ê-monotone quasi-λ-Cauchy sequence in X, λ. Since X, λ is Ê
-complete, there existsx ∈ X such that xÊxnfor all n ∈ Æ note that the above limit x may
not be unique. Following the same argument as in the proof ofTheorem 3.1, we obtain the conclusion.
Remark 3.5. InTheorem 3.1, ifX, λ is λ-complete not necessarily quasi-λ-complete, F has
λ-closed values not necessarily quasi-λ-closed values, and if we assume further that the
vector space E is a normed space, then the condition ii ofTheorem 3.1can be replaced by the following condition.
ii
For every sequence{xn} with xn1∈ Fxn, we have limn → ∞ρFxn 0.
Proof. Let λ: X × X → Ê be defined as
λx, yλx, y ∀x,y ∈ X. 3.7
Then, a λ-Cauchy sequence inX, λ is a λ-Cauchy sequence in X, λ.
For it, let{xn} be a λ-Cauchy sequence inX, λ. Since c 0 means that c ∈ int P,
for each fixed c ∈ E with c 0, there exists ε > 0 such that Bc, ε ⊂ int P . Since
limn → ∞ρFxn 0, there exists a positive integer N such that λx, y < ε for all
x, y ∈ Fxn whenever n ≥ N. Then, c − λx, y ∈ int P for all x, y ∈ Fxn whenever n ≥ N,
and hence λx, y c for all x, y ∈ Fxn whenever n ≥ N. Therefore, sequence {xn} is a
λ-Cauchy sequence in X, λ.
SinceX, λ is λ-complete, a λ-Cauchy sequence inX, λ is λ-convergent in X, λ. For any fixedx ∈ X, by the same argument as in the proof ofTheorem 3.1, there exists
x∗∈ X such that x∗∈ Fxn for all n ∈Æ with x1 x. Since limn → ∞ρFxn 0, we have
n∈
Fxn {x∗}. 3.8
Indeed, if there exists y ∈ n∈Fxn with y / x
∗, let c λx∗, y > 0, then c λx∗, y ≤ ρFx
n ∀n ∈Æ. 3.9
Since limn → ∞ρFxn 0, 0 < c ≤ 0. This leads to a contradiction. Therefore, ∅ / Fx∗ ⊆
n∈Fxn {x
∗} for all λ-limit points x∗ of{x
n}. Then, following the same argument as
Lemma 3.6. Let X, λ be a λ-space, and let E be a normed vector space with an ordering cone P. Let {An} be a sequence of subsets of X such that An1 ⊆ An for all n ∈ Æ. Then, limn → ∞ρAn 0
implies δAn−→ 0 of the second type in X, λ. Moreover, the converse holds if P is a normal cone.λ
Proof. Let limn → ∞ρAn 0. For any fixed c ∈ E with c 0, there exists ε > 0 such that
Bc, ε ⊂ int P . Since limn → ∞ρAn 0, there exists a positive N such that
λ x, y< ε ∀x, y ∈ An whenever n ≥ N. 3.10 Then, c − λx, y∈ int P ∀x, y ∈ An whenever n ≥ N, 3.11 and hence λx, y c ∀x, y ∈ Anwhenever n ≥ N. 3.12
Therefore, δAn−→ 0 of the second type in X, λ.λ
Conversely, assume that E is ordered by a normal cone P and δAn−→ 0 of the secondλ
type inX, λ. For any fixed c∈ E with c 0, there exists a positive Nsuch that
λx, y< c ∀x, y ∈ An whenever n ≥ N. 3.13
Then,
λ
x, y ≤ Kc ∀x,y ∈ An whenever n ≥ N, 3.14
where K is the normal constant of P . Since cwas arbitrary, the proof is completed.
Lemma 3.7. Let E be a normed vector space ordered by a normal cone P, then the following statements
are equivalent.
i {xn} is a λ-Cauchy sequence in X, λ.
ii For every ε > 0, there exists a positive N such that λxi, xj < ε for all i, j ≥ N.
