Volume 2010, Article ID 276294,15pages doi:10.1155/2010/276294
Research Article
Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications
S. Eshghinezhad and M. Fakhar
Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan 81745-163, Iran
Correspondence should be addressed to M. Fakhar,majid [email protected] Received 23 June 2010; Accepted 2 November 2010
Academic Editor: A. T M. Lau
Copyrightq2010 S. Eshghinezhad and M. Fakhar. 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.
By using a Dane˘s’ drop theorem in locally convex spaces we obtain a vectorial form of Ekeland- type variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk’s fixed-point theorem, Takahashi’s nonconvex minimization theorem, and Oettli-Th´era’s theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given.
1. Introduction and Preliminaries
A very important result in nonlinear analysis about the existence result for an approximate minimizer of a lower semicontinuous and bounded below function was first presented by Ekeland 1. Known nowadays as Ekeland’s variational principle in short, EVP, it has significant applications in the geometry theory of Banach spaces, optimization theory, game theory, optimal control theory, dynamical systems, and so forth; see1–11and references therein. It is well known that EVP is equivalent to many famous results, namely, the Caristi- Kirk fixed-point theorem, the petal theorem, Phelp’s lemma, Danˆes’ drop theorem, Oettli- Th´era’s theorem and Takahashi’s theorem, see, for example, 4, 6, 7, 10, 12–19. Many authors have obtained EVP on complete metric spaces1,10,19,20and in locally convex spaces20–24. Along with the development of vector optimization and motivated by the wide usefulness of EVP, many authors have been interested in obtaining this principle for vector-valued functions and set-valued mappings; see 3, 5, 8, 9, 11–15, 21, 22, 25.
Recently, this principle has been obtained for bifunctions and applied to solve equilibrium problem in nonconvex setting10,12,14–16,20,26,27. Our goal in this paper is to obtain Ekeland’s variational principle for vector-valued bifunctions in locally convex spaces. By
using this result we derive the existence of solution of vector equilibrium problem in the setting of seminormed spaces. Also, we obtain vectorial Caristi-Kirk’s fixed-point theorem, vectorial Takahashi’s nonconvex minimization theorem and vectorial Oettli-Th´era’s theorem.
Moreover, we show that these results and Dane˘s’ drop theorem are equivalent to each other.
Let us, introduce some known definitions and results which will be used in the sequel.
LetEbe a Hausdorfflocally convex real vector space. A subsetBofEis said to be a disc, ifBis bounded and absolutely convex. Let spB be the vector subspace spanned by B, andpBbe the Minkowski functional ofB, thenEB : spB, pBis a normed space. IfEB
is a Banach space, thenBis called a Banach disc. A sequence{xn}inEis said to be locally convergent to an elementxif there is a discBinEsuch that the sequence{xn}is convergent to xinEBand{xn}is said to be locally Cauchy if there is a discBinEsuch that{xn}is a Cauchy sequence inEB. We say thatEis a locally complete space if every locally Cauchy sequence is locally convergent. This is equivalent to that each bounded subset ofEis contained in a certain Banach disc. A nonempty subsetX ofEis said to be locally complete if every locally Cauchy sequence inXis locally convergent to a point inX. The subsetXis said to be locally closed if for any locally convergent sequence inX, its local limit point belongs to X. It is well known that every sequentially complete locally convex space is locally complete and the converse is not true; see28,29.
LetY, Cbe a locally convex space ordered by the nontrivial closed convex coneCas follows:
x≤Cy⇐⇒y−x∈C. 1.1
For everyx, y∈Y we write
x /≤Cy⇐⇒x−y /∈ −C. 1.2
Definition 1.1. LetX be a nonempty subset of a locally convex spaceE,Y, Cbe a locally convex space ordered by the nontrivial closed convex coneC. A vector-valued functionφ : X → Y is said to be
1 e, C-locally lower semicontinuous if for everyr∈Rthe set{x∈X:φx≤Cre}is locally closed inX;
2C-upper semicontinuous atx0∈Xif for any neighborhoodUofφx0, there exists a neighborhood V of x0 such that φx ∈ U−C, for all x ∈ V. If φ isC-upper semicontinuous at each point ofX, thenφis said to beC-upper semicontinuous on X;
3C-bounded from below, if there existsb∈Y such thatb≤Cφxfor allx∈X.
