Best Approximations Theorem for a Couple in Cone Banach Space

(1)

Volume 2010, Article ID 784578,9pages doi:10.1155/2010/784578

Research Article

Best Approximations Theorem for a Couple in

Cone Banach Space

Erdal Karapınar

1

_{and Duran T ¨urko ˘glu}

2

1_{Department of Mathematics, At}_ı_l_ı_{m University, 06836 Incek, Ankara, Turkey} 2_{Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey}

Correspondence should be addressed to Erdal Karapınar,erdalkarapinar@yahoo.com

Received 23 March 2010; Accepted 8 June 2010

Academic Editor: A. T. M. Lau

Copyrightq2010 E. Karapınar and D. T ¨urko ˘glu. 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.

The notion of coupled fixed point is introduced by Bhaskar and Lakshmikantham,2006. In this manuscript, some result of Mitrovi´c,2010extended to the class of cone Banach spaces.

1. Introduction and Preliminaries

Banach, valued metric space was considered by Rzepecki1, Lin2, and lately by Huang and Zhang3. Basically, for nonempty setX, the definition of metricd:X×X → R 0,∞

is replaced by a new metric, namely, by an ordered Banach space E: d : X × X → E. Such metric spaces are called cone metric spaces in short CMSs. In 1980, by using this idea Rzepecki 1 generalized the fixed point theorems of Maia type. Seven years later, Lin2extends some results of Khan and Imdad4by considering this new metric space construction. In 2007, Huang and Zhang3discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces: any mappingT of a complete cone metric spaceXinto itself that satisfies, for some 0≤k <1, the inequality

dTx, Ty≤kdx, y 1.1

(2)

Throughout this paperEstands for real Banach space. LetP : PE always be closed

subset ofE.Pis calledconeif the following conditions are satisfied:

C1P /∅,

C2axby∈Pfor allx, y∈Pand nonnegative real numbersa, b,

C3P∩−P {0}andP /{0}.

For a given coneP, one can define a partial orderingdenoted by≤or≤Pwith respect toP

byx ≤ yif and only ify−x ∈ P. The notationx < yindicates thatx ≤ yandx /ywhile

x ywill showy−x∈intP, where intP denotes the interior ofP. It can be easily shown that intPintP ⊂ intP andλintP ⊂intP where 0< λ ∈R. Throughout this manuscript intP /∅.

The conePis called

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

0≤x≤y⇒ x ≤Ky; 1.2

Rregularif 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 limn→ ∞xn−x0.

InN, the least positive integerKsatisfying1.2is called the normal constant ofP. Note that, in3,5, normal constantKis considered a positive real number,K >0, although it is proved that there is no normal cone forK <1 insee e.g., Lemma 2.1,5.

Lemma 1.1see e.g.,13. One has the following.

iEvery regular cone is normal.

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

iiiThe cone P is regular if every decreasing sequence which is bounded from below is

convergent.

Definition 1.2see14. 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.

Example 1.3. LetE C0,1with the supremum norm andP {f ∈ E : f ≥ 0}.Since the

sequencexn _{is monotonically decreasing, but not uniformly convergent to 0, thus,}_P _{is not}

strongly minihedral.

Definition 1.4. LetXbe 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.

M4dx, y dy, xfor allx, y∈X.

(3)

Example 1.5. LetER3_and_P _{_{x, y, z}_∈_E_:_{x, y, z}_≥_0}_and_X_R_{. Define}_d_:_X_×_X _→ _E

bydx,x α|x−x|, β|x−x|, γ|x−x|, whereα, β, γ are positive constants. ThenX, dis a CMS. Note that the conePis normal with the normal constantK1.

It is quite natural to consider Cone Normed SpacesCNSs.

Definition 1.6see e.g.,9,15,16. LetXbe a vector space overR. Suppose that the mapping · P :X → Esatisfies the following:

N1xP >0 for allx∈X, N2xP 0 if and only ifx0,

N3xyP ≤ xP yP, for allx, y∈X. N4kxP |k|xP for allk∈R.

Then · P is called cone norm onX, and the pairX, · Pis called a cone normed space CNS.

Note that each CNS is CMS. Indeed,dx, y x−yP.

Definition 1.7. LetX, · Pbe a CNS,x∈X,and{xn}n≥1a sequence inX. Then one has the

following.

i{xn}n≥1 converges to x whenever for every c ∈ E with 0 c there is a natural

numberN, such thatxn−xP cfor alln≥N. It is denoted by limn→ ∞xnxor

xn → x.

ii{xn}n≥1is aCauchy sequencewhenever for everyc∈Ewith 0cthere is a natural

numberN, such thatxn−xP cfor alln, m≥N.

iii X, · Pis a complete cone normed space if every Cauchy sequence is convergent.