Proof. It follows by taking An {xk}∞kninLemma 3.6.
Remark 3.8. Note that the topological space X, d in 28 is assumed to be a cone metric
space. Therefore,Lemma 3.7generalizes28, Lemma 4Example 3.9; see below. Further, as a consequence ofRemark 3.4andLemma 3.6, if E is assumed to be a normed vector space with a normal ordering cone P , X, λ is λ-complete not necessarily quasi-λ-complete and the values of mapping F is λ-closed not necessarily quasi-λ-closed. Then, the condition ii ofTheorem 3.1can be replaced by the following.
ii For every sequence {xn} with xn1 ∈ Fxn, we have δFxn −→ 0 of the secondλ
Example 3.9. InExample 3.2,X, λ is a λ-space but not a cone metric space. ByLemma 3.7, {1/n}n∈is a λ-Cauchy sequence but 28, Lemma 4 is not applicable.
Now, we establish a critical point theorem in the setting of cone metric spaces. Theorem 3.10. Let X, d be a complete cone metric space, E a normed vector space, and F : X → 2X
a multivalued map with nonempty closed values. Assume that
i for all x, y ∈ X, y ∈ Fx implies Fy ⊆ Fx,
ii for every sequence {xn} with xn1∈ Fxn, one has limn → ∞dxn, xn1 0.
Then, for eachx ∈ X, there exists x∗∈ Fx such that Fx∗ {x∗}.
Proof. Without loss of generality, we may assume that for each x ∈ X, Fx is bounded; that
is, ρFx exists. For any given ε > 0 and any fixed element x ∈ X, let x1 x and choose
x2∈ Fx1 such that
dx1, x2 > ρFx1
2 −
ε
2. 3.15
Continuing in this way, we obtain a sequence{xn}n∈such that xn1∈ Fxn and
dxn, xn1 >
ρFxn
2 −
ε
2n, ∀n ∈Æ. 3.16
Since limn → ∞dxn, xn1 0 and ε is arbitrary positive number, we have
lim
n → ∞ρFxn 0. 3.17
By Lemma 3.7, sequence {xn}n∈ is a Cauchy sequence inX, d. Since X, d is complete,
{xn} converges to some x∗ ∈ X. By hypothesis, Fxn is closed and Fxn1 ⊆ Fxn for all
n ∈Æ. Then, x
∗∈ Fx
n for all n ∈Æ. Since limn → ∞ρFxn 0, we have
n∈ Fxn lim n → ∞Fxn {x ∗}. 3.18 Indeed, if there exists y ∈ n∈Fxn with y / x
∗. Then,
0 <dx∗, y ≤ ρFxn, ∀n ∈Æ,
ρFxn −→ 0 as n −→ ∞.
3.19
This leads to a contradiction. Therefore,∅ / Fx∗ ⊆ n∈Fxn {x
∗}, and hence Fx∗ {x∗}.
Remark 3.11. We observe that if E is a normed space and {An}n∈is a sequence of subsets in
a cone metric spaceX, d with An1 ⊆ An for all n ∈ Æ, then the following statements are
equivalent:
a limn → ∞ρAn 0,
b for any sequence {an} with an∈ An, limn → ∞dan, an1 0.
The following result characterizes the λ-completeness of a λ-space.
Theorem 3.12. Let X, λ be a λ-space, E a normed space ordered by a normal cone P, and F : X → 2Xa multivalued map with nonempty λ-closed values. Assume that
i for all x, y ∈ X, y ∈ Fx implies Fy ⊆ Fx,
ii for every sequence {xn} with xn1 ∈ Fxn, one has δFxn −→ 0 of the second type inλ
X, λ,
iii for all x, y, z ∈ X, λx, z ≤ λx, y λy, z.
Then, the multivalued map F has a critical point in X if and only if X, λ is λ-complete.