Assume that the interior of CintCis nonempty and g : X ×X → Y is a vector-valued function. The vector equilibrium problemin short, VEPis to findx∈Xsuch that
g x, y
/∈ −intC ∀y∈X. 1.3
It is well known that VEP includes fundamental mathematical problems like vector optimization, vector variational inequality, and vector complementarity problem. For further details on VEP, one can refer to23,24,30–32.
Lete∈intC. Recall the definition of the Gerstewitz function33:
ξe
y :inf
r ∈R:y∈re−C
, ∀y∈Y. 1.4
The following lemma describes some properties of the Gerstewitz function and it will be used in the sequel. For its proof we refer the reader to5,6,33.
Lemma 1.2. For eachr∈Randy∈Y, the following statements are satisfied:
iξey≤r⇔y≤Cre.
iiξey> r⇔y /≤Cre.
iiiξey≥r⇔y /∈re−intC.
ivξey< r⇔y∈re−intC.
vξe·is positively homogeneous and continuous onY. viξey1y2≤ξey1 ξey2, for ally1, y2∈Y.
viiξe·is monotone, that is, ify2 ∈y1C, thenξey1≤ξey2.
In order to obtain a vectorial form of Ekeland-type variational principle we need the following result.
Theorem 1.3see17. LetAbe a locally closed subset of a locally convex spaceEandBa locally closed, bounded convex subset ofEwith 0/∈clA−B. If eitherAorBis locally complete, then for eachx0 ∈A, there existsa∈Dx0, B∩Asuch thatDa, B∩A{a}, whereDa, Bdenotes the convex hull of{a} ∪B.
2. Vectorial Ekeland-Type Variational Principle
Recently, Qiu18obtained some versions of Ekeland’s variational principle in locally convex spaces, which only need to assume local completeness of some related sets. Motivated by this paper we obtain some versions of EVP for vector-valued bifunctions in locally convex spaces.
These results extend Qiu’s results to vector-valued bifunctions.
Throughout this section E is a locally convex space, X is locally closed subset of E, {pλ}λ∈Λ is a family of seminorms generating the locally convex topology onE,Y is a Hausdorfflocally convex space ordered by a closed convex coneCwith intC /∅ande∈intC.
We consider a vector-valued bifunctionf : X×X → Y, a family of positive real numbers {αλ}λ∈Λand the following assumptions:
A1fx, x 0 for allx∈X.
A2fz, x≤Cfz, y fy, xfor anyx, y, z∈X.
A3y → fx, yis C-bounded from below for allx∈X.
A4y → fx, yise, C-locally lower semicontinuous for anyx∈X.
A5There existsx0∈Xsuch that the set{x∈X:fx0, x≤C0}is locally complete.
A6The set
λ∈Λ{x∈E:αλpλx≤1}is locally complete.
Notice that if assumptionsA1andA2hold, thenf is called half distance. The following result is a vectorial form of Ekeland-type variational principle.
Theorem 2.1. Suppose that assumptions (A1)–(A4) are satisfied. If either assumption (A5) or assumption (A6) holds, then for anyε >0, there existsz∈Xsuch that
ifx0, z εαλpλz−x0e≤C0, for anyλ∈Λ;
iiFor anyx /z, there existsμ∈Λsuch thatfz, x εαμpμx−ze /≤C0.
Proof. Without loss of generality, we may assume thatε 1 and putM : E×Rwith the product topology, then the topology can be generated by a family{qλ:λ∈Λ}of seminorms, whereqλx, t pλx |t|, for allx, t∈M. IfA{x, t∈X×R:fx0, x≤Cte, t≤0}and m: inf{t :x, t ∈ A}, then sincey → fx, yisC-bounded from below for allx ∈X we have−∞< m≤0. Take any fixed real numberr < mand putB:{x, r∈M:αλpλx−x0≤
−r, ∀λ ∈ Λ}. ThenK : coneBis exactly the set{y, t ∈ M : αλpλy−x0 ≤ −t, for all λ∈Λ}. If the set{x∈X:fx0, x≤C0}is locally complete, thenAis locally complete and if
λ∈Λ{x∈E:αλpλx≤ 1}is locally complete, thenBis locally complete. Furthermore,Bis bounded closed convex subset ofMandqλA−B≥m−r >0. Hence, byTheorem 1.3, there exists
z, s∈A∩D0,0, B⊂A∩K 2.1
such that
A∩Dz, s, B {z, s}. 2.2
According to2.1, we havez, s∈A∩K, so
fx0, z≤Ces 2.3
and for eachλ∈Λ
αλpλz−x0e≤C −es. 2.4
Therefore, by2.3and2.4, we have
αλpλz−x0e≤C −es≤C −fx0, z, ∀λ∈Λ. 2.5 Hence, the part i holds. We show that the point z satisfies in the part ii. Let δ ξefx0, z−r/s−r. Since re≤Cme≤Cfx0, z≤Cse and ξe is monotone, then 0 ≤ δ ≤ 1.
