Quasicone Metric Spaces and Generalizations of Caristi Kirk’s Theorem

Volume 2009, Article ID 574387,9pages doi:10.1155/2009/574387

Research Article

Quasicone Metric Spaces and Generalizations of Caristi Kirk’s Theorem

Thabet Abdeljawad

and Erdal Karapinar

1Department of Mathematics, C¸ankaya University, 06530 Ankara, Turkey

2Department of Mathematics, Atılım University, 06836 Ankara, Turkey

Correspondence should be addressed to Thabet Abdeljawad,[email protected] Received 4 July 2009; Accepted 3 December 2009

Recommended by Hichem Ben-El-Mechaiekh

Cone-valued lower semicontinuous maps are used to generalize Cristi-Kirik’s fixed point theorem to Cone metric spaces. The cone under consideration is assumed to be strongly minihedral and normal. First we prove such a type of fixed point theorem in compact cone metric spaces and then generalize to complete cone metric spaces. Some more general results are also obtained in quasicone metric spaces.

Copyrightq2009 T. Abdeljawad and E. Karapinar. 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.

1. Introduction and Preliminaries

In 2007, Huang and Zhang 1 introduced the notion of cone metric spaces CMSs by replacing real numbers with an ordering Banach space. The authors there gave an example of a function which is contraction in the category of cone metric spaces but not contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that series of articles about cone metric spaces started to appear. Some of those articles dealt with the extension of certain fixed point theorems to cone metric spacessee, e.g.,2–5, and some other with the structure of the spaces themselves see, e.g., 3, 6. Very recently, some authors have used regular cones to extend some fixed point theorems. For example, in7a result about Meir-Keeler type contraction mappings has been extended to regular cone metric spaces. In other works, some results about fixed points of multifunctions on cone metric spaces with normal cones have been obtained as well8. For the use of lower semicontinuous functions in obtaining fixed point theorems in cone metric spaces we refer to9.

In this manuscript, we use cone-valued lower semicontinuous functions to extend some of the results in Caristi 10 and Ekeland 11 to CMS and quasicone metric space QCMS. The cones under consideration are assumed to be strongly minihedral and normal

and hence regular. In particular the coneP 0,∞in the real lineRis strongly minihedral and normal; hence the results mentioned in the above references are recovered.

Throughout this paperEstands for a real Banach space. LetP :P_Ealways be a closed subset ofE.Pis called cone if the following conditions are satisfied:

C1P /∅,

C2axby∈Pfor allx, y∈Pand non-negative real numbersa, b, C3P∩−P {0}andP /{0}.

For a given coneP, one can define a partial orderingdenoted by≤: or≤Pwith respect toPbyx≤yif and only ify−x∈P. The notationx < yindicates thatx≤yandx /ywhile x ywill showy−x ∈ intP, where intP denotes the interior ofP. From now on, it is assumed that intP/∅.

The conePis called

Nnormal if there is a numberK≥1 such that for allx, y∈E,

0≤x≤y⇒ x ≤Ky; 1.1

Rregular if every increasing sequence which is bounded from above is convergent.

That is, if {xn}_n≥1 is a sequence such thatx₁ ≤ x₂ ≤ · · · ≤ yfor somey ∈ E, then there isx∈Esuch that lim_{n→ ∞}xn−x0.

InN, the least positive integerK, satisfying1.1, is called the normal constant ofP.

Note that, in1,2, normal constantKis stated a positive real number,K > 0. However, later on and in2, Lemma 2.1it was proved that there is no normal cone with constantK <1.

Lemma 1.1. iEvery regular cone is normal.

iiFor eachk >1, there is a normal cone with normal constantK > k.

iii The cone P is regular if every decreasing sequence which is bounded from below is convergent.

The proof ofiandiiwere given in2and the last one just follows from definition.

Example 1.2see2. LetE C¹0,1with the normf f_∞f_∞, and consider the coneP {f ∈E:f≥0}.

For eachk≥1, putfx xandgx x^2k.Then, 0≤g ≤f,f2 andg2k1.

Sincekf<g,kis not normal constant ofPand henceP is a nonnormal cone.

Definition 1.3. LetXbe a nonempty set. Suppose that the mappingd: X×X → Esatisfies the following:

M10≤dx, yfor allx, y∈X, M2dx, y 0 if and only ifxy,

M3dx, y≤dx, z dz, y, for allx, y∈X.

Thendis said to be a quasicone metric onX, and the pairX, dis called a quasicone metric spaceQCMS. Additionally, ifdalso satisfies

M4dx, y dy, xfor allx, y∈X,

thendis called a cone metric onX, and the pairX, dis called a cone metric spaceCMS.

