Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

Volume 2010, Article ID 170253,17pages doi:10.1155/2010/170253

Research Article

Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

Zoran Kadelburg,

Stojan Radenovi ´c,

and Vladimir Rako ˇcevi ´c

1Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia

2Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

3Department of Mathematics, Faculty of Sciences and Mathematics, University of Niˇs, Viˇsegradska 33, 18000 Niˇs, Serbia

Correspondence should be addressed to Stojan Radenovi´c,sradenovic@mas.bg.ac.rs Received 18 December 2009; Revised 14 July 2010; Accepted 19 July 2010

Academic Editor: Hichem Ben-El-Mechaiekh

Copyrightq2010 Zoran Kadelburg 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 develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory ofnormed-valuedcone metric spaces. Examples are given to distinguish our results from the known ones.

1. Introduction

Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method1–4and in optimization theory5.K-metric and K-normed spaces were introduced in the mid-20th century2, see also3,4,6by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric.

Huang and Zhang7reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. These and other authorssee, e.g.,8–22proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces and cone uniform spaces.

In some of the mentioned papers, results were obtained under additional assumptions about the underlying cone, such as normality or even regularity. In the papers23,24, the authors tried to generalize this approach by using cones in topological vector spacestvs instead of Banach spaces. However, it should be noted that an old resultsee, e.g.,3shows

that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space. So, proper generalizations when passing from norm-valued cone metric spaces of 7 to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones.

In the present paper we develop further the theory of topological vector space valued cone metric spaceswith nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory ofnormed-valuedcone metric spaces.

Examples are given to distinguish our results from the known ones.

2. Tvs-Valued Cone Metric Spaces

LetEbe a real Hausdorfftopological vector space tvs for shortwith the zero vectorθ. A proper nonempty and closed subsetPofEis called aconvex cone ifPP ⊂P,λP ⊂P for λ≥0 andP∩−P θ. We will always assume that the cone P has a nonempty interior int P such cones are called solid.

Each coneP induces a partial orderonEbyx y ⇔ y−x ∈P.x ≺ ywill stand forx yandx /y, whilex ywill stand fory−x ∈ intP. The pairE, Pis an ordered topological vector space.

For a pair of elementsx, yinEsuch thatxy, put x, y

z∈E:xzy

. 2.1

The sets of the formx, yare called order intervals. It is easily verified that order-intervals are convex. A subsetAofEis said to be order-convex ifx, y⊂A, wheneverx, y∈Aandxy.

Ordered topological vector spaceE, Pis order-convex if it has a base of neighborhoods ofθconsisting of order-convex subsets. In this case the conePis said to be normal. In the case of a normed space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is a number k such that x, y ∈ Eand 0 x y implies that x ≤ky. Another equivalent condition is that

inf

xy:x, y∈P andxy1

>0. 2.2

It is not hard to conclude from2.2thatP is a nonnormal cone in a normed spaceEif and only if there exist sequencesun, vn∈Psuch that

0ununvn, unvn−→0 butun0. 2.3 Hence, in this case, the Sandwich theorem does not hold.

Note the following properties of bounded sets.

If the coneP is solid, then each topologically bounded subset ofE, Pis also order- bounded, that is, it is contained in a set of the form−c, cfor somec∈int P.

If the cone P is normal, then each order-bounded subset of E, P is topologically bounded. Hence, if the cone is both solid and normal, these two properties of subsets ofE coincide. Moreover, a proof of the following assertion can be found, for example, in3.

Theorem 2.1. If the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.

Example 2.2. see5LetEC¹_R0,1withx x∞x∞, and letP {x∈E:xt≥ 0 on0,1}. This cone is solidit has the nonempty interiorbut is not normal. Consider, for example,x_nt 1−sinnt/n2andy_nt 1sinnt/n2. Sincexn yn 1 andxny_n2/n2 → 0, it follows thatPis a nonnormal cone.

Now consider the space E C¹_R0,1 endowed with the strongest locally convex topologyt^∗. ThenPis alsot^∗-solidit has the nonemptyt^∗-interior, but nott^∗-normal. Indeed, if it were normal then, according toTheorem 2.1, the spaceE, t^∗would be normed, which is impossible since an infinite-dimensional space with the strongest locally convex topology cannot be metrizablesee, e.g.,25.