On the other hand we haveδs 1−δrξefx0, z. Hence, z, ξe
fx0, z
z, δs 1−δr δz, s 1−δz, r. 2.6 But from2.5we have
αλpλz−x0e≤C −fx0, z≤C −re, ∀λ∈Λ. 2.7
Therefore,z, r ∈B. Thus,z, ξefx0, z∈Dz, s, B.Also, clearlyz, ξefx0, z∈A.
Hence, we have
z, ξe
fx0, z
∈A∩Dz, s, B. 2.8
Therefore, by2.1,z, ξefx0, z {z, s}and sosξefx0, z.
Supposing thatx∈Xandx /z, we consider the following two cases.
Case 1. Ifx, ξefx0, x/∈A, thenξefx0, x> 0. Sinceξeis monotone, then for allλ ∈Λ we have
ξe
fx0, x αλpλz−x e≥ξe
fx0, x
>0≥ξe
fx0, z
. 2.9
Butfis half distance andξeis sublinear, thus ξe
fz, x αλpλz−xe
≥ξe
fx0, x−fx0, z αλpλz−xe
≥ξe
fx0, x αλpλz−xe
−ξefx0, z
>0. 2.10
Hence, by the partiiofLemma 1.2;
fz, x αλpλz−xe /≤C0. 2.11
Case 2. Letx, ξefx0, x ∈ A, we will show that x, ξefx0, x/∈z, s K. If not, we assume thatx−z, ξefx0, x−s∈K, that is,
s−ξe
fx0, x
≥αλpλx−z, ∀λ∈Λ. 2.12 Sincex /zand{pλ}λ∈Λseparates points inX, we conclude that there existsμ∈Λsuch that pμx−z>0 thuss−ξefx0, x>0. Put
η s−ξe
fx0, x
s−r , 0< η <1. 2.13
SinceKis a cone,
x−z η ,ξe
fx0, x
−s η
∈K 2.14
that is,
x−z η , r−s
∈K. 2.15
By2.1,
z, s∈K. 2.16
SinceKis a convex cone, by2.15and2.16we have
z, s x−z η , r−s
∈K, 2.17
so
zx−z η , r
∈K∩E× {r} B. 2.18
It is easy to verify that 1−η
sηr ξe
fx0, x
−r
s−r ss−ξe
fx0, x s−r ξe
fx0, x
. 2.19
Hence,
x, ξe
fx0, x
∈Dz, s, B∩A{z, s}. 2.20
Therefore,x, ξefx0, x z, sand soxz, which it is a contradiction. This shows that x, ξefx0, x/∈z, s K. Thus, there existsμ∈Λsuch that
ξe
fx0, x
−ξe
fx0, z
αμpμx−z>0. 2.21
On the other hand byA2we have ξe
fz, x
≥ξe
fx0, x
−ξe
fx0, z
. 2.22
Hence,
ξe
fz, x αμpμx−ze
>0. 2.23
Therefore,
fz, x αμpμx−ze /≤C0. 2.24
Remark 2.2. In the above theorem, if assumptionA5holds, then instead of assumptionA3, we can assume thaty →fx0, yisC-bounded from below. Also, if assumptionA6holds, assumptionA3can be replaced by the following assumption:y → fx, y isC-bounded from below for somex∈X.
As a consequence of the above theorem we can obtain the following result which is a vectorial version of Theorem 3.1 of18.