Complete cone-normed spaces will be called cone Banach spaces.

Lemma 1.8. LetX, · Pbe a CNS, letPbe a normal cone with normal constantK, and let{xn}be a sequence inX. Then, one has the following:

ithe sequence{xn}converges toxif and only ifxn−xP → 0, asn → ∞, iithe sequence{xn}is Cauchy if and only ifxn−xmP → 0asn, m → ∞,

iiithe sequence {xn} converges to x and the sequence {yn} converges to y and then

xn−ynP → x−yP.

The proof is direct by applying Lemmas 1, 4, and 5 in3to the cone metric space

X, d, wheredx, y x−yP, for allx, y∈X.

Lemma 1.9see, e.g.,6,7. LetX,·Pbe a CNS over a conePinE. Then (1)IntPIntP⊆

IntPandλIntP⊆IntP, λ >0. (2) Ifc0,then there existsδ > 0such thatb< δimplies

(4)

Definition 1.10. LetX, dbe a CNS and letI 0,1be the closed unit interval. A continuous mappingW :X×X×I :→ X is said to be a convex structure onX if for allx, y ∈ Xand

t∈I

u−Wx, y, t_P ≤tu−x_P 1−tu−y_P 1.3

holds for allu∈X. A CNSX, dtogether with a convex structure is said to be convex CNS. A subsetY ⊂Xis convex, ifWx, y, t∈Y holds for allx, y∈Xandt∈I.

Definition 1.11. LetXbe a CNS, andKandCthe nonempty convex subsets ofX. A mapping

g:K → Xis said to bealmost quasiconvex with respecttoCif

gtx 1−ty−z≤cg

tx 1−ty, z, 1.4

where cgtx 1 − ty, z ∈ {gx−z_P,gy−z_P} for all x, y ∈ K, z ∈ C, and

0< t <1.

2. Couple Fixed Theorems on Cone Metric Spaces

Let X, d be a CMS and X2 _: _X _× _X_{. Then the mapping} _ρ _: _X2 _× _X2 _:_→ _E _such

that ρx1, y1,x2, y2 : dx1, x2 dy1, y2 forms a cone metric on X2. A sequence {xn},{yn} ∈ X2 is said to be a double sequence of X. A sequence {xn},{yn} ∈ X2 is

convergent tox,y∈ X2_{if, for every}_c _∈_int_P_{, there exists a natural number}_{M >} _{0 such}

thatρxn, yn,x, ycfor alln > M.

Lemma 2.1. Letzn xn, yn∈X2andz x, y∈X2. Then,zn → zif and only ifxn → xand

yn → y.

Proof. Suppose zn → z. Thus, for any c ∈ intP, there exist M > 0 such that

ρxn, yn,x, y dxn, x dyn, y c for all n > M. Hence, dxn, x c and

dyn, ycfor alln > M, that is,xn → xandyn → y.

Conversely, assume xn → x and yn → y. Thus, for any c ∈ intP, there exist

M0, M1 > 0 such that dxn, x c/2 for all n > M0, and also dyn, y c/2 for

all n > M1. Hence, ρxn, yn,x, y dxn, x dyn, y c for all n > M, where

M:max{M0, M1}.

Definition 2.2. Let X, d be a CMS. A function f : X → X is said to be sequentially

continuous if dxn, x → 0 implies that dfxn, fx → 0. Analogously, a function

F : X × X :→ X is sequentially continuous if ρxn, yn,x, y → 0 implies that

dFxn, yn, Fx, y → 0.

Lemma 2.3see6. LetX, dbe a CNS. Thenf:X, d → X, dis continuous if and only iff

(5)

Definition 2.4see10,17,18. LetX,be partially ordered set andF : X ×X → X.F

is said to have mixed monotone property ifFx, yis monotone nondecreasing inxand is monotone nonincreasing iny, that is, for anyx, y∈X,

x1x2⇒F

x1, y

Fx2, y

, forx1, x2∈X,

y1y2⇒F

x, y2

Fx, y1

, fory1, y2∈X.

2.1

Note that this definition reduces the notion of mixed monotone function onR2_where

represents usual total order≤inR2_.

Definition 2.5see10,17,18. An elementx, y∈X×Xis said to be a couple fixed point of the mappingF:X×X → Xif

Fx, yx, Fy, xy. 2.2

Throughout this paper, letX,be partially ordered set and letdbe a cone metric on

X such thatX, dis a complete CMS over the normal coneP with the normal constantK. Further, the product spacesX×Xsatisfy the following:

u, vx, y⇐⇒ux, yv; ∀x, y,u, v∈X×X. 2.3

Definition 2.6see3. 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 2.7see19. Every cone metric spaceX, dis a topological space which is denoted

byX, τc. Moreover, a subsetA⊂Xis sequentially compact if and only ifAis compact.

