R E S E A R C H
Open Access
Fixed point theorems for
(
ξ
,
α
)-expansive
mappings in complete metric spaces
Priya Shahi
*, Jatinderdeep Kaur and SS Bhatia
*Correspondence:
[email protected] School of Mathematics and Computer Applications, Thapar University, Patiala, 147004, India
Abstract
In this paper, we introduce a new, simple and unified approach to the theory of expansive mappings. We present a new category of expansive mappings called (
ξ
,α
)-expansive mappings and study various fixed point theorems for such mappings in complete metric spaces. Further, we shall use these theorems to derive coupled fixed point theorems in complete metric spaces. Our new notion complements the concept ofα
-ψ
-contractive type mappings introduced recently by Sametet al.(Nonlinear Anal. 2011, doi:10.1016/j.na.2011.10.014). The presented theorems extend, generalize and improve many existing results in the literature. Some comparative examples are constructed which illustrate our results in comparison to some of the existing ones in literature.
Keywords: fixed point; complete metric space; expansive mapping
1 Introduction
The advancement and the rich growth of fixed point theorems in metric spaces have im-portant theoretical and practical applications. The development has been tremendous in the last three decades. Although the concept of a fixed point theorem may appear as an abstract notion in metric spaces, it has remarkable influence on applications such as the theory of differential and integral equations [], the game theory relevant to the military, sports and medicine as well as economics []. Besides, it has applications in physics, engi-neering, boundary value problems and variational inequalities (see, for instance, Beg [] and El Naschie []). In , Wanget al.[] presented some interesting work on expansion mappings in metric spaces which correspond to some contractive mappings in []. Fur-ther, Khanet al.[] generalized the result of [] by using functions. Also, Rhoades [] and Taniguchi [] generalized the results of Wang [] for a pair of mappings. Kang [] gener-alized the result of Khanet al.[], Rhoades [] and Taniguchi [] for expansion mappings. Recently, Sametet al.[] introduced a new concept ofα-ψ-contractive type mappings and established fixed point theorems for such mappings in complete metric spaces.
In this paper, we introduce a new notion of (ξ,α)-expansive mappings and establish various fixed point theorems for such mappings in complete metric spaces. The presented theorems extend, generalize and improve many existing results in the literature. Some examples are considered to illustrate the usability of our obtained results.
2 Preliminaries
We need the following definitions and results, consistent with [, ].
Wanget al.[] defined expansion mappings in the form of the following theorem.
Theorem .[] Let(X,d)be a complete metric space. If f is a mapping of X into itself and if there exists a constant q> such that
df(x),f(y)≥qd(x,y)
for each x,y∈X and f is onto, then f has a unique fixed point in X.
Recently, Sametet al.[] considered the following family of functions and presented the new notions ofα-ψ-contractive andα-admissible mappings.
Definition .[] Letϕdenote the family of all functionsψ: [, +∞)→[, +∞) which satisfy the following:
(i) +∞n=ψn(t) < +∞for eacht> , whereψnis thenth iterate ofψ; (ii) ψis non-decreasing.
Definition .[] Let (X,d) be a metric space andT:X→Xbe a given self mapping.T
is said to be anα-ψ-contractive mapping if there exist two functionsα:X×X→[, +∞) andψ∈ϕsuch that
α(x,y)d(Tx,Ty)≤ψd(x,y) for allx,y∈X.
Definition .[] LetT:X→Xandα:X×X→[, +∞).Tis said to beα-admissible if
x,y∈X, α(x,y)≥ ⇒ α(Tx,Ty)≥.
Now, we present some examples ofα-admissible mappings.
Example . LetXbe the set of all non-negative real numbers. Let us define the mapping
α:X×X→[, +∞) by
α(x,y) =
⎧ ⎨ ⎩
, ifx≥y, , ifx<y
and define the mappingT:X→XbyTx=xfor allx∈X. ThenT isα-admissible.
Example . LetXbe the set of all non-negative real numbers. Let us define the mapping
α:X×X→[, +∞) by
α(x,y) =
⎧ ⎨ ⎩
ex–y, ifx≥y,
, ifx<y
In what follows, we present the main results of Sametet al.[].
Theorem . [] Let (X,d) be a complete metric space and T :X→X be anα-ψ -contractive mapping satisfying the following condition\s:
(i) Tisα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥; (iii) Tis continuous.
