ISSN 2291-8639
Volume 4, Number 2 (2014), 100-106 http://www.etamaths.com
SOME COMMON FIXED POINT THEOREMS IN GENERALIZED VECTOR METRIC SPACES
RAJESH SHRIVASTAVA1, RAJENDRA KUMAR DUBEY2, PANKAJ TIWARI1,∗
Abstract. In this paper we give some theorems on point of coincidence and common fixed point for two self mappings satisfying some general contractive conditions in generalized vector spaces. Our results generalize some well-known recent results in this direction.
1. Introduction and preliminaries
In 2003, Mustafa and Sims [6] introduced a more appropriate and robust notion of a generalized metric space as follows.
Definition 1.1. [6]. Let X be a nonempty set, and letG:X×X×X →[0,∞) be a function satisfying the following axioms:
(1) G(x, y, z) = 0 if and only ifx=y=z; (2) G(x, x, y)>0, for allx6=y;
(3) G(x, y, z)≥G(x, x, y), for allx, y, z∈X;
(4) G(x, y, z) =G(x, z, y) =G(z, y, x) =· · · (symmetric in all three variables); (5) G(x, y, z)≤G(x, w, w) +G(w, y, z), for allx, y, z, w∈X.
Then the functionGis called a generalized metric, or, more specifically aG-metric onX, and the pair (X, G) is called aG-metric space.
A Riesz space is an ordered vector space and a lattice. LetE be a Riesz space
with the positive coneE+={x∈E:x≥0}. If{an} is a decreasing sequence inE
such that infan=a, writean↓a.
Definition 1.2. The Riesz spaceE is said to be Achimedean if 1
nan↓0 holds for everyE+.
Definition 1.3. A sequence{bn}is said to be order convergent (or o-convergent) to b if there is a sequence{an} in E satisfyingan ↓ 0 and|bn−b| ≤an for all n, and writtenbn
o
−→boro−limbn=b, where|a|= sup{a,−a}for anya∈E.
Definition 1.4. A sequence{bn}is said to be order-Cauchy (or o-Cauchy) if there exists a sequence{an} in Esuch thatan ↓ 0 and|bn−bn+p| ≤an holds for alln andp.
Definition 1.5. The Riesz space E is said to beo−Cauchy complete if every
o−Cauchy sequenceino−convergent.
For notion and other facts regarding Riesz spaces we refer to [1].
2010Mathematics Subject Classification. 47H10.
Key words and phrases. Reisz space, Generalized vector space, coincidence point, fixed point.
c
2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
2. Vector G-metric spaces
In this section we introduce the following concepts and properties of Vector G-metric spaces.
Definition 2.1. LetX be a non-empty set andEbe a Riesz space. The function
G:X×X×X →Eis said to be vector G−metric if it is satisfying the following
properties :
(VGM-1) G(x, y, z) = 0 if and only ifx=y=z, (VGM-2) G(x, y, z)≤G(x, w, w) +G(w, y, z),
for allx, y, z, w∈X. Also the triple (X, G,E) is said to be vectorG−metric space.
For arbitrary elementsx, y, z, w∈X of a vectorG−metric space, the following statements are satisfied
(1) G(x, x, y)>0, for allx6=y;
(2) G(x, y, z)≥G(x, x, y), for allx, y, z∈X;
(3) G(x, y, z) =G(x, z, y) =G(z, y, x) =· · · (symmetric in all three variables).
Example 2.2. A Riesz spaceEis a vectorG−metric space withG:E×E×E→E defined by G(x, y, z) =|x−y|+|y−z|+|z−x|. This vector G−metric space is called to be absolute valuedG−metric space onE.
It is well known thatR2 is a Riesz space with coordinatwise ordering defined by
(x1, y1)≤(x2, y2)⇐⇒x1≤x2 and y1≤y2
for (x1, y1),(x2, y2)∈R2.
AgainR2 is a Riesz space with lexicographical ordering defined by
(x1, y1)≤(x2, y2)⇐⇒x1< x2 or x1=x2 and y1≤y2.