Example 1.4. LetER³andP {x, y, z∈E:x, y, z≥0}andXR. Defined:X×X → E bydx,x α|x −x|, β|x −x|, γ|x −x|, where α, β, γ are positive constants. ThenX, dis a CMS. Note that the conePis normal with the normal constantK1.

Definition 1.5. LetX, dbe a CMS,x∈X, and let{xn}_n≥1be a sequence inX. Then

i{xn}_n≥1 converges toxif for everyc ∈ Ewith 0 cthere is a natural numberN, such thatdxn, xcfor alln≥N. It is denoted by lim_{n→ ∞}x_n xorx_n → x;

ii{xn}_n≥1is a Cauchy sequence if for everyc∈Ewith 0cthere is a natural number N, such thatdxn, x_mcfor alln, m≥N;

iii X, dis a complete cone metric space if every Cauchy sequence inXis convergent inX.

Lemma 1.6see1. LetX, dbe a CMS, letPbe a normal cone with normal constantK, and let {xn}be a sequence inX. Then,

ithe sequence{xn}converges toxif and only if d (xn,x)→ 0or equivalentlydxn, x → 0;

iithe sequence{xn}is Cauchy if and only ifdxn, xm → 0 (or equivalentlydxn, xm → 0);

iiithe sequence{xn}converges toxand the sequence{yn}converges toythendxn, yn → dx, y.

Lemma 1.7see1,2. LetX, dbe a CMS over a coneP inE. Thenone has the following.

1IntP IntP⊆IntPandλIntP⊆IntP,λ >0.

2Ifc0, then there existsδ >0 such thatb< δimpliesbc.

3For any givenc0 andc00 there existsn0∈Nsuch thatc0/n0c.

4Ifa_n,b_nare sequences inEsuch thata_n → a,b_n → banda_n ≤b_nfor alln≥1, then a≤b.

Definition 1.8see12. P is called minihedral cone if sup{x, y}exists for all x, y ∈ E, and strongly minihedral if every subset ofEwhich is bounded from above has a supremum.

It is easy to see that every strongly minihedral normal cone is regular.

Example 1.9. LetE C0,1with the supremum norm andP {f ∈ E : f ≥ 0}.ThenP is a cone with normal constantM 1 which is not regular. This is clear, since the sequence xⁿ is monotonicly decreasing, but not uniformly convergent to 0. Thus, P is not strongly minihedral. It is easy to see that the cone mentioned inExample 1.4is strongly minihedral.

Definition 1.10see1. LetX, dbe a CMS andA⊂X.Ais said to be sequentially compact if for any sequence{xn}inAthere is a subsequence{xnk}of{xn}such that{xnk}is convergent inA.

Remark 1.11see6. Every cone metric spaceX, dis a topological space which is denoted byX, τc. Moreover, a subsetA⊂Xis sequentially compact if and only ifAis compact.

2. Main Results

LetX, dbe a CMS,C⊂X, andϕ:C → Ea function onX. Then, the functionϕis called a lower semicontinuous (l.s.c) onCwhenever

nlim→ ∞xnx⇒ϕx≤ lim

n→ ∞infϕxn:sup

n≥1inf

m≥nϕxm. 2.1

Also, letT :C → Cbe an arbitrary selfmapping onCsuch that

dx, Tx≤ϕx−ϕTx ∀x∈X. 2.2

Then,T is called a Caristi map onX, d.

The following Lemma will be used to prove the next results.

Lemma 2.1. If{cn}is a decreasing sequence (via the partial ordering obtained by the closed coneP) such thatc_n → u, thenuinf{cn:n∈N}.

Proof. Since{cn}is an increasing sequence,cm−cn∈P, forn≥mandcm−cn → cm−u,for all m. Then closeness ofPimplies thatu≤c_mfor allm. To see thatuis the greatest lower bound of{cn}, assume that somev∈Esatisfiesc_m ≥vfor allm. Fromcm−v → u−vand the closeness ofPwe getu−v∈Porv≤uwhich shows thatuinf{cn:n∈N}.

Proposition 2.2. LetX, dbe a compact CMS,Pa strongly minihedral cone, andϕ:X → P ⊂E a lower semicontinuousl.s.cfunction. Then,ϕattains a minimum onX.

Proof. Letuinf{ϕx:x∈X}which exists by strong minihedrality. For eachn∈N, there is anx_n∈Xsuch thatϕxn−uc/n, wherec∈intP. SinceXis compact, then{xn}has a convergent subsequence. Let{yn}be this sequence and letylimy_n.