Following7,23,24we give the following.

Definition 2.3. LetXbe a nonempty set andE·Pan ordered tvs. A functiond:X×X → E is called a tvs-cone metric andX, dis called a tvs-cone metric, space if the following conditions hold:

C1θdx, yfor allx, y∈Xanddx, y θif and only ifxy;

C2dx, y dy, xfor allx, y∈X;

C3dx, zdx, y dy, zfor allx, y, z∈X.

Letx∈Xand{xn}be a sequence inX. Then it is said the following.

i{xn}tvs-cone converges toxif for everyc ∈ Ewithθ cthere exists a natural numbern₀such thatdxn, x cfor alln > n₀; we denote it by lim_{n→ ∞}x_n xor xn → xasn → ∞.

ii{xn} is a tvs-cone Cauchy sequence if for everyc ∈ Ewith 0 cthere exists a natural numbern₀such thatdxm, x_n cfor allm, n > n₀.

iii X, dis tvs-cone complete if every tvs-Cauchy sequence is tvs-convergent inX.

Taking into accountTheorem 2.1, proper generalizations when passing from norm- valued cone metric spaces of7to tvs-cone metric spaces can be obtained only in the case of nonnormal cones.

We will prove now some properties of a real tvsEwith a solid coneP and a tvs-cone metric spaceX, dover it.

Lemma 2.4. (a) Letθxn → θinE, P, and letθ c. Then there existsn0such thatxn cfor eachn > n₀.

(b) It can happen thatθxn cfor eachn > n0, butxnθinE, P.

(c) It can happen that xn → x,yn → y in the tvs-cone metricd, but thatdxn, yn dx, yinE, P. In particular, it can happen thatx_n → xindbut thatdxn, x θ(which is impossible if the cone is normal).

(d)θu cfor eachc∈intPimplies thatuθ.

(e)x_n → x∧x_n → y(in the tvs-cone metric) implies thatxy.

(f) Each tvs-cone metric space is Hausdorffin the sense that for arbitrary distinct pointsxand ythere exist disjoint neighbourhoods in the topologytchaving the local base formed by the sets of the formK_cx {z∈X:dx, z c},c∈intP.

Proof. aIt follows fromxn → θthatxn ∈ int−c, c intP−c∩c−intPforn > n0. Fromx_n∈c−intP, it follows thatc−x_n∈intP, that is,x_n c.

bConsider the sequencesx_nt 1−sinnt/n2andy_nt 1sinnt/n2 fromExample 2.2. We know that in the ordered Banach spaceC¹_R0,1

θx_nx_ny_n 2.4

and thatxnyn → θin the norm ofEbut thatxn θin this norm. On the other hand, sincex_nx_ny_n → θandx_n x_ny_n c, it follows thatx_n c. Then alsox_n θin the tvsE, t^∗ the strongest locally convex topologybutxn calso considering the interior with respect tot^∗.

We can also consider the tvs-cone metricd : P×P → Edefined bydx, y xy, x /y, anddx, x θ. Then for the sequence{xn}we have thatdxn, θ x_nθx_n → θin the tvs-cone metric, sincexn c, butxnθin the tvsE, t^∗for otherwise it would tend to θin the norm of the spaceE.

cTake the sequence{xn}frombandy_n θ. Thenx_n → θ, andy_n → θin the cone metricdsincedxn, θ xnθ xn canddyn, θ ynθ θθ θ c, but dxn, y_n x_ny_n x_n θdθ, θinE, t^∗. This means that a tvs-cone metric may be a discontinuous function.

dThe proof is the same as in the Banach case. For an arbitraryc∈intP, it isθu 1/ncfor eachn∈N, and passing to the limit inθ −u 1/ncit follows thatθ −u, that is,u∈ −P. SinceP is a cone it follows thatuθ.

eFromdx, y dx, xn dxn, y c/2c/2 cfor eachn > n0it follows that dx, y cfor arbitraryc∈intP, which, byd, means thatxy.

fSuppose, to the contrary, that for the given distinct pointsxandy there exists a point z ∈ Kcx∩Kcy. Then dx, y dx, z dz, y c/2c/2 c for arbitrary c∈intP, implying thatxy, a contradiction.