Then T has a fixed point, that is, there exists x*∈X such that Tx*=x*.
Theorem . [] Let (X,d) be a complete metric space and T :X→X be anα-ψ -contractive mapping satisfying the following conditions:
(i) Tisα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥;
(iii) if{xn}is a sequence inXsuch thatα(xn,xn+)≥for allnandxn→x∈Xas n→+∞, thenα(xn,x)≥for alln.
Then T has a fixed point.
Sametet al.[] added the following condition (H) to the hypotheses of Theorem . and Theorem . to assure the uniqueness of the fixed point:
(H): For allx,y∈X, there existsz∈Xsuch thatα(x,z)≥andα(y,z)≥.
Further, Sametet al.[] derived coupled fixed point theorems in complete metric spaces using the previously obtained results.
Theorem .[] Let(X,d)be a complete metric space and F:X×X→X be a given mapping. Suppose that there existsψ∈ϕand a functionα:X×X→[, +∞)such that
α(x,y), (u,v)dF(x,y),F(u,v)≤ ψ
d(x,u) +d(y,v),
for all(x,y), (u,v)∈X×X. Suppose also that (i) for all(x,y), (u,v)∈X×X, we have
α(x,y), (u,v)≥ ⇒ αF(x,y),F(y,x),F(u,v),F(v,u)≥;
(ii) there exists(x,y)∈X×Xsuch that α(x,y),
F(x,y),F(y,x)
≥ and αF(y,x),F(x,y)
, (y,x)
≥;
(iii) Fis continuous.
Then F has a coupled fixed point, that is, there exists(x*,y*)∈X×X such that x*=F(x*,y*) and y*=F(y*,x*).
Theorem .[] Let(X,d)be a complete metric space and F:X×X→X be a given mapping. Suppose that there existsψ∈ϕand a functionα:X×X→[, +∞)such that
α(x,y), (u,v)dF(x,y),F(u,v)≤ ψ
d(x,u) +d(y,v),
(i) for all(x,y), (u,v)∈X×X, we have
α(x,y), (u,v)≥ ⇒ αF(x,y),F(y,x),F(u,v),F(v,u)≥;
(ii) there exists(x,y)∈X×Xsuch that α(x,y),
F(x,y),F(y,x)
≥ and αF(y,x),F(x,y)
, (y,x)
≥;
(iii) if{xn}and{yn}are sequences inXsuch that
α(xn,yn), (xn+,yn+)
≥ and α(yn+,xn+), (yn,xn)
≥,
xn→x∈Xandyn→y∈Xasn→+∞, then α(xn,yn), (x,y)
≥ and α(y,x), (yn,xn)
≥ for alln∈N.
Then F has a coupled fixed point.
Sametet al.[] added the following condition (H) to the hypotheses of Theorem . and Theorem . to assure the uniqueness of the coupled fixed point:
(H): For all(x,y), (u,v)∈X×X, there exists(z,z)∈X×Xsuch that α(x,y), (z,z)
≥, α(z,z), (y,x)
≥
and
α(u,v), (z,z)
≥, α(z,z), (v,u)
≥.
3 Main results
We shall make use of the standard notations and terminologies of nonlinear analysis throughout this paper. We introduce here a new notion of (ξ,α)-expansive mappings as follows.
Letχdenote all functionsξ: [, +∞)→[, +∞) which satisfy the following properties:
(i) ξ is non-decreasing;
(ii) +∞n=ξn(a) < +∞for eacha> , whereξnis thenth iterate ofξ; (iii) ξ(a+b) =ξ(a) +ξ(b),∀a,b∈[, +∞).
Lemma .[] Ifξ: [, +∞)→[, +∞)is a non-decreasing function, then for each a> ,
limn→+∞ξn(a) = impliesξ(a) <a.
Definition . Let (X,d) be a metric space andT :X→Xbe a given mapping. We say thatT is an (ξ,α)-expansive mapping if there exist two functionsξ∈χandα:X×X→
[, +∞) such that
Remark . IfT:X→Xis an expansion mapping, thenTis an (ξ,α)-expansive mapping, whereα(x,y) = for allx,y∈Xandξ(a) =kafor alla≥ and somek∈[, ).
We now prove our main results.
Theorem . Let(X,d)be a complete metric space and T:X→X be a bijective,(ξ,α) -expansive mapping satisfying the following conditions:
(i) T–isα-admissible;
(ii) there existsx∈Xsuch thatα(x,T–x)≥; (iii) Tis continuous.