From the definition of lower semicontinuity andLemma 2.1it follows that

ϕ y

≤ lim

n→ ∞infϕ y_n

lim

n→ ∞inf

u c n

u. 2.3

But then, by the definition ofu,ϕx0≤ϕxfor allx∈X. This completes the proof.

Theorem 2.3. LetX, dbe a CMS,Ca compact subset ofX,Pa strongly minihedral normal cone, and ϕ : C → P ⊂ Ea lower semicontinuousl.s.cfunction. Then, each selfmapT : C → C satisfying2.2has a fixed point inX.

Proof. ByProposition 2.2,ϕattains its minimum at some point ofC, sayu∈C. Sinceuis the minimum point ofϕ, we haveϕTu≥ϕu. By2.2,

0≤du, Tu≤ϕu−ϕTu≤0. 2.4

Thus,du, Tu 0 and soTuu.

The following theorem is an extension of the result of Caristi10, Theorem 2.1. Theorem 2.4. LetX, dbe a complete CMS,P a strongly minihedral normal cone, andϕ :X → P ⊂Ea lower semicontinuousl.s.cfunction. Then, each selmapT :X → Xsatisfying2.2has a fixed point inX.

Proof. LetP have the normal constantK. LetSx : {z ∈ X : dx, z ≤ ϕx−ϕz}and αx:inf{ϕz:z∈Sx}for allx∈X. Sincex∈Sx,Sx/∅and so 0≤αx≤ϕx.

Forx ∈ X, setx1 : xand construct a sequencex1, x2, x3, . . . , xn, . . . in the following way: letx_n1∈Sxnbe such thatϕx_n1≤αxn c₀/n, wherec₀ ∈IntP/∅. Thus, one can observe that

idxn, x_n1≤ϕxn−ϕx_n1, iiαxn≤ϕxn1≤αxn c0/n

for alln≥1. Note that,iimplies that the sequence{ϕxn}is a decreasing sequence inEand P is regular cone. So, the sequence{ϕxn}is convergent. Thus, for eachε > 0, there exists Nεsuch thatϕxm−ϕxn< ε/Kfor alln,m≥Nε. Form≥n, the triangular inequality implies that

dxn, xm≤^m−1

jn

d

xj, x_j1

≤ϕxn−ϕxm. 2.5

Hence,dx, y ≤Kϕxn−ϕxm< Kε/K ε. ByLemma 1.6,dxn, x_m → 0 yields that the sequence{xn}is a Cauchy inX. Completeness ofX, dimplies that the sequence {xn}is convergent to some point inX, sayy.

By2.5,ϕxn−ϕxm−dxm, x_n∈P and so

ϕxm≤ϕxn−dxm, x_n 2.6

for allm≥n. By regarding2.6,Lemma 1.6, and lower semicontinuity of the functionϕ, one can obtain that

ϕ y

≤ lim

m→ ∞infϕxm≤ lim

m→ ∞inf ϕxn−dxm, x_n

ϕxn−d x_n, y

2.7

for alln≥1. Thus,

0≤d xn, y

≤ϕxn−ϕ y

2.8

for alln≥ 1. Hence,y∈Sxnand it is trivial thatϕxn≤ϕyfor alln ≥1. Note thatii implies that

α: lim

n→ ∞αxn lim

n→ ∞ϕxn. 2.9

Thus,α≤ϕxnfor alln≥1. On the other hand, by lower semicontinuity ofϕand2.9, one can obtain that

ϕ y

≤ lim

n→ ∞infϕxn α. 2.10

Therefore,αϕy.

Sincey∈Sxnfor eachn≥1 andTy∈Sy, the following inequalities are obtained:

d xn, Ty

≤d xn, y

d y, Ty

≤ϕxn−ϕ y

ϕ y

−ϕ Ty

ϕxn−ϕ Ty

. 2.11 Hence,Ty∈Sxnfor alln≥1. This implies thatαxn≤ϕTyfor alln≥1.

By2.9,ϕTy≥αis obtained. AsϕTy≤ϕyis observed by2.2and thatϕy α, then

ϕ y

α≤ϕ Ty

≤ϕ y

2.12

is achieved. Hence,ϕTy ϕy. Finally, by2.2we haveTyy.

The following theorem is a generalization of the result in11.