The following properties, which can be proved in the same way as in the normed case, will also be needed.

Lemma 2.5. (a) Ifuvandv w, thenu w.

(b) Ifu vandvw, thenu w.

(c) Ifu vandv w, thenu w.

(d) Letx ∈ X,{xn}and{bn}be two sequences in X andE, respectively,θ c, and 0 dxn, xb_nfor alln∈N. Ifb_n → 0, then there exists a natural numbern₀such thatdxn, x c for alln≥n₀.

3. Fixed Point and Common Fixed Point Results

Theorem 3.1. LetX, dbe a tvs-cone metric space and the mappingsf, g, h:X → Xsatisfy d

fx, gy pd

hx, hy qd

hx, fx rd

hy, gy sd

hx, gy td

hy, fx

, 3.1

for allx, y∈X, wherep, q, r, s, t≥0,pqrst <1, andqrorst. IffX∪gX⊂hX andhXis a complete subspace ofX, thenf,g, andhhave a unique point of coincidence. Moreover, iff, handg, hare weakly compatible, thenf,g, andhhave a unique common fixed point.

Recall that a pointu ∈ X is called a coincidence point of the pairf, gandvis its point of coincidence iffuguv. The pairf, gis said to be weakly compatible if for each x∈X,fxgximplies thatfgxgfx.

Proof. Letx0∈Xbe arbitrary. Using the conditionfX∪gX⊂hXchoose a sequence{xn} such thathx_2n1 fx2n andhx_2n2 gx_2n1 for alln ∈N0. Applying contractive condition 3.1we obtain that

dhx2n1, hx_2n2 d

fx2n, gx_2n1

pdhx2n, hx_2n1 qdhx2n, hx_2n1 rdhx_2n1, hx_2n2 sdhx2n, hx_2n2 tdhx2n1, hx_2n1

pdhx2n, hx_2n1 qdhx2n, hx_2n1 rdhx2n1, hx_2n2 sdhx2n, hx_2n1 dhx_2n1, hx_2n2.

3.2

It follows that

1−r−sdhx2n1, hx_2n2

pqs

dhx2n, hx_2n1, 3.3

that is,

dhx2n1, hx_2n2 pqs

1−rsdhx2n, hx_2n1. 3.4

In a similar way one obtains that

dhx2n2, hx_2n3 pqt 1−

qt· pqs

1−rsdhx2n, hx_2n1. 3.5 Now, from3.4and3.5, by induction, we obtain that

dhx2n1, hx_2n2 pqs

1−rsdhx2n, hx_2n1 pqs

1−rs· prs 1−

qtdhx_2n−1, hx_2n pqs

1−rs· prs 1−

qt· pqs

1−rsdhx2n−2, hx_2n−1 · · · pqs

1−rs

prt 1−

qt · pqs 1−rs

n

dhx0,hx₁,

dhx2n2, hx_2n3 prt 1−

qtdhx2n1, hx_2n2 · · ·

prt 1−

qt· pqs 1−rs

n1

dhx0, hx1.

3.6

Let

A pqs

1−rs, B prt 1−

qt. 3.7

In the caseqr,

AB pqs 1−

qs· prt 1−

qt pqs 1−

qt· prt

1−rs <1·11, 3.8 and ifst,

AB pqs

1−rs · prs 1−

qt <1·11. 3.9

Now, forn < m, we have

dhx2n1, hx_2m1dhx2n1, hx_2n2 · · ·dhx2n, hx_2m1

A

m−1

in

AB^{i m}

in1

ABⁱ dhx0, hx₁

AABⁿ

1−AB ABⁿ¹

1−AB dhx0, hx₁ 1BAABⁿ

1−AB dhx0, hx₁.

3.10

Similarly, we obtain

dhx2n, hx_2m11AABⁿ

1−ABdhx0, hx1, dhx2n, hx2m1AABⁿ

1−ABdhx0, hx1, dhx_2n1, hx_2m1BAABⁿ

1−AB dhx0, hx₁.

3.11

Hence, forn < m dhxn, hx_mmax

1B AABⁿ

1−AB ,1AABⁿ 1−AB

dhx0, hx₁ λ_ndhx0, hx₁, 3.12

whereλ_n → 0, asn → ∞.