Proof. The sufficiency follows fromRemark 3.5andLemma 3.6condition iii is not used.
We now show the necessity. Let {xn}n∈ be a λ-Cauchy sequence in X, λ. If x ∈
{xn}n∈ the λ-closure of {xn}n∈inX, λ, then by condition iii, either x xn for some
n ∈Æ or x is a λ-limit point of {xn}
n∈. Let An {xk}
∞
knthe λ-closure of {xk}∞kninX, λ.
For each x ∈{xn}n∈, we consider the following two cases:
i x ∈ {xn}n∈, then x xk∈ Ak/ ∅ for some k ∈ Æ,
ii x ∈ {xn}n∈\ {xn}n∈, then x ∈ Anfor all n ∈
Æ. Indeed, if x ∈ {xn}
n∈\ {xn}n∈,
then x is a λ-limit point of {xn}n∈, and hence x ∈ An for all n ∈
Æ. In this case,
x ∈ n∈An/ ∅.
Define a multivalued mapping F : {xn} → 2{xn}by
Fx ⎧ ⎪ ⎨ ⎪ ⎩ An if x xn for some n ∈Æ, n∈ An otherwise. 3.20
Then, x ∈ Fx / ∅ for all x ∈ {xn}n∈.
Since{xn}n∈is a λ-Cauchy sequence in X, λ, then for each c 0, there exists N ∈ Æ
such that λxn, xm c/2 for all m, n ≥ N. Then, by condition iii, for each n > N and
x, y ∈ An, we have λx, y≤ λx, xn λ xn, y c 2 c 2 c. 3.21
Then, δAn−→ 0 of the second type in X, λ. For each {uλ n}n∈in{xn} with un1∈ Fun for
Case 1. {un}n∈is a subsequence of{xn}n∈.
Case 2. there exists n ∈Æ such that un∈ {xn}
n∈\ {xn}n∈.
In Case1, δFun δFxnk−→ 0 of the second type for some subsequence {xλ nk} of
{xn}n∈. In Case2, Fun
k∈Akfor some n ∈
Æ. Since δAn
λ
−→ 0 of the second type in X, λ, δFun−→ 0 of the second type. Then, the condition ii is satisfied.λ
For each x, y ∈{xn}n∈with y ∈ Fx, we consider the following two cases.
Case 1. x xnfor some n ∈Æ, then y xmfor some m ≥ n or y is a λ-limit point of {xn}
n∈.
Then Fx Anand Fy Amfor some m ≥ n or Fy
n∈An.
Case 2. x is a λ-limit point of {xn}n∈, then y is also a λ-limit point of {xn}n∈. Then, Fx
Fy n∈An.
In either cases, we have that Fy ⊆ Fx and condition i is satisfied. By assumption, there exists a critical point of F, say x∗∈ X. Then, x∗∈ Anfor all n ∈Æ. Indeed, if x
∗ x
nfor
some n ∈ Æ, then{xn} {x
∗} Fx∗ Fx
n An. Then, xm x∗ xn for all m ≥ n and
hence x∗ ∈ An
k∈Ak. If x
∗/ x
nfor all n ∈Æ, then x
∗ ∈
n∈An. In either cases, we have
that x∗ ∈ n∈An. Since P is normal and δAn
λ
−→ 0 of the second type in X, λ, for each
ε > 0, there exists N ∈Æ such thatλxn, x
∗ < ε for all n ≥
Æ byLemma 3.6. Since for every
fixed c 0, there exists δ > 0 such that Bc, δ ⊆ int P and so c − λxn, x∗ ∈ int P as n large
enough. Hence, λxn, x∗ c as n large enough. Therefore, the sequence {xn} λ-converges to
x∗inX, λ.