Definition 2.8. LetKbe a nonempty subset of a CNSX, d. A set-valued mapH:K → 2X_is

called KKM map if for every finite subset{x1, x2, . . . , xn}ofK

co{x1, x2, . . . , xn} ⊂ n

i1

Hxi, 2.4

where co denotes the convex hull.

Lemma 2.9. Let X be a topological vector space, let K be a nonempty subset of X, and let H :

K → 2X _{be called KKM map with closed values. If}_H_x_{is compact for at least one}_x _∈ _K,_then

(6)

Theorem 2.10. LetX, · Pbe a CNS over strongly minidhedral coneP, and letKbe a nonempty

convex compact subset ofX. IfF:K×K → Xis continuous mapping andg :K → Xis continuous

almost quasiconvex mapping with respect toFK×K, then there existsx0, y0∈K×Ksuch that

gx0−Fx, y_Pgy0−Fy, x_P

inf

x,y∈K×K

_g_x₋_F

x, y_Pgy−Fy, x_P. 2.5

Proof. LetH:K×K → 2K×K_by

Hu, v x, y∈K×K:gx−Fx0, y0_Pg

y−Fy0, x0_P

≤gu−Fx0, y0_Pgv−Fy0, x0_P

2.6

for eachu, v∈K×K. Sinceu, v∈Hu, v, thenHu, v/∅. Regarding that the mappings Fandgare continuous,Hu, vis closed for eachu, v. SinceKis compact, thenHu, vis compact for eachu, v. Thus,His a KKM map.

Letui, vj∈K×K,i∈I, j ∈JwhereIandJare finite subsets ofN. Then, there exists

u0, v0∈co

ui, vj

:i, j∈I×J, so thatu0, v0/∈ ∪

Hui, vj

:i, j∈I×J. 2.7

From the first expression in 2.7, one can get that there exist tij ≥ 0,i, j ∈ I ×J such

that u0, v0 i,j∈I×Jtijui, vj and i,j∈I×Jtij 1. Set ti j∈Jtij and zj i∈Itij

then_j_∈_Jzj 1,

i∈Iti 1 and

j∈Jzjvj v0,

i∈Itiui u0. Regarding thatg is almost

quasiconvex with respect toF :K×K → Xyields

gu0−Fu0, v0_P ≤cgu0, Fu0, v0, _g_v₀₋_F_v₀_{, u}₀

P ≤cgv0, Fv0, v0.

2.8

where cgu0, Fu0, v0 ∈ {gui−Fu0, v0_P : i ∈ I} and cgv0, Fv0, u0 ∈ {gvj−

Fv0, u0P : j∈J}.

Thus

_g_u₀₋_F_u₀_{, v}₀

P gv0−Fv0, u0P

≤infgui−Fu0, v0_P :i∈I

infgvj

−Fv0, u0_P :j ∈J

.

2.9

Taking2.7into account, one can get

_g_u₀₋_F_u₀_{, v}₀

(7)

for alli, j∈I×Jwhich is a contradiction. HenceHis a KKM mapping. It follows that there existsx0, y0∈K×Ksuch thatx0, y0∈Hx, yfor allx, y∈K×K. Thus,

gx0−F

x0, y0_P g

y0

−Fy0, x0_P

≤gx−Fx0, y0_Pg

y−Fy0, x0_P, ∀

x, y∈K×K.

2.11

convex compact subset ofX. IfF:K×K → Xis continuous mapping andg :X → Kis continuous

almost quasiconvex mapping with respect toFK×Ksuch thatFK×K⊂gK, thenF andg

have a coupled coincidence point.

Proof. Due toTheorem 2.10, there existsx0, y0∈K×Ksuch that

gx0−F

x0, y0g

y0

−Fy0, x0_P

inf

x,y∈K×K

_g_x₋_F

x0, y0_P g

y−Fy0, x0_P

. 2.12

SinceFK×K⊂gK,

inf x,y∈K×K

gx−Fx0, y0_P g

y−Fy0, x0_P

0, _2.13

thengx0−Fx0, y0Pgy0−Fy0, x0P 0.

Thus,gx0 Fx0, y0andgy₀ Fy0, x0.