Theorem 2.5. Let ϕ : X → E be a l.s.c function on a complete CMS, where P is a strongly minihedral normal cone. Ifϕis bounded below, then there exitsy∈Xsuch that

ϕ y

< ϕx d y, x

∀x∈X withx /y. 2.13

Proof. It is enough to show that the pointy, obtained inTheorem 2.4, satisfies the statement of the theorem. Following the same notation in the proof ofTheorem 2.4, it is needed to show thatx /∈Syforx /y. Assume the contrary that for somez /y, we havez /∈Sy. Then, 0<

dy, z≤ϕy−ϕzimpliesϕz< ϕy α. By triangular inequality, dxn, z≤d

xn, y d

y, z

≤ϕxn−ϕ y

ϕ y

−ϕz ϕxn−ϕz, 2.14 which implies thatz ∈ Sxnand thusαxn ≤ ϕyfor alln ≥ 1. Taking the limit whenn tends to infinity, one can obtainα ≤ ϕz, which is in contradiction withϕz < ϕy α.

Thus, for anyx∈X,x /yimpliesx /∈Sy, that is, x /y⇒d

y, x

> ϕ y

−ϕx. 2.15

Letd_x:X → Ebe defined byd_xy:dx, y.

Theorem 2.6. LetX, dbe a sequentially complete QCMS and letPbe a strongly minihedral normal cone. Assume that for eachx∈X, the functiond_xdefined above is continuous onXandFis a family of mappingsf :X → X. If there exists a l.s.c functionϕ:X → P such that

d

x, fx

≤ϕx−ϕ fx

, ∀x∈X, ∀f ∈ F, 2.16

then for eachx∈Xthere is a common fixed pointuofFsuch that

dx, u≤ϕx−s, wheresinf

ϕx:x∈X

. 2.17

Proof. Let P be strongly minihedral normal cone with normal constant K. First note that strong minihedrality ofPguarantees thatsexists. LetSx:{z∈X:dx, y≤ϕx−ϕz}

and αx : {ϕz : z ∈ Sx} for all x ∈ X. Note that x ∈ Sx, so Sx/∅ and also 0≤αx≤ϕx.

Forx∈X, setx₁ : xand construct a sequencex₁, x₂, x₃, . . . , x_n, . . .as in the proof of Theorem 2.4:x_n1 ∈Sxnsuch thatϕxn1 ≤ αxn c0/n,c0 0. Thus, one can observe that for eachn,

idxn, x_n1≤ϕxn−ϕx_n1, iiαxn≤ϕxn1≤αxn c0/n.

Similar to the proof ofTheorem 2.4,iiimplies that

α: lim

n→ ∞αxn lim

n→ ∞ϕxn. 2.18

Also, by using the same method in the proof ofTheorem 2.4, it can be shown that{xn} is a Cauchy sequence and converges to somey∈Xandϕy α.

We shall show thatfy yfor allf∈ F. Assume the contrary that there isf ∈ Fsuch thatfy/y. Then2.16withxyimplies thatϕfy< ϕy α.Thus, by definition of α, there isn∈Nsuch thatϕfy< αxn. Sincey∈Sxn,

d x_n, f

y

≤d x_n, y

d y, f

y

≤ ϕxn−ϕ y

ϕ y

−ϕ f

y

ϕxn−ϕ f

y , 2.19 which implies thatfy ∈ Sxn. Henceαxn ≤ ϕfywhich is in a contradiction with ϕfy< αxn.Thus,fy yfor allf∈ F. Sincey∈Sxn, we have

d x_n, y

≤ϕxn−ϕ y

≤ϕxn−inf

ϕz:z∈X

ϕx−s 2.20 is obtained.

The following theorem is a generalization of13, Theorem 2.2.

Theorem 2.7. Let Abe a set,X, das in Theorem 2.6,g : A → X a surjective mapping, and F{f}a family of arbitrary mappingsf:A → X. If there exists al.c.s.functionϕ:X → Psuch that

d

ga, fa

≤ϕ ga

−ϕ fa

, ∀f ∈ F 2.21

and eacha∈A, thengandFhave a common coincidence point, that is, for someb∈A,gb fb for allf∈ F.

Proof. Letxbe arbitrary andy ∈ X as inTheorem 2.6. Sinceg is surjective, for eachx ∈ X there is somea axsuch thatga x. Let f ∈ F be a fixed mapping. Define byf a mappinghhfofXinto itself such thathx fa, whereaax, that is,ga x. Let Hbe a family of all mappingshhf. Then,2.21yields that

dx, hx≤ϕx−ϕhx, ∀h∈ H. 2.22

Thus, byTheorem 2.6,yhyfor allh∈ H. Hencegb fbfor allf ∈ F, wherebby is such thatgb y.

Acknowledgment

This work is partially supported by the Scientific and Technical Research Council of Turkey.

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