Corollary 3.13. Let X, d be a cone metric space, E a normed vector space ordered by a normal cone
P , and F : X → 2Xbe a multivalued map with nonempty closed values inX, d. Suppose that
i for all x, y ∈ X, y ∈ Fx implies Fy ⊆ Fx, ii δAn−→ 0 of the second type in X, d.d
Then, F has a critical point in X if and only if X, d is d-complete.
We next establish another critical point theorem which generalizes the main result in 15.
Theorem 3.14. Let X, λ be a quasi-λ-complete space, and let F : X → 2Xbe a multivalued map
with nonempty quasi-λ-closed values. Suppose that
i for all x, y ∈ X, y ∈ Fx implies Fy ⊆ Fx,
ii {xn}n∈ is a sequence inX, λ with xn1 ∈ Fxn, and there exists {yn}n∈with yn ∈
Fxn such that δFxn−→ 0 of the first type w.r.t. {yλ n}n∈inX, λ,
iii every quasi-λ-convergent net in X, λ has a unique limit.
Then there exists x∗∈ Fx1 such that Fx∗ {x∗}.
Proof. Let{xn} and {yn} be the sequences given by condition ii. If m, n ∈Æ, m > n, by i, we
have ym∈ Fxm ⊆ Fxn. Since δFxn−→ 0 of the first type w.r.t. {yλ n}n∈inX, λ, for every
Therefore,{yn} is a quasi-λ-Cauchy sequence in X, λ. Since X, λ is quasi-λ-complete, {yn}
quasi-λ-converges to some x∗ ∈ X. Since Fxn is quasi-λ-closed and Fxn1 ⊆ Fxn for all
n ∈ Æ, we have x
∗ ∈ Fx
n for all n ∈ Æ. Since δFxn
λ
−→ 0 of the first type w.r.t. {yn} in
X, λ, we obtain Fx∗ {x∗}. Indeed, for each y ∈ Fx∗ ⊆
n∈Fxn, since δFxn
λ
−→ 0 of the first type w.r.t.{yn} in X, λ, both x∗and y are quasi-λ-limit points of {yn}n∈. Since
the limit is assumed to be unique, x∗ y and Fx∗ {x∗}.
Remark 3.15. The uniqueness assumption of the limit inTheorem 3.14holds if for each x ∈ X, one of the following conditions is satisfied:
A limn → ∞λxn, x 0 implies limn → ∞λx, xn 0 and for each y ∈ X, either λx, y ≤
λx, z λz, y or λy, x ≤ λy, z λz, x for all z ∈ X,
B λx, x 0 and for each y ∈ X, λx, y ≤ λz, x λz, y for all z ∈ X.
Indeed, it is obvious that conditionB implies the symmetricity of λ and thus implies conditionA. If condition A holds and both x and y are quasi-λ-limit of sequence {xn}n∈.
We have λxn, x q−λ −−−→ 0 and λxn, y q−λ −−−→ 0. Further, either λx, y≤ λx, xn λ xn, y q−λ −−−→ 0 as n −→ ∞, 3.22 or λy, x≤ λy, xn λxn, x q−λ −−−→ 0 as n −→ ∞. 3.23
In either cases, we have x y.
Now, we provide an example which shows that limit in a quasimetric spaces can be not unique.
Example 3.16. Let X 0, 1 and define d : X × X → Ê by
dx, y ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ y − x if y ≥ x, 1 y − x if y < x butx, y/ 1, 0, 1 ifx, y 1, 0. 3.24
Obviously, for each x, y ∈ X, we have dx, y ≥ 0 and dx, y 0 if and only if x y. We show that dx, y ≤ dx, z dz, y for all x, y, z ∈ X. Consider the following three cases:
a x < y,
b x > y but x, y / 1, 0, c x, y 1, 0.