Then T has a fixed point, that is, there exists u∈X such that Tu=u. Proof Let us define the sequence{xn}inXby
xn=Txn+, for alln∈N,
wherex∈Xis such thatα(x,T–x)≥. Now, ifxn=xn+for anyn∈N, one has thatxn
is a fixed point ofT from the definition{xn}. Without loss of generality, we can suppose xn=xn+for eachn∈N.
It is given thatα(x,x) =α(x,T–x)≥. Recalling thatT–isα-admissible, therefore,
we have
αT–x,T–x
=α(x,x)≥.
Using mathematical induction, we obtain
α(xn,xn+)≥, ()
for alln∈N. Using () and applying the inequality () withx=xnandy=xn+, we obtain d(xn,xn+)≤α(xn,xn+)d(xn,xn+)≤ξ
d(Txn,Txn+)
=ξd(xn–,xn)
. Therefore, by repetition of the above inequality, we have that
d(xn,xn+)≤ξn
d(x,x)
, for alln∈N. For anyn>m≥, we have
d(xm,xn)≤d(xm,xm+) +d(xm+,xm+)
+d(xm+,xm+) +· · ·+d(xn–,xn)
≤ξmd(x,x)
+· · ·+ξn–d(x,x)
.
Fromξn(a) < +∞for eacha> , it follows that{xn}is a Cauchy sequence in the
com-plete metric space (X,d). So, there existsu∈Xsuch thatxn→uasn→+∞. From the
continuity ofT, it follows thatxn=Txn+→Tuasn→+∞. By the uniqueness of the limit,
In what follows, we prove that Theorem . still holds forTnot necessarily continuous, assuming the following condition:
(P): If{xn}is a sequence inXsuch thatα(xn,xn+)≥for allnand{xn} →x∈Xasn→
+∞, then
αT–xn,T–x
≥ ()
for alln.
Theorem . If in Theorem . we replace the continuity of T by the condition (P), then the result holds true.
Proof Following the proof of Theorem ., we know that{xn}is a sequence inXsuch that α(xn,xn+)≥ for allnand{xn} →u∈Xasn→+∞. Now, from the hypothesis (), we
have
αT–xn,T–u
≥, ()
for alln∈N. Utilizing the inequalities (), () and the triangular inequality, we obtain
dT–u,u≤dT–u,xn+
+d(xn+,u)
=dT–xn,T–u
+d(xn+,u)
≤αT–xn,T–u
dT–xn,T–u
+d(xn+,u)
≤ξd(xn,u)
+d(xn+,u).
Continuity ofξatt= implies thatd(T–u,u) = asn→+∞. That is,T–u=u. Consider, Tu=T(T–u) = (TT–)u=u. This gives an end to the proof.
We now present some examples in support of our results.
Example . Let X = [, +∞) endowed with standard metric d(x,y) =|x–y| for all
x,y∈X. Define the mappingsT:X→Xandα:X×X→[, +∞) by
T(x) =
⎧ ⎨ ⎩
x–, x≥,
x
, x∈[, )
and
α(x,y) =
⎧ ⎨ ⎩
, x,y∈[, ), , otherwise.
Clearly, T is an (ξ,α)-expansive mapping with ξ(a) =a/ for alla≥. In fact, for all
x,y∈X, we have
Moreover, there existsx∈Xsuch thatα(x,T–x)≥. In fact, forx= , we have α,T–= .
Obviously,T is continuous, and so it remains to show thatT–isα-admissible. For this, letx,y∈Xsuch thatα(x,y)≥. This implies thatx≥ andy≥, and by the definitions ofT–andα, we have
T–x=x +
≥, T
–y=y
+
≥ and α
T–x,T–y= .
ThenT–isα-admissible.
Now, all the hypotheses of Theorem . are satisfied. Consequently,Thas a fixed point. In this example, and / are two fixed points ofT.
Remark . The expansion mapping theorem proved by Wanget al.[] cannot be applied in the above example since we have
d
T
,T() = <
=d(/, ).
Now, we give an example involving a functionTthat is not continuous.
Example . LetX= [, +∞) endowed with the standard metricd(x,y) =|x–y|for all
x,y∈X. Define the mappingsT:X→Xandα:X×X→[, +∞) by
Tx=
⎧ ⎨ ⎩
x, x≥, x
, ≤x<
and
α(x,y) =
⎧ ⎨ ⎩
, x,y∈[, ), , otherwise.
