R E S E A R C H
Open Access
Fixed point theorems for integral
G-contractions
Maria Samreen
1and Tayyab Kamran
2**Correspondence:
2Department of Mathematics,
Quaid-i-Azam University, Islamabad, Pakistan
Full list of author information is available at the end of the article
Abstract
We define the notion of an integralG-contraction for mappings on metric spaces and establish some fixed point theorems for such mappings. Our results generalize and unify some recent results by Jachymski, Branciari and those contained therein. As an application, we obtain a result for cyclic operators. Moreover, we provide an example to show that our results are substantial improvements of some known results in literature.
MSC: 47H10; 54H25
Keywords: fixed points; directed graph; Picard operator
1 Introduction
Branciari [] generalized the Banach contraction principle by proving the existence of a unique fixed point of a mapping on a complete metric space satisfying a general contrac-tive condition of integral type. Afterwards, many authors undertook further investigations in this direction (see,e.g., [–]). Ran and Reurings [] initiated the study of fixed points of mappings on partially ordered metric spaces. A number of interesting fixed point the-orems have been obtained by different authors for this setting; see, for example, [–]. Jachymski [] used the platform of graph theory instead of partial ordering and unified the results given by authors [, , ]. He showed that a mapping on a complete metric space still has a fixed point provided the mapping satisfies the contraction condition for pairs of points which form edges in the graph. Subsequently, Beget al.[] established a multivalued version of the main result of Jachymski []. Aydiet al.[] studied fixed point theorems for weaklyG-contraction mappings inG-metric spaces. Later on, Bojor [] ob-tained some results in such settings by weakening the condition of BanachG-contractivity and introducing some new type of connectivity of a graph.
In this paper, motivated by the work of Jachymski [] and Branciari [], we introduce two new contraction conditions for mappings on complete metric spaces and, using these contractive conditions, obtain some fixed point theorems. Our results generalize and unify some results by the above mentioned authors.
2 Preliminaries
Let (X,) be a partially ordered set. A mappingf :X→Xis said to be nonincreasing if x,y∈X,xy⇒f(x)f(y). A mappingf is said to be nondecreasing ifx,y∈X,xy⇒ f(x)f(y). A mappingf from a metric space (X,d) into (X,d) is called a Picard operator (PO) [] iffhas a unique fixed pointt∈Xandlimn→∞fnx=tfor allx∈X. Two sequences
{xn}and{yn}in a metric space (X,d) are said to be equivalent ifd(xn,yn)→. Moreover, if
each of them is Cauchy, then these are called Cauchy equivalent. A mapping from a metric space (X,d) into (X,d) is called orbitally continuous if for allx,y∈Xand any sequence{kn} of positive integers,fknx→yimpliesf(fknx)→fyasn→ ∞.
LetG= (V(G),E(G)) be a directed graph. ByG–we denote the graph obtained fromG by reversing the direction of edges, and by letterGwe denote the undirected graph ob-tained fromGby ignoring the direction of edges. It will be more convenient to treatGas a directed graph for which the set of its edges is symmetric,i.e.,E(G) =E(G)∪E(G–). If xandyare vertices in a graphG, then a path inGfromxtoyof lengthlis a sequence (xi)li=ofl+ vertices such thatx=x,xl=yand (xi–,xi)∈E(G) fori= , . . . ,l. A graph
Gis called connected if there is a path between any two vertices.Gis weakly connected ifGis connected. For a graphGsuch thatE(G) is symmetric andxis a vertex inG, the subgraphGxconsisting of all edges and vertices which are contained in some path
begin-ning atxis called component ofGcontainingx. In this caseV(Gx) = [x]G, where [x]Gis
the equivalence class of a relationRdefined onV(G) by the rule:yRzif there is a path inGfromytoz. Clearly,Gxis connected. A graphGis known as a (C)-graph inX[]
if for any sequence{xn}inXwithxn−→xand (xn,xn+)∈E(G) forn∈N, there exists a subsequence{xnk}of{xn}such that (xnk,x)∈E(G) fork∈N.
Subsequently, in this paper,Xis a complete metric space with metricd, andis the diagonal of the Cartesian productX×X.Gis a directed graph such that the setV(G) of its vertices coincides withX, and the setE(G) of its edges contains all loops,i.e.,E(G)⊇. Assume thatGhas no parallel edges. We may treatGas a weighted graph by assigning to each edge the distance between its vertices. A mappingf :X→Xis called orbitally G-continuous [] if for allx,y∈Xand any sequence (kn)n∈Nof positive integers,fknx→y
and (fknx,fkn+x)∈E(G),∀n∈Nimplyf(fknx)→fy.
We state, for convenience, the following definition and result.
Definition .[, Definition .] A mappingf :X→Xis called a BanachG-contraction or simply aG-contraction iff preserves edges ofG,i.e.,
∀x,y∈X(x,y)∈E(G)⇒(fx,fy)∈E(G), (.)
andf decreases weights of edges ofGin the following way:
∃c∈(, ), ∀x,y∈X(x,y)∈E(G)⇒d(fx,fy)≤cd(x,y). (.)