For Case (a)
If x < z < y, then dx, y y − x z − x y − z dx, z dz, y, 3.25 if z < x < y, then dx, y y − x < y − z dz, y≤ dx, z dz, y, 3.26 if x < y < z, then dx, y y − x < z − x dx, z ≤ dx, z dz, y. 3.27For Case (b)
If x > z > y, then dx, y 1 y − x < 1 z − x dx, z ≤ dx, z dz, y, 3.28 if z > x > y, then dx, y 1 y − x z − x 1 y − z dx, z dz, y, 3.29 if x > y > z, then dx, y 1 y − x 1 − x z y − z dx, z dz, y. 3.30For Case (c)
For any z ∈ 0, 1, dx, y 1 z 1 − z dx, z dz, y. 3.31Since d0, 1/3 1/3 / 2/3 d1/3, 0, X, d is a quasimetric space but not a metric space. Further, for every sequence{xn} ⊆ 0, 1 increasing to 1, we have
dxn, 1 1 − xn −→ 0, dxn, 0 1 − xn−→ 0. 3.32
Then, both 1 and 0 are limits of the convergent sequence{xn} in X, d, but 1 / 0 in X, d for
The following example shows that in the setting of quasimetric spaces conditionA inRemark 3.11is a reasonable assumption.
Example 3.17. Let X Êand define d : X × X → Ê by
dx, y ⎧ ⎨ ⎩ y − x if x ≤ y, −2y − x if x > y. 3.33 Then,X, d is a quasimetric space, but not a metric space. Further, it is easy to verity that conditionA inRemark 3.15holds.
Remark 3.18. In Theorems3.1and3.14, the assumptions “X, λ is quasi-λ-complete” and “F
has nonempty quasi-λ-closed values” is imposed to guarantee quasi-λ-limit x∗of a sequence {xn} ⊂ X such that x∗ ∈
n∈Fxn. Therefore, these assumptions can be replaced by the
following condition: there existsx ∈ X such that x ∈ Fx for all x ∈ X.
Indeed, for any fixed x ∈ X, take a sequence {xn}∞n1 with x1 x and xn1 ∈ Fxn
for all n ∈Æ, we have x ∈
n∈Fxn. By the same argument as in the proof ofTheorem 3.1
also,Theorem 3.14, we see that Fx {x}.
As a corollary ofTheorem 3.14andRemark 3.15, we have the following result, which generalizes one of the main tools in15,16.
Corollary 3.19. Let X, d be a quasimetric space having a unique limit of a convergent sequence, p
a weak τ-function on X, and F : X → 2Xa multivalued map. Assume that{x
n} ⊆ X converges to
x ∈ X so that the following conditions are satisfied:
i xn1∈ Fxn and Fxn1 ⊆ Fxn for all n ∈Æ,
ii lim supn → ∞{pxn, u : u ∈ Fxn} 0,
iii x ∈ Fxn for all n ∈Æ.
then, n∈Fxn {x}.
If, in addition,
iv Fx / ∅ and Fx ⊆ Fxn for all n ∈Æ,
then Fx {x}.
Proof. Let E Êwith P Ê and define a multivalued mapping G : Fx1 → 2
Fx1by Gx ⎧ ⎪ ⎨ ⎪ ⎩ Fxn, if x ∈ Fxn x /∈ Fxm ∀m > n, n∈ Fxn, if x ∈ n∈ Fxn. 3.34
As an application ofTheorem 3.14andRemark 3.15, it suffices to show that the conditions i
andii implies δFxn → 0 of the first type w.r.t. {xn} in X, d. First, we show that for
Indeed, since lim supn → ∞{pxn, u : u ∈ Fxn} 0, we have lim n → ∞pxn, un ≤ lim supn → ∞ pxn, u : u ∈ Fxn 0, lim sup n → ∞ pxn, xm : m > n ≤ lim sup n → ∞ pxn, u : u ∈ Fxn 0. 3.35 Byτ3, limn → ∞dxn, un 0.