Due to the discontinuity ofTat , Theorem . is not applicable in this case. Clearly,Tis an (ξ,α)-expansive mapping withξ(a) =a/ for alla≥. In fact, for allx,y∈X, we have
d(Tx,Ty)≥α(x,y)d(x,y).
Moreover, there existsx∈Xsuch thatα(x,T–x)≥. In fact, forx= , we have α,T–= .
Now, letx,y∈Xsuch thatα(x,y)≥. This implies thatx≥,y≥ and by the definition ofT–andα, we have
Finally, let{xn}be a sequence inXsuch thatα(xn,xn+)≥ for allnand{xn} →x∈Xas n→+∞. Sinceα(xn,xn+)≥ for alln, by the definition ofα, we havexn≥ for allnand x≥. Thenα(T–x
n,T–x) = .
Therefore, all the required hypotheses of Theorem . are satisfied, and soThas a fixed point. Here, and are two fixed points ofT.
Remark . As in the previous example, the expansion mapping theorem is not applicable in this case either.
To ensure the uniqueness of the fixed point in Theorems . and ., we consider the condition:
(U): For allu,v∈X, there existsw∈Xsuch thatα(u,w)≥andα(v,w)≥.
Theorem . Adding the condition (U) to the hypotheses of Theorem . (resp. Theo-rem .), we obtain the uniqueness of the fixed point of T .
Proof From Theorem . and Theorem ., the set of fixed points is non-empty. We shall show that ifuandvare two fixed points ofT, that is,T(u) =uandT(v) =v, thenu=v. From the condition (U), there existsw∈Xsuch that
α(u,w)≥ and α(v,w)≥. ()
Recalling theα-admissible property ofT–, we obtain from ()
αu,T–w≥ and αv,T–w≥, for alln∈N. () Therefore, by repeatedly applying theα-admissible property ofT–, we get
αu,T–nw≥ and αv,T–nw≥, for alln∈N. () Using the inequalities () and (), we obtain
du,T–nw≤αu,T–nwdu,T–nw
≤ξdu,T–n+w.
Repetition of the above inequality implies that
du,T–nw≤ξnd(u,w), for alln∈N.
Thus, we haveT–nw→uasn→+∞. Using the similar steps as above, we obtainT–nw→ vasn→+∞. Now, the uniqueness of the limit ofT–nwgives usu=v. This completes the
proof.
Definition .[] LetF:X×X→Xbe a given mapping. We say that (x,y)∈X×Xis a coupled fixed point ofFif
F(x,y) =x and F(y,x) =y.
We shall require the following lemma due to Sametet al.[] for the proof of our result.
Lemma .[] Let F:X×X→X be a given mapping. Define the mapping T:X×X→ X×X by
T(x,y) =F(x,y),F(y,x), for all(x,y)∈X×X. ()
Then(x,y)is a coupled fixed point of F if and only if(x,y)is a fixed point of T .
We, now prove the following results.
Theorem . Let(X,d)be a complete metric space and F:X×X→X be a given bijective mapping. Suppose that there existsξ∈χand a functionα:X×X→[, +∞)such that
ξdF(x,y),F(u,v)≥ α
(x,y), (u,v)d(x,u) +d(y,v) ()
for all(x,y), (u,v)∈X×X. Suppose also that (i) for all(x,y), (u,v)∈X×X, we have
α(x,y), (u,v)≥ ⇒ αF–(x),F–(u)≥;
(ii) there exists(x,y)∈X×Xsuch that α(x,y), (a,b)
≥ and α(b,a), (y,x)
≥,
whereF–(x
) = (a,b); (iii) Fis continuous.
Then F has a coupled fixed point, that is, there exists(x*,y*)∈X×X such that x*=F(x*,y*) and y*=F(y*,x*).
Proof For the proof of our result, we consider the mappingT given by () as a bijective mapping such that
T–(x,y) =F–(x).