Now, using propertiesaanddfromLemma 2.5and only the assumption that the underlying cone is solid, we conclude that{hxn}is a Cauchy sequence. Since the subspace hXis complete, there existu, v∈Xsuch thathx_n → vhun → ∞.

We will prove thathufugu. Firstly, let us estimate thatdhu, fu dv, fu. We have that

d hu, fu

dhu, hx2n1 d

hx_2n1, fu

dv, hx2n1 d

fu, gx_2n1

. 3.13

By the contractive condition3.1, it holds that d

fu, gx_2n1

pdhu, hx_2n1 qd hu, fu

rd

hx_2n1, gx_2n1 sd

hu, gx_2n1 td

hx_2n1, fu pd

v, fx2n

qd v, fu

rd

fx2n, gx_2n1 sd

v, gx_2n1 td

fx_2n, fu pd

v, fx_2n qd

v, fu rd

fx_2n, gx_2n1 sd

v, gx_2n1 td

fx2n, v td

v, fu .

3.14

Now it follows from3.13that 1−q−t

d v, fu

dv, hx_2n1 pd v, fx2n

rd

fx_2n, gx_2n1 sd

v, gx_2n1 td

fx_2n, v

. 3.15

that is, 1−q−t

d v, fu

1sd

v, gx_2n1

pt d

v, fx_2n rd

fx_2n, gx_2n1 , d

v, fu

1s 1−q−td

v, gx_2n1

pt 1−q−td

v, fx2n

r 1−q−td

fx2n, gx_2n1

. 3.16

Letc∈intP. Then there existsn0such that forn > n0it holds that d

v, gx_2n1 1−q−t 31sc, d

v, fx2n

1−q−t 3

ptc 3.17

anddfx2n, gx_2n1 1−q−t/3rc, that is,dv, fu cforn > n0. Sincec∈intPwas arbitrary, it follows thatdv, fu 0, that is,fuhuv.

Similarly using that d

hu, gu

dhu, hx2n1 d

hx_2n1, gu dhu, hx_2n1 d

fx_2n, gu

, 3.18

it can be deduced thathuguv. It follows thatvis a common point of coincidence forf, g, andh, that is,

vfuguhu. 3.19

Now we prove that the point of coincidence off, g, his unique. Suppose that there is another pointv₁∈Xsuch that

v1 fu1gu1hu1 3.20

for someu1∈X. Using the contractive condition we obtain that dv, v1 d

fu, gu₁ pdhu, hu1 qd

hu, fu rd

hu₁, gu₁ sd

hu, gu₁ td

hu₁fu pdv, v1 q·0r·0sdv, v1 tdv, v1

pst

dv, v1.

3.21

Sincepst <1, it follows thatdv, v1 0, that is,vv₁.

Using weak compatibility of the pairsf, handg, hand proposition 1.12 from16, it follows that the mappingsf, g, hhave a unique common fixed point, that is, fv gv hvv.

Corollary 3.2. LetX, dbe a tvs-cone metric space and the mappingsf, g, h:X → Xsatisfy d

fx, gy αd

hx, hy β

d hx, fx

d

hy, gy γ

d hx, gy

d

hy, fx

3.22 for allx, y∈X, whereα, β, γ≥0 andα2β2γ <1. IffX∪gX⊂hXandhXis a complete subspace ofX, thenf, g, andhhave a unique point of coincidence. Moreover, iff, handg, hare weakly compatible, thenf, g, andhhave a unique common fixed point.

Putting in this corollaryh i_X and taking into account that each self-map is weakly compatible with the identity mapping, we obtain the following.

Corollary 3.3. LetX, dbe a complete tvs-cone metric space, and let the mappingsf, g :X → X satisfy

d fx, gy

αd x, y

β d

x, fx d

y, gy γ

d x, gy

d y, fx

3.23 for allx, y ∈X, whereα, β, γ ≥0 andα2β2γ <1. Thenf andg have a unique common fixed point inX. Moreover, any fixed point offis a fixed point ofg, and conversely.

In the case of a cone metric space with a normal cone, this result was proved in14.

Now put firstg f inTheorem 3.1and thenh g. Choosing appropriate values for coefficients, we obtain the following.