We will show that limn → ∞supu∈Fxndxn, u 0. Suppose to the contrary that there
exists ε > 0 and sequence {unk} with unk ∈ Fxnk such that for each nk, we have dxnk, unk ≥
ε. This contradicts the fact that limk → ∞dxnk, unk 0. Since Gxn ⊆ Fxn for all n ∈Æand
limn → ∞supu∈Fxndxn, u 0, then δGxn → 0 is of the first type w.r.t. {xn}n∈inX, d
and xn ∈ Gxn for all n ∈Æ. By the definition of mapping G, for all x, y ∈ X with y ∈ Gx,
we have Gy ⊆ Gx. Indeed, Gx Fxn and Gy Fxk for some n, k ∈ Æ. Since
y ∈ Gx Fxn, then by the definition of mapping G, we have that k ≥ n. Then, Gy
Fxk ⊆ Fxn Gx. Then, byTheorem 3.14andRemark 3.15., there exists x∗∈ X such that
Gx∗ {x∗}. Since x ∈ n∈Fxn ⊆ Gx
∗, then x x∗. Therefore
n∈Fxn {x}.
Remark 3.20. In Corollary 3.19., if X, d is a metric space, then Corollary 3.19. reduces to Lemma 3.4 in15. In the proof ofCorollary 3.19, we have shown that the conditionii of
Corollary 3.19implies conditionii ofTheorem 3.14. Khanh and Quy16 showed that if a
sequence{xn} is asymptotic by p, then {xn} satisfies condition ii ofCorollary 3.19.
4. Ekeland Type Variational Principles
Definition 4.1. Let X, d be a metric space. An extended real-valued function f : X →
−∞, ∞ is said to be lower semicontinuous from above in short, lsca at x0 ∈ X if for any sequence {xn} in X with xn → x0 and fx1 ≥ fx2 ≥ · · · ≥ fxn ≥ · · · imply that
fx0 ≤ limn → ∞fxn. The function f is said to be lsca on X if f is lsca at every point of
X.
Definition 4.2. Let E be a normed vector space ordered by a cone P . Then, we have the
following.
i E is called well-normed with respect to P if there exists S ∈Ê such that
n k1 vk ≤ S n k1 vk ∀n ∈Æ, where vk∈ P ∀k ∈Æ. 4.1
The least positive number S satisfying inequality 4.1 is called well-normed constant
of P .
ii E satisfies condition L if for all n ∈Æ,
n
k1
vk≤ v for some v ∈ E, where vk∈ P ∀k ∈Æ, 4.2
Example 4.3. For each n ∈ Æ, letÊ
n be ordered by the cone P
Ê
n
. Then,Ê
n is not only
well-normed with well norm constant S √n but also satisfies condition L.
Definition 4.4see 20. Let X be a topological space, E be a topological vector space ordered
by a cone P , and F : X → 2Ea multivalued map and domF {x ∈ X : Fx / ∅}. F is said
to be
i upper P-continuous at x ∈ domF if for any neighborhood V of the origin 0 of
E, there exists a neighborhood U of x such that Fx ⊆ Fx V P for all x ∈ U ∩ domF,
ii lower P-continuous at x ∈ domF if for any neighborhood V of the origin 0 of
E, there exists a neighborhood U of x such that Fx ⊆ Fx V − P for all x ∈ U ∩ domF.
Remark 4.5see 20. Let X be a topological space, and let f : X → Êbe a function. It can be
easily verified that the followings statements are equivalent: i f is lower semicontinuous at x,
ii f is upperÊ-continuous at x,
iii f is lowerÊ-continuous at x.
The following lemma plays an important role in this section. For the sake of completeness of the paper, we give a detailed proof of this lemma.
Lemma 4.6 see 20. Let X be a topological space, E a topological vector space ordered by a cone P
and G, and H : X → 2Emultivalued maps with nonempty values. If G is a lower −P -continuous
map and H is an upper P -continuous map with compact values, then the set S {x ∈ X : Gx ⊆ Hx P } is a closed set.