Note that R2 is Archimedean with coordinatwise ordering but not with
lexico-graphical ordering.
Example 2.3. LetG:R2×R2×R2→R2 defined by
G((x1, y1),(x2, y2),(x3, y3)) = (αx∗, βy∗)
wherex∗=|x1−x2|+|x2−x3|+|x3−x1|andy∗=|y1−y2|+|y2−y3|+|y3−y1| alsoα, βare positive real numbers. ThenGis a vectorG−metric space.
LetG:R×R×R→R2 defined by
G(x, y, z) = (αw, βw)
wherew=|x−y|+|y−z|+|z−x|,α, β≥0 andα+β >0. ThenGis a vector
G−metric space.
Definition 2.4. A sequence{xn} in a vectorG−metric space (X, G,E) vectorial
G−convergence to somex∈E, writtenxn G,E
−−→x,if there is a sequence{an} inE such thatan↓0 and satisfying,
(3) G(xn, xm, x)≤an, for alln.
Definition 2.5. A sequence{xn}is calledGE−Cauchy sequencewhenever there exists a sequence{an}in Esuch thatan ↓0 andG(xn, xm, xm)≤an holds for all
nandm.
Definition 2.6. A vector G-metric space is said to be complete if each GE−
Cauchy sequencein X isE−convergens to a limit inX.
Using the above definitions, we have the following properties.
Ifxn G,E
−−→x, then (1) The limitxunique,
(2) Every subsequence of{xn}E−converges tox,
(3) If also yn G,E
−−→y andzn G,E
−−→z, thenG(xn, yn, zn) o
−→G(x, y, z).
WhenE=Rthen the concepts of vectorialGE−convergenceandconvergence inG−metric space are the same also the concepts ofGE−Cauchy sequence and
G−Cauchy sequenceare the same.
Remark 2.7. ItEis a Riesz space anda≤kawherea∈E+ k∈[0,1), thena= 0.
Proof. The conditiona≤ka means that−(1−k)a=ka−a∈E+. Sincea∈E+ and 1−k >0, then also (1−k)a∈E+. Thus we have (1−k)a= 0 anda= 0.
3. Main Results
Theorem 3.1. LetX be a vectorG−metric space withEis Archimedean. Suppose the mappingsS, T :X →X satisfying the following conditions,
(i) for allx, y, z∈X andα, β, γ, δ∈[0,1)such that 0≤α+β+γ+δ <1
G(T x, T y, T z(3.1))≤αG(Sx, Sy, Sz) +βG(Sx, T x, T x) +γG(Sy, T y, T y) +δG(Sz, T z, T z)
or
G(T x, T y, T z(3.2))≤αG(Sx, Sy, Sz) +βG(Sx, Sx, T x) +γG(Sy, Sy, T y) +δG(Sz, Sz, T z)
(ii) T(X)⊆S(X),
(iii) T(X)or S(X)is complete subspace ofX.
ThenS andT have a unique point of coincidence in X. Moreover, ifS andT are weakly compatible, then they have a unique common fixed point inX.
Proof. Letx0 be an arbitrary point in X, sinceT(X)⊆S(X) so we can choose a pointx1∈X such thatSx1=T x0. In general we can chooseSxn+1 =T xn=yn for alln.
Now, form 3.1 we have
G(Sxn, Sxn+1, Sxn+1) = G(T xn−1, T xn, T xn)
≤ (α+β)G(Sxn−1, Sxn, Sxn) + (γ+δ)G(Sxn, xn+1, xn+1)
G(Sxn, Sxn+1, Sxn+1) ≤
α+β
1−(γ+δ)G(Sxn−1, Sxn, Sxn). (3.3)
Letq= 1−α+β(γ+δ), then 0≤q <1 since 0≤α+β+γ+δ <1. So
Continuing in the same way, we have
G(Sxn, Sxn+1, Sxn+1) ≤ qnG(Sx0, Sx1, Sx1). (3.5)
Therefore, for alln, m∈N, n < m, we have by (VGM-2)
G(yn, ym, ym) ≤ G(yn, yn+1, yn+1) +G(yn+1, yn+2, yn+2) +...+G(ym−1, ym, yn) ≤ (qn+qn+1+qn+2+...+qm−1)G(y0, y1, y1)
≤ q
n
1−qG(y0, y1, y1).