Corollary 3.4. Let X, d be a tvs-cone metric space. Suppose that the mappingsf, g : X → X satisfy the contractive condition

d fx, fy

λ·d gx, gy

, 3.24

d fx, fy

λ· d

fx, gx d

fy, gy

, 3.25

or

d fx, fy

λ· d

fx, gy d

fy, gx

, 3.26

for allx, y∈X, whereλis a constant (λ∈0,1in3.24andλ∈0,1/2in3.25and3.26). If fX⊂gXandgXis a complete subspace ofX, thenfandghave a unique point of coincidence inX. Moreover, iffandgare weakly compatible, thenfandghave a unique common fixed point.

In the case when the spaceEis normed and the coneP is normal, these results were proved in9.

Similarly one obtains the following.

Corollary 3.5. LetX, dbe a tvs-cone metric space, and letf, g : X → X be such thatfX ⊂ gX. Suppose that

d

fx, fy αd

fx, gx βd

fy, gy γd

gx, gy

, 3.27

for allx, y∈X, whereα, β, γ ∈0,1andαβγ <1, and letfxgximply thatfgxggxfor eachx∈X. IffXorgXis a complete subspace ofX, then the mappingsf andg have a unique common fixed point inX. Moreover, for anyx₀∈X, thef-g-sequence{fxn}with the initial pointx₀ converges to the fixed point.

Here, an f-g-sequence also called a Jungck sequence {fxn} is formed in the following way. Letx₀ ∈ X be arbitrary. Since fX ⊂ gX, there existsx₁ ∈ X such that fx0gx1. Having chosenxn∈X,x_n1∈Xis chosen such thatgx_n1fxn.

In the case when the spaceEis normed and under the additional assumption that the coneP is normal, these results were firstly proved in10.

Corollary 3.6. LetX, dbe a complete tvs-cone metric space. Suppose that the mappingf:X → X satisfies the contractive condition

d fx, fy

λ·d x, y

, 3.28

d fx, fy

λ· d

fx, x d

fy, y

, 3.29

or

d fx, fy

λ· d

fx, y d

fy, x

3.30 for allx, y ∈X, whereλis a constant (λ ∈0,1in3.28andλ ∈0,1/2in3.29and3.30).

Thenfhas a unique fixed point inX, and for anyx∈X, the iterative sequence{fⁿx}converges to the fixed point.

In the case when the space Eis normed and under the additional assumption that the coneP is normal, these results were firstly proved in7. The normality condition was removed in8.

Finally, we give an example of a situation whereTheorem 3.1can be applied, while the results known so far cannot.

Example 3.7see26, Example 3.3. LetX {1,2,3}, E C¹_R0,1 with the coneP as in Example 2.2 and endowed with the strongest locally convex topologyt^∗. Let the metricd : X×X → Ebe defined bydx, yt 0 ifxyandd1,2t d2,1t 6e^t,d1,3t d3,1t 30/7e^t, andd2,3t d3,2t 24/7e^t. Further, letf, g:X → Xbe given by,fx1,x∈Xandg1g31,g23. Finally, lethIX.

Takingpqrs0,t5/7, all the conditions ofTheorem 3.1are fulfilled. Indeed, sincef1g1f3g31, we have only to check that

d f3, g2

0·d3,2 0·d 3, f3

0·d 2, g2

0·d 3, g2

5 7d

2, f3

, 3.31

which is equivalent to 30

7 e^t≤ 5 7d

2, f3 t 5

7d2,1t 5

7·6e^t 30

7 e^t. 3.32

Hence, we can apply Theorem 3.1 and conclude that the mappings f, g, h have a unique common fixed pointu1.

On the other hand, since the space E, P, t^∗is not an ordered Banach space and its cone is not normal, neither of the mentioned results from7–10,14can be used to obtain such conclusion. Thus,Theorem 3.1and its corollaries are proper extensions of these results.

Note that an example of similar kind is also given in24.

The following example shows that the condition “pqorst” inTheorem 3.1cannot be omitted.