Proof. Let x0 ∈ T Scthe complement of S and z0 ∈ Gx0 such that z0/∈ Hx0 P. Since
Hx0 P is closed, there exists a balanced neighborhood V of 0 ∈ E such that
z0 V ∩ Hx0 V P ∅. 4.3
Since P is a cone, it follows that
z0 V − P ∩ Hx0 V P ∅. 4.4
By the upper P -continuity of H, there exists a neighborhood U of x0such that
Hx ⊆ Hx0 V P ∀x ∈ U, 4.5
and hence
By lower−P-continuity of G, we may assume that U is such that
Gx0 ⊆ Gx V P ∀x ∈ U. 4.7
Since z0∈ Gx0, this shows that for all x ∈ U, we can find z ∈ Gx such that z ∈ z0− V − P. Then, for each x ∈ U, we can find z ∈ Gx such that z /∈ Hx P. Thus, U is contained in T, and S is closed.
We present a version of Ekeland type variational principle in the setting of complete cone metric spaces.
Theorem 4.7. Let E be a well-normed vector space ordered by a normal cone P. Let X, d be a
complete cone metric space, and let f : X → E be a lower −P -continuous function, bounded from below by l. Then, for every ε > 0 and for every x ∈ X, there exists x∗∈ X such that
i fx∗ εdx∗, x ≤ fx,
ii εdx∗, x/≤fx∗ − fx for all x ∈ X \ {x∗}.
Proof. Without loss of generality, we may assume that ε 1. Let
Fx y ∈ X : fy εdx, y≤ fx ∀x ∈ X, 4.8
and take
x1 x, xn1∈ Fxn ∀n ∈Æ. 4.9
It is easy to verify that for each fixed x ∈ X, dx, · is an upper P -continuous function on X and thus−dx, · is upper −P-continuous. Since f is a lower −P-continuous function, by
Lemma 4.6, F is a multivalued map with nonempty closed values. Since E is well-normed, we have ∞ k1 dxk, xk1 lim n → ∞ n k1 dxk, xk1 ≤ lim n → ∞K n k1 fxk − fxk1 ≤ lim n → ∞SK n k1 fxk − fxk1 ≤ SK2fx1 − l< ∞. 4.10
Then,dxi, xj → 0 as i, j → ∞. Therefore, the conclusion follows fromTheorem 3.10.
Now, we establish the following Ekeland type variational principle for multivalued bifunctions in the setting of complete cone metric spaces.
Theorem 4.8. Let X, d be a complete cone metric space, E be a normed vector space with an ordering
cone P that satisfies condition L, and let F : X × X → 2Eand H : X → 2Xbe multivalued maps.
For each x ∈ X, suppose that the following conditions hold:
i Hx is a closed subset of X and Hy ⊆ Hx for all y ∈ Hx, ii There exists y ∈ Hx such that Fx, y dx, y ⊆ −P,
iii Either Fx, · is bounded from below on Hx or y∈HxFx, y / ∅,
iv Fx, z ⊆ Fx, y Fy, z − P for all y ∈ Hx and z ∈ Hy; v the map Fx, · is lower P-continuous on Hx.
Then, for every ε > 0 and for every x ∈ X, there exists x∗∈ Hx such that
a x∗∈ Hx∗,
b Fx, x∗ εdx, x∗ ⊆ −P,
c Fx∗, x εdx∗, x/⊆ − P for all x ∈ Hx∗ \ {x∗}.
Proof. Without loss of generality, we may assume that ε 1. For all x ∈ X, let
Gx y ∈ X : Fx, y dx, y⊆ −P , Sx Gx ∩ Hx. 4.11
It is easy to verify that for each fixed x ∈ X, dx, · is an upper P -continuous function on
X, thus −dx, · is upper −P -continuous. By condition v andLemma 4.6, Sx is a closed subset of Hx for all x ∈ X. Since Hx is a closed subset of X for all x ∈ X, Sx is also a closed subset of X for all x ∈ X. For every y ∈ Sx,
Fx, y dx, y⊆ −P. 4.12
For every z ∈ Sy,
Fy, z dy, z⊆ −P. 4.13
Then,
Fx, z dx, z ⊆ Fx, y Fy, z dx, y dy, z− P ⊆ −P. 4.14
Hence, Sy ⊆ Sx if y ∈ Sx.