Corollary 2.3. Letφ :X → Y be a function such thatφisC-bounded from below andφise, C- locally lower semicontinuous. Furthermore, let assumption (A6) holds or there existsx0∈Xsuch that the set{x∈X :φx≤Cφx0}is locally complete. Then there existsz∈Xsuch that
iφz εαλpλz−x0e≤Cφx0, for anyλ∈Λ;
iifor anyx /z, there existsμ∈Λsuch thatφx εαμpμx−ze /≤Cφz.
Proof. It is enough inTheorem 2.1to considerfx, y φy−φxfor allx, y∈X.
In the following theorem we show that the previous results are equivalent to each other.
Theorem 2.4. Corollary 2.3impliesTheorem 2.1.
Proof. Letφ:X → Ybe defined as follows:
φx fx0, x ∀x∈X. 2.25
It is an easy task to derive the assumptions ofCorollary 2.3for the above function from the assumptions ofTheorem 2.1. Therefore, there existsz ∈ Xwhich satisfies the conditionsi andiiofCorollary 2.3. Hence,
ifx0, z εαλpλz−x0e≤C0, for anyλ∈Λ;
iifor anyx /z, there existsμ∈Λsuch thatfx0, x−fx0, z εαμpμx−ze /≤C0.
Also, by assumptionA2we havefx0, x≤Cfx0, z fz, x. Thus,
fz, x fx0, z−fx0, x fx0, x−fx0, z εαμpμx−ze
fz, x εαμpμx−ze /≤C0. 2.26
LetSbe a convex subset ofEcontaining 0. The Minkowski functional ofSis defined as follows:
pSx
⎧⎨
⎩
inf{t >0 :x∈tS} if there existst >0 such thatx∈tA
∞, otherwise. 2.27
We extendYby an additional element∞∗such that∞∗y /≤C0 for ally∈Yandc×∞ ∞∗ for allc∈intC.
By usingTheorem 2.1 we obtain another version of vectorial form of Ekeland-type variational principle in which the perturbation function is the Minkowski functional of a bounded set.
Theorem 2.5. Suppose that assumptions (A1)–(A4) are satisfied. Let S ⊆ E be a locally closed, bounded convex set containing 0 and α be a positive real number. Let S be locally complete or assumption (A5) holds. Then, for anyε >0, there existsz∈Xsuch that:
ifx0, z εαpSz−x0e≤C0;
iiFor anyx /z,fz, x εαpSx−ze /≤C0.
Proof. Suppose thatTis the absolutely convex hull of the setS∪{x0}, thenXT, pTis a normed space. Assume that
D:
x∈XT:fx0, x εαpSx−x0e≤C0
. 2.28
Sincex0 ∈ T, thenx0 ∈ D. Also,fx0,· ise, C-locally lower semicontinuous andpS is locally lower semicontinuous, thenDis closed inXT, pT. Suppose thatg is restrictedf to D×D. IfSis locally complete thenT is a Banach disk andXT, pTis a Banach space. If the set{x ∈X :fx0, x≤C0}is locally complete, then{x∈ D :gx0, x≤C0} D∩ {x ∈X : fx0, x≤C0}is a complete set inXT, pT. Therefore, byTheorem 2.1there existsz∈Dsuch that:
agx0, x εαpTz−x0e≤C0.
bFor anyx∈Dandx /z,
gz, x εαpTx−ze /≤C0. 2.29
Sincez∈D, then the partaholds. Now, we show that the partbholds. Ifx /zandx∈D, then2.29becomes
fz, x εαpSx−ze /≤C0. 2.30
Letx /zandx /∈XT, thenpSx−z ∞, so the partbholds. Letx /z,x∈XT,x∈X\D andfz, x εαpSz−xe≤C0. Sincefx0, z εαpSz−x0e≤C0, then
fx0, x εαpSx−x0e≤Cfx0, z εαpsz−x0efz, x εαpSz−xe≤C0. 2.31
Therefore,x∈Dwhich is a contradiction. Hence,
fz, x εαpSz−xe /≤C0, ∀x∈X\D. 2.32
Assuming thatE is a locally complete locally convex space, the condition on local completeness of some related subsets is automatically satisfied. However, we give the following examples of spaces which are not locally complete but the condition on local completeness of some related subsets is satisfied.