Example 3.8see26, Example 3.4. LetX {x, y, u, v}, where x 0,0,0,y 4,0,0, u 2,2,0, andv 2,−2,1. Letdbe the Euclidean metric inR³, and let the tvs-cone metric d1:X×X → EE,P, andt^∗are as in the previous examplebe defined in the following way:

d₁a, bt da, b·ϕt, whereϕ∈Pis a fixed function, for example,ϕt e^t. Consider the mappings

f

x y u v u v v u

, g

x y u v y x y x

, 3.33

and lethi_X. By a careful computation it is easy to obtain that d

fa, gb

≤ 3 4max

da, b, d a, fa

, d b, gb

, d a, gb

, d b, fa

, 3.34

for alla, b ∈X. We will show thatfandgsatisfy the following contractive condition: there existp, q, r, s, t≥0 withpqrst <1 andq /r,s /tsuch that

d₁ fa, gb

pd₁a, b qd₁ a, fa

rd₁ b, gb

sd₁ a, gb

td₁ b, fa

3.35 holds true for alla, b∈X. Obviously,fandgdo not have a common fixed point.

Taking3.34into account, we have to consider the following cases.

1In cased1fa, gb 3/4d1a, b, then3.35holds for p 3/4, r t 0 and qs1/9.

2In cased₁fa, gb3/4d1a, fa, then3.35holds forq3/4,prt0 and s1/5.

3In cased1fa, gb3/4d1b, gb, then3.35holds forr 3/4,pqt0 and s1/5.

4In cased₁fa, gb3/4d1a, gb, then3.35holds fors3/4,pr t0 and q1/5.

5In cased₁fa, gb3/4d1b, fa, then3.35holds fort3/4,pr s0 and q1/5.

4. Quasicontractions in Tvs-Cone Metric Spaces

Definition 4.1. LetX, dbe a tvs-cone metric space, and letf, g:X → X. Then,fis called a quasi-contractionresp., ag-quasi-contractionif for some constantλ∈0,1and for allx, y∈X, there exists

u∈C x, y

d

x, y , d

x, fx , d

x, fy , d

y, fy , d

y, fx , resp., u∈C

g;x, y

d gx, gy

, d gx, fx

, d gx, fy

, d

gy, fy , d

gy, fx

, 4.1 such that

d fx, fy

λ·u. 4.2

Theorem 4.2. LetX, dbe a complete tvs-cone metric space, and letf, g : X → Xbe such that fX ⊂ gX andgXis closed. Iff is ag-quasi-contraction withλ ∈ 0,1/2, thenf and g have a unique point of coincidence. Moreover, if the pairf, gis weakly compatible or, at least, occasionally weakly compatible, thenfandghave a unique common fixed point.

Recall that the pairf, gof self-maps onX is called occasionally weakly compatible see27or28if there existsx∈Xsuch thatfxgxandfgxgfx.

Proof. Let us remark that the conditionfX⊂gXimplies that starting with an arbitraryx₀ ∈ X, we can construct a sequence{yn}of points inXsuch thatyn fxn gx_n1for alln≥0.

We will prove that{yn}is a Cauchy sequence. First, we show that

d

yn, y_n1 λ

1−λd y_n−1, yn

4.3

for alln≥1. Indeed,

d

yn, y_n1 d

fxn, fx_n1

≤λun, 4.4

where u_n∈

d

gx_n, gx_n1 , d

gx_n, fx_n , d

gx_n1, fx_n1 , d

gx_n, fx_n1 , d

gx_n1, fx_n

d y_n−1, yn

, d y_n−1, yn

, d

yn, y_n1 , d

y_n−1, y_n1 , d

yn, yn

d

y_n−1, y_n , d

y_n, y_n1 , d

y_n−1, y_n1 , θ

.

4.5

The following four cases may occur:

1First,dyn, y_n1λdy_n−1, y_nλ/1−λdy_n−1, y_n.

2Second,dyn, y_n1λdyn, y_n1and sodyn, y_n1 θ. In this case,4.3follows immediately, becauseλ < λ/1−λ.

3Third,dyn, y_n1λdy_n−1, y_n1λdy_n−1, yn λdyn, y_n1. It follows that4.3 holds.

4Fourth,dyn, y_n1λ·θθand sodyn, y_n1 θ. Hence,4.3holds.