If we takeg : K → X as an identity, gx x, in Theorem 2.11, then we get the following result.

convex compact subset ofX. IfF :K×K → Kis continuous mapping, thenF has a coupled fixed

point.

convex compact subset ofX. IfF :K×K → Xis continuous mapping, then eitherFhas a coupled

fixed point or there existsx0, y0∈∂K×K∪K×∂Ksuch that

0<x0−Fx0, y0_P y0−Fy0, x0_P ≤x−Fx0, y0_Py−Fy0, x0_P 2.14

(8)

Proof. IfFhas a coupled fixed point, then we are done. Suppose thatFhas no coupled fixed points. Due toTheorem 2.10, there existsx0, y0∈K×Ksuch that

_g_x₀₋_F_x₀_{, y}₀

P gy0−Fy0, x0P

inf

x,y∈K×K

gx−Fx0, y0_P g

y−Fy0, x0_P

.

2.15

Takegx xwhich implies2.14. It is suﬃcient to show thatx0, y0∈∂K×K∪K×∂K.

The inequality2.14implies that eitherFx0, y0/∈KorFy0, x0/∈K.

Consider the first case:Fx0, y0/∈K. Supposex0 ∈ intK. SinceK is convex, then

there existst∈0,1such thatxtx0 1−tFx0, y0∈K. Thusx−Fx0, y0P tx0−

Fx0, y0P and

inf

x∈K

_x₋_F_x₀_{, y}₀

P ≤tx0−Fx0, y0P <x0−Fx0, y0P. 2.16

This is a contradiction. Analogously one can get the contradiction from the caseFy0, x0/∈K.

Thus,x0, y0∈∂K×K∪K×∂K.

convex compact subset ofX. Suppose thatF : K×K → Xis continuous mapping. Then Fhas a

coupled fixed point if one of the following conditions is satisfied for allx0, y0∈∂K×K∪K×∂K

such thatx, y/ Fx, y, Fy, x:

ithere exists au, v∈K×Ksuch that

_u₋_F_{x, y}

P <x−Fx, yP, v−Fy, xP <y−Fy, xP, 2.17

iithere exists ant∈0,1such that

K∩BFx, y, tx−Fx, y_P/∅, K∩BFy, x, ty−Fy, x_P/∅

2.18

iii{Fx, y, Fy, x} ⊂K.

(9)

References

1 B. Rzepecki, “On fixed point theorems of Maia type,”Publications de l’Institut Math´ematique, vol. 28, pp. 179–186, 1980.

2 S.-D. Lin, “A common fixed point theorem in abstract spaces,”Indian Journal of Pure and Applied Mathematics, vol. 18, no. 8, pp. 685–690, 1987.

3 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.

4 M. S. Khan and M. A. Imdad, “A common fixed point theorem for a class of mappings,”Indian Journal of Pure and Applied Mathematics, vol. 14, no. 10, pp. 1220–1227, 1983.

5 Sh. Rezapour and R. Hamlbarani, “Some notes on the paper: “cone metric spaces and fixed point theorems of contractive mappings”,”Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719–724, 2008.

6 D. Turkoglu and M. Abuloha, “Cone metric spaces and fixed point theorems in diametrically contractive mappings,”Acta Mathematica Sinica, vol. 26, no. 3, pp. 489–496, 2010.

7 D. Turkoglu, M. Abuloha, and T. Abdeljawad, “KKM mappings in cone metric spaces and some fixed point theorems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 348–353, 2010.

8 C. Di Bari and P. Vetro, “φ-pairs and common fixed points in cone metric spaces,”Rendiconti del Circolo Matematico di Palermo, vol. 57, no. 2, pp. 279–285, 2008.

9 E. Karapınar, “Fixed point theorems in cone Banach spaces,”Fixed Point Theory and Applications, Article ID 609281, 9 pages, 2009.

10 E. Karapınar, “Couple fixed point theorems for nonlinear contractions in cone metric spaces,” Computers and Mathematics with Applications, vol. 59, no. 12, pp. 3656–3668, 2010.

11 S. Radenovi´c and B. E. Rhoades, “Fixed point theorem for two non-self mappings in cone metric spaces,”Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1701–1707, 2009.

12 Z. D. Mitrovi´c, “A coupled best approximations theorem in normed spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 11, pp. 4049–4052, 2010.

13 T. Abdeljawad and E. Karapınar, “Quasicone metric spaces and generalizations of Caristi Kirk’s theorem,”Fixed Point Theory and Applications, Article ID 574387, 9 pages, 2009.

14 K. Deimling,Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

15 T. Abdeljawad, “Completion of cone metric spaces,”Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 1, pp. 67–74, 2010.

16 T. Abdeljawad, D. Turkoglu, and M. Abuloha, “Some theorems and examples of cone Banach spaces,” Journal of Computational Analysis and Applications, vol. 12, no. 4, pp. 739–753, 2010.

17 T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,”Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.

18 V. Lakshmikantham and L. ´Ciri´c, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.