Example 2.6. LetC0,1be the space of all continuous functions defined on0,1. By Corollary 11-7-3, 11-7-4 of 29, C0,1 with weak-topology is quasi barreled but it is not barreled.
Therefore, by Proposition 11-2-5 of29,M0,1with weak∗-topology is not locally complete.
Moreover,
x∈C0,1
{x∗∈M0,1:|x∗, x| ≤1}{x∗∈M0,1:x∗ ≤1}. 2.33
But by Banach-Alaoglu theorem this set is weak∗-compact. AlsoC0,1is separable, so weak∗- topology on unit ballM0,1is metrizable. Hence, this set is locally complete.
Example 2.7. LetC10,1, ∞be the space of all differentiable functions whose derivative is continuous. ThenC10,1, ∞is not a complete space. Therefore, by Proposition 5.1.928 C10,1,∞is not a locally complete space.
Also, the set {x ∈ C10,1 : x∞ ≤ 1} is not locally complete. Suppose that f:C10,1×C10,1 → C0,1is defined as follows:
f x, y
y2−x2 ∀x, y∈C10,1, 2.34 whereC0,1is ordered by the coneC{x∈C0,1:xt≥0}. If we choosex00, then the set{x∈C10,1:x2t≤x20t 0, ∀t∈0,1}{0}is locally complete.
3. Caristi-kirk’s Fixed-Point Theorem,
Takahashi’s Nonconvex Minimization Theorem, and Oettli-Th ´era’s Theorem and Equilibrium Problem
In this section, we obtain an existence result for solution of vector equilibrium problem in nonconvex setting. Also, some new versions of the vectorial Caristi-Kirk fixed-point theorem, vectorial Takahashi’s nonconvex minimization theorem and the vectorial Oettli- Th´era theorem are given.
Theorem 3.1. LetX be a weakly compact subset of a semi normed space Z, p. Suppose thatf : X×X → Y is a function satisfying assumptions (A1)–(A5) together with somex0 ∈Xandf·, y isC-upper semicontinuous for everyy∈X. Then, there existsx∈Xsuch that
f x, y
/∈ −intC, ∀y∈X. 3.1
Proof. Assume thatS {x ∈Z :px≤ 1}, thenpSx pxfor allx∈ Z. Takingε 1/n andα1, fromTheorem 2.5, we find a sequence{xn}, such that
f xn, y
1 np
xn−y
e /≤C0, ∀y /xn. 3.2
By the weakly compactness of X, we can assume that {xn} weakly converges tox ∈ X.
Suppose thatfx, y ∈ −intCfor a suitabley ∈X. Take a neighborhoodUoffx, ysuch thatU⊂ −intC. ByC-upper semicontinuity off·, y, there exists a natural numberNsuch thatfxn, y∈U−C, forn≥N. Moreover, ifnis big enough,1/npxn−yeU⊆ −intC, thus
f xn, y
1 np
xn−y
e∈U−C 1 np
xn−y
e⊆ −intC. 3.3
This is a contradiction.
In the above theorem, when X is not necessarily weakly compact we have the following result. Since its proof is similar to Theorem 4 of20, we omit it.
Theorem 3.2. Let X be a nonempty subset of a reflexive semi normed space Z, p. Suppose that f : X×X → Y is a function satisfying assumptions (A1)–(A5) together with somex0 ∈ X and f·, yisC-upper semicontinuous for everyy∈X. Let the following coercivity condition holds:
There exists a nonempty closed bounded subsetKofXsuch that for allx∈X\Kthere exists y∈Xwithpy< pxsatisfyingfx, y≤C0.
Then, there existsx∈Xsuch that
f x, y
/∈ −intC, ∀y∈X. 3.4
As a consequence of Theorems 2.1and 2.5we can obtain two versions of vectorial Caristi-Kirk’s fixed-point theorem.
Theorem 3.3. Suppose that all of the conditions ofTheorem 2.1are satisfied. Assume thatT :X → 2Xis a set-valued mapping with nonempty valued and the following property holds:
f x, y
αλpλ
y−x
e≤C0; ∀λ∈Λ, ∀x∈X, ∀y∈Tx. 3.5
Then, there existsz∈Tx0such thatTz{z}.
Proof. Let z be a point which satisfies in the parts i and ii with ε 1. We show that Tz {z}. If there existsx ∈ Tzandx /z, then byTheorem 2.1there is aμ ∈ Λsuch that fz, x αμpμx−ze /≤C0 and it is a contradiction.