Now, sinceEis Archimedean then{yn}is anGE−Cauchy sequenceinX. Since the range ofScontains the range ofT and the range of at least one isGE−complete,
so there is w∈X such that Sxn G,E
−−→ w. Hence there exists a sequence{an} ∈E
such thatan ↓0 and G(Sxn, w, w)≤an. On the other hand, we can find u∈X such thatSw=u.
Let us show thatT w=u, we have
G(T w, u, u) ≤ G(T w, T xn, T xn) +G(T xn, u, u)
≤ αG(Sw, Sxn, Sxn) +βG(Sw, T w, T w) + (γ+δ)G(Sxn, T xn, T xn) +an+1 ≤ (α+β+γ+δ+ 1)an+1.
Since the infimum of the sequence on the right side of the above inequality are zero, then T w = u. Therefore, w is a point of coincidence of T and S. If w1 is another point of coincidence then there isw1 ∈ X withw1 =T w1 =Sw1. Now from 3.1 it follows thatG(w, w1, w1) = 0, that isw=w1.
IfS andT are weakly compatible, then it is obvious thatw is unique common fixed point ofT andS inX.
If S and T satisfies condition 3.2, then the argument is similar to that above. However to show that the sequence{xn}isGE−Cauchy sequence, we start with
G(Sxn, Sxn, Sxn+1) = G(T xn−1, T xn−1, T xn)
≤ (α+β+γ)G(Sxn−1, Sxn−1, Sxn) +δG(Sxn, xn+1, xn+1)
G(Sxn, Sxn+1, Sxn+1) ≤
α+β+γ
1−δ G(Sxn−1, Sxn, Sxn).
(3.6)
Letq= α+β+γ1−δ , then 0≤q <1 since 0≤α+β+γ+δ <1. So
G(Sxn, Sxn, Sxn+1) ≤ qG(Sxn−1, Sxn−1, Sxn). (3.7)
Continuing in the same way, we have
G(Sxn, Sxn, Sxn+1) ≤ qnG(Sx0, Sx0, Sx1). (3.8)
Then for alln, m∈N, n < m, we have by (VGM-2) we prove the remaining part
of the proof.
(i) for allx, y, z∈X andα, β, γ, δ∈[0,1)such that 0≤α+β+γ+δ <1
G(Tmx, Tmy, Tmz)≤αG(Smx, Smy, Smz) +βG(Smx, Tmx, Tmx)
+γG(Smy, Tmy, Tmy) +δG(Smz, Tmz, Tmz)
or
G(Tmx, Tmy, Tmz)≤αG(Smx, Smy, Smz) +βG(Smx, Smx, Tmx)
+γG(Smy, Smy, Tmy) +δG(Smz, Smz, Tmz)
(ii) T(X)⊆S(X),
(iii) T(X)or S(X)is complete subspace ofX.
ThenS andT have a unique point of coincidence in X. Moreover, ifS andT are weakly compatible, then they have a unique common fixed point inX, alsoTmand
Sm areGE−continuousatu. .
Proof. From Theorem 3.1, we see that Tm and Sm have a unique common fixed point (sayu), that is,Tm(u) =u. butT(u) =T(Tm(u)) =Tm+1(u) =Tm(T(u)), so T(u) is another fixed point forTmand by uniquenessT u=u.Similarly we can
show thatSu=u.