For arbitrary x ∈ X, take the sequence {xn}n∈ with xn1 ∈ Sxn for all n ∈ Æ and x1 x. We have Fxn, xn1 dxn, xn1 ⊆ −P ∀n ∈Æ, 4.15 therefore, n k1 Fxk, xk1 n k1 dxk, xk1 ⊆ −P. 4.16
By conditioniv, Fx1, xn1 n k1 dxk, xk1 ⊆ n k1 Fxk, xk1 n k1 dxk, xk1 − P ⊆ −P. 4.17
Hence, by conditioniii, there exists l ∈ E such thatnk1dxk, xk1 ∈ l − P. By condition
L, we have limn → ∞dxk, xk1 0. ByTheorem 3.10, there exists x∗ ∈ Sx such that
Sx∗ {x∗}.
If Hx X for all x ∈ X, then we deduce the following corollary fromTheorem 4.8. Corollary 4.9. Let X, d be a complete cone metric space, E a normed vector space with an ordering
cone P that satisfies condition (L), and F : X × X → 2E be a multivalued map. For each x ∈ X,
suppose that the following conditions hold:
i there exists y ∈ X such that Fx, y dx, y ⊆ −P,
ii either Fx, · is bounded from below on X or y∈XFx, y / ∅,
iii Fx, z ⊆ Fx, y Fy, z − P for all y, z ∈ X, iv the map Fx, · is lower P-continuous on X.
Then, for every ε > 0 and for every x ∈ X, there exists x∗∈ X such that
a Fx, x∗ εdx, x∗ ⊆ −P,
b Fx∗, x εdx∗, x/⊆ − P for all x ∈ X \ {x∗}.
As an application ofTheorem 3.12, we improve Theorem 5.1 in17.
Theorem 4.10. Let X, d be a metric space, f : X → Ê a lsca function, ϕ : −∞, ∞ → Ê a
nondecreasing function, and p a τ-function on X. Define a binary relationÊon X by
yÊx ⇐⇒ p
x, y≤ ϕfxfx − fy. 4.18 Suppose that H : X → 2X is a multivalued map with nonempty values, X isÊ-complete and
f is bounded from below on X, and for each x ∈ X, there exists y ∈ Hx such that px, y ≤ ϕfxfx − fy. Then, for each u ∈ X, there exists v ∈ X such that
i pu, v ≤ ϕfufu − fv,
ii pv, y > ϕfvfv − fy for all y ∈ X, y / v, iii v ∈ Hv.
Proof. Let Fx {y ∈ X : yÊx}, x1 u and choose xn1Êxn for all n ∈Æ. Then, it is easy
to verify thatÊis a transitive relation and hence conditioni of Theorem 3.12is satisfied.
Let limn → ∞xn v. Since f is lsca, the transitive relationÊ is lower closed. By the same
arguments in the proof ofTheorem 4.7, we see that{xn} is asymptotic by p. As an application
ofTheorem 3.12and Remarks3.3and3.18, we complete the proof.
Remark 4.11. In17, Theorem 5.1, the function p is assumed to be a strong τ-function i.e.,
Theorem 4.10 ofTheorem 3.12, the assumptions above are not necessary and the proof is much easier.
Remark 4.12. Theorems4.7–4.10are generalizations of EVP2. To the best of our knowledge,
there is almost no EVP results in the setting of cone metric spaces.
Acknowledgments
In this research, the third author was partially supported by the SABIC/Fast Track Research Project no. SB080004 of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. His part was done during his stay at KFUPM, Dhahran.
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