The proof of the following results is similar to that ofTheorem 3.3and we omit it.
Theorem 3.4. Suppose that all of the conditions ofTheorem 2.5are satisfied andT : X → 2X is a set-valued mapping with nonempty valued and the following condition holds:
f x, y
αpS y−x
e≤C0; ∀x∈X, y∈Tx. 3.6
Then, there existsz∈Tx0such thatTz{z}.
In the following we give two versions of vectorial Takahashi’s nonconvex minimiza- tion.
Theorem 3.5. Suppose that all of conditions ofTheorem 2.1 are satisfied and for eachy ∈ X with fx0, X∩fx0, y−C/{fx0, y}, there existsx /ysuch thatfy, xeαλpλx−y≤C0 for any λ∈Λ. Then, there existsx∈Xsuch thatfx0, x∈Minfx0, Xthat isfx0, X∩fx0, x−C {fx0, x}.
Proof. ByTheorem 2.1there existsz∈Xsuch that for eachx /zthere existsμ∈Λsuch that fz, x eαμpμx−z/≤C0. Iffx0, X∩fx0, z−C/{fx0, z}, then by assumption there existsx /zsuch thatfz, x eαμpμx−z≤C0, and this is a contradiction.
The proof of the following results is similar to that ofTheorem 3.5and we omit it.
Theorem 3.6. Suppose that all of the conditions ofTheorem 2.5are satisfied and for eachy∈Xwith fx0, X∩fx0, y−C/{fx0, y}, there existsx /ysuch thatfy, x epSx−y≤C0. Then, there existsx∈Xsuch thatfx0, x∈Minfx0, X.
In the final step we obtain a vectorial form of Oettli-Th´era type theorem.
Theorem 3.7. Assume that all of the conditions ofTheorem 2.1 are satisfied andS0 {x ∈ X : fx0, x εαλpλx−x0e≤C0}. LetΨ⊆Xsuch that for everyx∈S0\Ψthere existsx∈Xsuch thatx /xandfx, x εαμpμx−xe≤C0. ThenS0∩Ψ/∅.
Proof. ByTheorem 2.1there existsz∈S0such that satisfies the conditioniiinTheorem 2.1.
It is easy to see thatz∈S0∩Ψ.
4. Equivalences
In this section, we show that Dane˘s’ drop theoremTheorem 1.3, two versions of vectorial form of Ekeland-type variational principle, vectorial Caristi-Kirk’s fixed-point theorem and vectorial Takahashi’s nonconvex minimization theorem are equivalent. In order to show that Theorems 1.3and 2.1are equivalent to each other we need the following definition which was introduced by Cheng et al.34.
Definition 4.1. Two nonempty subsetsAandBof the locally convex spaceEare said to be strongly Minkowski separated if and only if there exist a continuous seminormpandz∈E such that either
inf
paz:a∈A
>sup
pbz:b∈B
4.1
or
sup
paz:a∈A
<inf
pbz:b∈B
. 4.2
Theorem 4.2. Theorems2.1and1.3are equivalent to each other.
Proof. It is only enough to show thatTheorem 2.5impliesTheorem 1.3. Since 0/∈clA−B, then there existη∈Λandδ >0 such that
pηa−b≥δ, ∀a∈A, b∈B. 4.3
Therefore, by Lemma 1 of7AandBare strongly Minkowski separated. Hence, there exist a continuous seminormpandz∈Esuch that
inf
paz:a∈A
>sup
pbz:b∈B
. 4.4
Without loss of generality we may assume thatz0 and put d1:inf
px−p y
:x∈A, y∈b
. 4.5
Also, B is bounded, thus d2 : sup{px−y : x, y ∈ Dx0, B} is finite. Now, we apply Theorem 2.5for the setX Dx0, B∩A, function fx, y : epy−px,ε 1 and a positive number αwhichα ≤ d1/d2. It is easy to see that the assumptions A1–A4are satisfied. IfAorBis locally complete, thenXis locally complete, so{x∈X:fx0, x≤C0}is locally complete. Therefore, byTheorem 2.5there exists a pointz∈Xsuch that
fz, x αpx−ze /≤C0, ∀x∈X, x /z. 4.6
Letx∈Dz, B∩A, thenxtz 1−tb, whereb∈Band 0≤t≤1. Hence, ξe
fz, x αpx−ze
ptz 1−tb−pz αp1−tb−z
≤tpz 1−tpb α1−tpb−z−pz
≤tpz 1−tpz−d1αd2−pz
≤0.