Thus, by puttinghλ/1−λ<1, we have thatdyn, y_n1hdyn−1, yn. Now, using 4.3, we have

d

yn, y_n1 hd

y_n−1, yn

· · · hⁿd y0, y1

, 4.6

for alln≥1. It follows that d

y_n, y_m d

y_n, y_n−1 d

y_n−1, y_n−2

· · ·d

y_m1, y_m

hⁿ⁻¹hⁿ⁻². . .h^m d

y0, y1

h^m

1−hd y₀, y₁

−→θ, asm−→ ∞.

4.7

Using properties a and d fromLemma 2.5, we obtain that{yn} is a Cauchy sequence.

Therefore, sinceXis complete andgXis closed, there existsz∈Xsuch that

y_nfx_ngx_n1−→gz, asn−→ ∞. 4.8 Now we will show thatfzgz.

By the definition ofg-quasicontraction, we have that d

fxn, fz

λ·un, 4.9

whereun∈{dgxn, gz, dgxn, fxn, dgz, fz, dgz, fxn, dgxn, fz}. Observe thatdgz, fz dgz, fxn dfxn, fzanddgxn, fz dgxn, fx_n dfxn, fz. Now let 0 cbe given.

In all of the possible five cases there existsn₀ ∈ Nsuch thatusing4.9one obtains that dfxn, fz c:

1dfxn, fzλ·dgxn, gz λc/λ c;

2dfxn, fzλ·dgxn, fx_n λc/λ c;

3dfxn, fzλ·dgz, fzλdgz, fxn λdfxn, fz; it follows thatdfxn, fz λ/1−λdgz, fxn λ/1−λ1−λc/λ c;

4dfxn, fzλ·dgz, fxn λc/λ c;

5dfxn, fzλ·dgxn, fzλdgxn, fx_nλdfxn, fz; it follows thatdfxn, fz λ/1−λdgxn, fxn λ/1−λ1−λc/λ c.

It follows thatfxn → fzn → ∞. The uniqueness of limit in a cone metric space implies that fz gz t. Thus, z is a coincidence point of the pair f, g, and t is its point of coincidence. It can be showed in a standard way that this point of coincidence is unique. Using lemma 1.6 of27one readily obtains that, in the case when the pairf, gis occasionally weakly compatible, the pointtis the unique common fixed point offandg.

In the normed case and assuming that the cone is normalbut lettingλ∈0,1, this theorem was proved in11.

PutinggiXinTheorem 4.2we obtain the following.

Corollary 4.3. LetX, dbe a complete tvs-cone metric space, and let the mappingf :X → X be a quasi-contraction withλ∈0,1/2. Thenfhas a unique fixed point inX, and for anyx∈X, the iterative sequence{fⁿx}converges to the fixed point.

In the case of normed-valued cone metric spaces and under the assumption that the underlying conePis normaland withλ∈0,1, this result was obtained in12. Normality condition was removed in13.

FromTheorem 4.2, as corollaries, among other things, we again recover and extend the results of Huang and Zhang7and Rezapour and Hamlbarani8. The following three corollaries follow in a similar way.

In the next corollary, we extend the well-known result29,9’.

Corollary 4.4. LetX, dbe a complete tvs-cone metric space, and letf, g : X → X be such that fX⊂gXandgXis closed. Further, let for some constantλ∈0,1and everyx, y∈Xthere exists

uu x, y

∈ d

gx, gy , d

gx, fx , d

gy, fy

4.10

such that

d fx, fy

λ·u. 4.11

Thenf andg have a unique point of coincidence. Moreover, if the pairf, gis occasionally weakly compatible, then they have a unique common fixed point.

We can also extend the well-known Bianchini’s result29,5

Corollary 4.5. LetX, dbe a complete tvs-cone metric space, and letf, g : X → X be such that fX⊂gXandgXis closed. Further, let for some constantλ∈0,1and everyx, y∈X, there exists

uu x, y

∈ d

gx, fx , d

gy, fy

4.12

such that

d fx, fy

≤λ·u. 4.13

Thenf andg have a unique point of coincidence. Moreover, if the pairf, gis occasionally weakly compatible, then they have a unique common fixed point.

In the next corollary, we extend the well-known result of Jungck30, Theorem 1.1.