Theorem 3.3. LetX be a vectorG−metric space withEis Archimedean. Suppose the mappingsS, T :X →X satisfying the following conditions,
(i) for allx, y, z∈X andα∈[0,1)such that,
G((3.9)T x, T y, T z)≤α{G(Sx, Sy, Sz), G(Sx, T x, T x), G(Sy, T y, T y), G(Sz, T z, T z)}
or
G((3.10)T x, T y, T z)≤α{G(Sx, Sy, Sz), G(Sx, Sx, T x), G(Sy, Sy, T y), G(Sz, Sz, T z)}
(ii) T(X)⊆S(X),
(iii) T(X)or S(X)is complete subspace ofX.
ThenS andT have a unique point of coincidence in X. Moreover, ifS andT are weakly compatible, then they have a unique common fixed point in X, also T and
S are GE−continuousatu.
Proof. Letx0 be an arbitrary point in X, sinceT(X)⊆S(X) so we can choose a pointx1∈X such thatSx1=T x0. In general we can chooseSxn+1 =T xn=yn for alln.
Now, form 3.9 we have
G(Sxn, Sxn+1, Sxn+1) = G(T xn−1, T xn, T xn)
≤ α{G(Sxn−1, Sxn, Sxn), G(Sxn, Sxn+1, Sxn+1)}
G(Sxn, Sxn+1, Sxn+1) ≤ αG(Sxn−1, Sxn, Sxn)
G(Sxn, Sxn+1, Sxn+1) ≤ αG(Sxn−1, Sxn, Sxn). (3.11)
Continuing in the same way, we have
G(Sxn, Sxn+1, Sxn+1) ≤ αnG(Sx0, Sx1, Sx1). (3.12)
G(yn, ym, ym) ≤ G(yn, yn+1, yn+1) +G(yn+1, yn+2, yn+2) +...+G(ym−1, ym, yn) ≤ (αn+αn+1+αn+2+...+αm−1)G(y0, y1, y1)
≤ α
n
1−αG(y0, y1, y1).
Now, sinceEis Archimedean then{yn}is anGE−Cauchy sequenceinX. Since the range ofScontains the range ofT and the range of at least one isGE−complete,
so there is w∈X such that Sxn G,E
−−→ w. Hence there exists a sequence{an} ∈E
such thatan ↓0 and G(Sxn, w, w)≤an. On the other hand, we can find u∈X such thatSw=u.
Let us show thatT w=u, we have
G(T w, u, u) ≤ G(T w, T xn, T xn) +G(T xn, u, u)
≤ αmax{G(Sw, Sxn, Sxn), G(Sw, T w, T w),
G(Sxn, Sxn+1, Sxn+1), G(Sxn, Sxn+1, Sxn+1)}+an+1
≤ (α+ 1)an
Since the infimum of the sequence on the right side of the above inequality are zero, then T w = u. Therefore, w is a point of coincidence of T and S. If w1 is another point of coincidence then there isw1 ∈ X withw1 =T w1 =Sw1. Now from 3.9 it follows thatG(w, w1, w1) = 0, that isw=w1.
IfS andT are weakly compatible, then it is obvious thatw is unique common fixed point ofT andS inX.
IfS andT satisfies condition 3.10, then the argument is similar to that above. However to show that the sequence{xn}isGE−Cauchy sequence, we start with
G(Sxn, Sxn, Sxn+1) = G(T xn−1, T xn−1, T xn)
≤ αmax{G(Sxn−1, Sxn−1, Sxn), G(Sxn, xn+1, xn+1)}
G(Sxn, Sxn+1, Sxn+1) ≤ αG(Sxn−1, Sxn−1, Sxn).
Continuing in the same way, we have
G(Sxn, Sxn, Sxn+1) ≤ αnG(Sx0, Sx0, Sx1).
Then for alln, m∈N, n < m, we have by (VGM-2) we prove the remaining part of the proof.
4. Acknowledgement
The authors thank the referees for their careful reading of the manuscript and for their suggestions.
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1
Department of Mathematics, Govt. Science & Commerce College, Benazir Bhopal-India
2
Department of Mathematics, Govt. Science P.G. College, Reewa -India
∗