4.7
Therefore,fz, x αpx−ze∈ −Cand so by4.6, we conclude thatxz.
Theorem 4.3. Theorems2.1,2.5,3.3,3.5and3.7are mutually equivalent.
Proof. 1Theorem 2.5⇔Theorem 2.1.
It is enough to show thatTheorem 2.5⇒Theorem 2.1. Choose
S
λ∈Λ
x∈X :αλpλx≤1
. 4.8
ThenS⊂X is a bounded, closed absolutely convex set. IfpSis the Minkowski functional of S, then
pSx sup
λ∈Λαλpλx, ∀x∈X. 4.9
Now, if we applyTheorem 2.5for the setS, then we obtainTheorem 2.1.
2Theorem 3.3⇔Theorem 2.1.
It is enough to show thatTheorem 3.3⇒Theorem 2.1. DefineT :X → 2Xas follows:
Tx
y∈X:f x, y
αλpλ
y−x
e≤C0, ∀λ∈Λ
. 4.10
Obviously, for anyx∈X,Tx /∅. And for eachx∈Xandy∈Tx, f
x, y
αλpλ y−x
e≤C0, ∀λ∈Λ. 4.11
ByTheorem 3.3, there existsz ∈ Tx0 such thatTz {z}. Therefore,x /∈Tzfor anyx ∈ X, x /z. Thus, there existsμ ∈ Λ, such thatfz, x αμpμx−ze /≤C0. Hence, the partiiin Theorem 2.1holds.
3Theorem 3.5⇔Theorem 2.1.
It is enough to show thatTheorem 3.5⇒Theorem 2.1. Without loss of generality we assume thatε1. Let
X0
x∈X :fx0, x αλpλx−x0e≤C0, ∀λ∈Λ
. 4.12
Sincex0 ∈X0, thenX0is nonempty. Also, for anyx ∈ X,fx,·ise, C-locally lower semi continuous, thusX0is locally closed. Hence, if{x∈X:fx0, x≤C0}is locally complete, then {x∈X0:fx0, x≤C0}is locally complete. Suppose that the partiiofTheorem 2.1does not hold, that is, for allx∈X0there existsw /xsuch that
fx, w αλpλw−xe≤C0, ∀λ∈Λ. 4.13 Therefore,
fx0, w αλpλw−x0e
≤Cfx0, w αλpλw−xeαλpλx−x0e
≤Cfx, w fx0, x αλpλw−xeαλpλx−x0e
≤C0.
4.14
Hence,w ∈X0, so byTheorem 3.5, there existsx ∈X0such thatfx0, X∩fx0, x−C {fx0, x}.
However, there exists w ∈ X0 such that w /x which satisfies 4.13. Therefore, fx, w≤C0 and sofx0, , w≤Cfx0, x fx, w≤Cfx0, x, which is a contradiction.
4Theorem 3.7⇔Theorem 2.1.
It is enough to show thatTheorem 3.7⇒Theorem 2.1. Suppose thatT : X → 2X is defined as follows:
Tx
y∈X:f x, y
εαλpλ y−x
e≤C0, ∀λ∈Λ
. 4.15
Choose Ψ {x ∈ X : Tx {x}}. Ifx /∈Ψ, then there exists y ∈ Tx such that y /x.
Therefore, assumption ofTheorem 3.7is satisfied. Hence, there existsz ∈ S0 ∩Ψ. From the definition ofΨthe resultsiandiiofTheorem 2.1are satisfied.
Remark 4.4. aBy the same proof as that ofTheorem 4.3, one can show that Theorems2.5, 3.4, and3.6are equivalent to each other.
b By the same proof as that of Theorems 5.2 and 5.3 in 18 one can prove that Theorem 2.1and the Phelp’s lemma18, Theorem 5.1are equivalent.
Acknowledgment
M. FAKHAR was partially supported by the Center of Excellence for Mathematics, University of Isfahan.
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