Corollary 4.6. LetX, dbe a complete tvs-cone metric space, and letf, g : X → X be such that fX⊂gXandgXis closed. Further, let for some constantλ∈0,1and everyx, y∈X,

d fx, fy

λ·d gx, gy

. 4.14

Thenf andg have a unique point of coincidence. Moreover, if the pairf, gis occasionally weakly compatible, then they have a unique common fixed point.

Remark 4.7. Note that in the previous three corollaries it is possible that the parameterλtakes values from0,1 and not only in0,1/2as inTheorem 4.2. Namely, it is possible to show that the sequence{yn}used in the proof, is a Cauchy sequence because the condition onuis stronger.

Now, we prove the main result of Das and Naik31in the frame of tvs-cone metric spaces in which the cone need not be normal.

Theorem 4.8. LetX, dbe a complete tvs-cone metric space. Letgbe a self-map onXsuch thatg² is continuous, and letfbe any self-map onXthat commutes withg. Further letfandgsatisfy

fgX⊂g²X, 4.15

and letfbe ag-quasi-contraction. Thenfandghave a unique common fixed point.

Proof. By 4.15, starting with an arbitrary x0 ∈ gX, we can construct a sequence {xn} of points infXsuch thaty_n fx_n gx_n1,n≥0as inTheorem 4.2. Nowgy_n1 gfx_n1 fgx_n1fyn zn,n≥1. It can be proved as inTheorem 4.2that{zn}is a Cauchy sequence and hence convergent to somez∈X. Further, we will show thatg²zfgz. Since

nlim→ ∞gyn lim

n→ ∞gfxn lim

n→ ∞fgxn lim

n→ ∞fyn lim

n→ ∞znz, 4.16

it follows that

nlim→ ∞g⁴xn lim

n→ ∞g³fxn lim

n→ ∞fg³xng²z, 4.17

becauseg²is continuous. Now, we obtain d

g²z, fgz d

g²z, g³fx_n d

g³fx_n, fgz d

g²z, g³fx_n

λ·u_n, 4.18

where un∈

d

g⁴xn, f²z , d

g⁴xn, fg³xn

, d

g²z, fgz , d

g⁴xn, fgz , d

g²z, fg³xn

. 4.19

Letθ cbe given. Sinceg³fxn → g²zandg⁴xn → g²z, choose a natural numbern0such that for alln ≥ n0 we havedg²z, g³fxn c1−λ/2 anddg⁴xn, fg³xn 1−λc/2λ.

Again, we have the following cases:

a

d

g²z, fgz d

g²z, g³fxn

λd

g⁴xn, g²z c 2λ c

2λ c. 4.20

b d

g²z, fgz d

g²z, g³fxn

λd

g⁴xn, fg³z d

g²z, g³fxn

λd

g⁴xn, g²z λd

g²z, fg³xn

1λd

g²z, g³fxn

λd

g⁴xn, g²z 1λc1−λ

2 λ1−λc

2λ c.

4.21

c

d

g²z, fgz

d

g²z, g³fx_n λd

g²z, fgz

. Hence, d

g²z, fgz 1

1−λd

g²z, g³fx_n 1 1−λ

c1−λ

2 c.

4.22

d d

g²z, fgz d

g²z, g³fx_n λd

g⁴x_n, fgz d

g²z, g³fx_n λd

g⁴x_n, g²z d

g²z, fgz

. Hence, d

g²z, fgz 1

1−λd

g²z, g³fxn

λ 1−λd

g⁴xn, g²z 1

1−λ

c1−λ

2 λ

1−λ

1−λc 2λ c.

4.23

e

d

g²z, fgz d

g²z, g³fx_n λd

g²z, fg³x_n c 2 λ c

2λ c. 4.24

Therefore,dg²z, fgz cfor allθ c. By propertydofLemma 2.4,g²z fgz, and sofgzis a common fixed point forfandg. Indeed, putting in the contractivity condition xfgz, ygz, we getffgz fgz. Sinceg²zfgz, that is,ggz fgz, we have that gfgz fg²zffgz fgz.

Acknowledgments

The authors are very grateful to the referees for the valuable comments that enabled them to revise this paper. They are thankful to the Ministry of Science and Technological Development of Serbia.

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