Also, consider the complete metric space (Y,ρ), whereY=X×X andρ((x,y), (u,v)) =
d(x,u) +d(y,v) for all (x,y), (u,v)∈X×X. Using the inequality (), we have
ξdF(x,y),F(u,v)≥ α
(x,y), (u,v)d(x,u) +d(y,v) () and
ξdF(v,u),F(y,x)≥ α
Define the functionη:Y×Y→[, +∞) by
η(μ,μ), (ν,ν)
=minα(μ,μ), (ν,ν)
,α(ν,ν), (μ,μ)
() for allμ= (μ,μ),ν= (ν,ν)∈Y.
Summing up the inequalities ()-() and using (), we get
ξdF(μ,μ),F(ν,ν)
+ξdF(ν,ν),F(μ,μ)
≥η(μ,ν)ρ(μ,ν), ()
for allμ= (μ,μ),ν= (ν,ν)∈Y.
Using the property (ξ(a+b) =ξ(a) +ξ(b)) of the functionξ, we obtain
ξρ(Tμ,Tν)≥η(μ,ν)ρ(μ,ν), () for allμ= (μ,μ),ν= (ν,ν)∈Y.
Clearly,Tis continuous and (ξ,α)-expansive mapping.
Letμ= (μ,μ),ν= (ν,ν)∈Y such thatη(μ,ν)≥. Using the condition (i), we obtain η(T–μ,T–ν)≥. ThenT–isη-admissible.
Moreover, from the condition (ii) of the hypothesis of the theorem, we find that there exists (x,y)∈Ysuch that
η(x,y),T–(x,y)
≥.
So, we have transformed the problem to the complete metric space (Y,ρ). Therefore, all the hypotheses of Theorem . are satisfied, and so we deduce the existence of a fixed point ofT as well asT–. Now, Lemma . gives us the existence of a coupled fixed point ofF.
In what follows, we prove that Theorem . remains valid if we replace the continuity condition ofFwith the following condition:
(P): if{xn}and{yn}are sequences inXsuch that
α(xn,yn), (xn+,yn+)
≥ and α(yn+,xn+), (yn,xn)
≥,
xn→x∈Xandyn→y∈Xasn→+∞, then αT–(xn,yn),T–(x,y)
≥ and αT–(y,x),T–(yn,xn)
≥ for alln∈N.
Theorem . If we replace the continuity of F in Theorem . by the condition (P), then the result holds true.
Proof We employ the same notations of the proof of Theorem .. Let{xn,yn}be a
se-quence inY such thatη((xn,yn), (xn+,yn+))≥ and (xn,yn)→(x,y) asn→+∞. Using
the condition (P), we have
ηT–(xn,yn),T–(x,y)
Then all the hypotheses of Theorem . are satisfied. Thus, we deduce the existence of a fixed point ofT that gives us from Lemma . the existence of a coupled fixed point ofF.
To ensure the uniqueness of the coupled fixed point, we consider the following condi-tion:
(U): For all(x,y), (x,y)∈X×X, there exists(x,y)∈X×Xsuch that
α(x,y), (x,y)
≥, α(y,x), (y,x)
≥
and
α(x,y), (x,y)
≥, α(y,x), (y,x)
≥.
Theorem . Adding the condition (U) to the hypotheses of Theorem . (resp. Theo-rem .) we obtain the uniqueness of the coupled fixed point of F.
Proof Clearly, under the hypothesis (U),T andηsatisfy the hypothesis (U). Therefore, from Theorem . and Lemma ., the result follows immediately.
Example . LetX=Nequipped with the standard metricd(x,y) =|x–y|for allx,y∈N. Then (X,d) is a complete metric space. Define the mappingF:X×X→Xby
F(x,y) = x–·(y– ).
Clearly,Fis a continuous and bijective mapping. Defineα:X×X→[, +∞) by
α(x,y), (u,v)=
⎧ ⎨ ⎩
, ifx≥y,u≥v, , otherwise.
It is easy to show that for all (x,y), (u,v)∈X×X, we have
ξdF(x,y),F(u,v)≥ α
(x,y), (u,v)d(x,u) +d(y,v).
Then () is satisfied withξ(a) =a/ for alla≥. On the other hand, the condition (i) of Theorem . holds and the condition (ii) of the same theorem is also satisfied with (x,y) = (, ). All the required hypotheses of Theorem . are true and so we deduce the
existence of a coupled fixed point ofF. Here, (, ) is a coupled fixed point ofF.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Acknowledgements
The first author would like to thank the University Grants Commission for financial support during the preparation of this manuscript.
Received: 24 April 2012 Accepted: 31 August 2012 Published: 19 September 2012
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