Letdenote the class of all mappingsφ: [, +∞)→[, +∞) which are Lebesgue in-tegrable, summable on each compact subset of [, +∞), nonnegative and for each> ,
φ(s)ds> .
Theorem .[, Theorem .] Let(X,d)be a complete metric space,c∈(, ),and let f :X→X be a mapping such that for each x,y∈X,
d(fx,fy)
φ(s)ds≤c
d(x,y)
φ(s)ds, (.)
3 Main results
We begin this section, motivated by Jachymski [] and Branciari [], by introducing the following definition.
Definition . A mappingf :X→Xis called an integralG-contraction iff preserves edges (see (.)) and
∀x,y∈X, (x,y)∈E(G) implies
d(fx,fy)
φ(s)ds≤c
d(x,y)
φ(s)ds (.)
for somec∈(, ) andφ∈.
Remark . Note that iff :X→Xsatisfies (.), thenf is an integralG-contraction whereG= (X,X×X). Moreover, every BanachG-contraction is an integralG-contraction (takeφ(x) = ), but the converse may not hold.
Proposition . Let f :X→X be an integral G-contraction with contraction constant c∈(, )andφ∈,then:
(i) f is both an integralG–-contraction and an integralG-contraction with the same contraction constantc∈(, )andφ.
(ii) [x]Gisf-invariant andf|[x]Gis an integralGx-contraction provided that there exists somex∈Xsuch thatfx∈[x]G.
Proof (i) is a consequence of symmetry ofd.
(ii) Letx∈[x]G. Then there is a pathx=z,z, . . . ,zm=xbetweenxandxinG. Sincef is an integralG-contraction, then (fzi–,fzi)∈E(G),∀i= , , . . . ,l. Thusfx∈[fx]G= [x]G.
Suppose that (x,y)∈E(Gx), then (fx,fy)∈E(G) asf is an integralG-contraction. But
[x]G isf invariant, so we conclude that (fx,fy)∈E(Gx). Furthermore, (.) is satisfied
automatically becauseGxis a subgraph ofG.
Lemma . Let f :X→X be an integral G-contraction and y∈[x]G,then
lim n→∞d
fnx,fny= . (.)
Proof Letx∈Xandy∈[x]G, then there existsN∈Nsuch thatx=x,xN =yand (xi–,xi)∈
E(G) for alli= , , . . . ,N. By Proposition . it follows that (fnxi–,fnxi)∈E(G) and
d(fnxi–,fnxi)
φ(s)ds≤c
d(fn–xi–,fn–xi)
φ(s)ds (.)
holds for alln∈Nandi= , , , . . . ,N. Denotedn=d(fnxi–,fnxi) for alln∈N. Ifdm=
for somem∈N, then it follows from (.) thatdn= for alln∈Nwithn>m. Therefore, in
this case,limn→∞dn= . Now, assume thatdn> . We claim that{dn}is a non-increasing
sequence. Otherwise, there existsn◦∈Nsuch thatdn◦>dn◦–. Now, using the properties ofφ, it follows from (.) that
<
dn◦–
φ(s)ds≤
dn◦
φ(s)ds≤c
dn◦–
Since, <c< , this yields a contradiction. Therefore,limn→∞dn=r≥. Letr> , then it
follows from (.) that
r
φ(s)ds= lim n→∞
dn
φ(s)ds≤c lim n→∞
dn–
φ(s)ds
=c
r
φ(s)ds, (.)
which implies ( –c)rφ(s)ds≤ and it further implies thatrφ(s)ds= , a contradic-tion. Thus,limn→∞dn=limn→∞d(fnxi–,fnxi) = ,∀i= , , . . . ,N, in both cases. From the
triangular inequality, we haved(fnx,fny)≤N
i=d(fnxi–,fnxi), and lettingn→ ∞gives
limn→∞d(fnx,fny) = .
Now we define a subclass of integralG-contractions. We call this a class of sub-integral G-contractions. Let us denote bythe class of all mappingsφ∈satisfying following:
α+β
φ(s)ds≤
α
φ(s)ds+
β
φ(s)ds (.)
for everyα,β≥. Note that every constant functionφ(x) =k> belongs to the class.
Example . Defineφ,φ: [,∞)→[,∞) by
φ(x) =
x+ , φ(x) =
√x, x= ,
, otherwise.
It is easy to see thatφ,φsatisfy (.) and thus belong to the class.
Definition . We say that an integralG-contraction is a sub-integralG-contraction if φ∈.
Lemma . Let f :X→X be a sub-integral G-contraction and y∈[x]G.Then there exists
p(x,y)≥such that
d(fnx,fny)
φ(s)ds≤cnp(x,y), ∀n∈N. (.)
Proof Letx∈Xandy∈[x]G, then there existsN∈Nsuch thatx=x,xN =yand (xi–,xi)∈
E(G) for alli= , , . . . ,N. Sinceφ∈, using the triangular inequality, it follows that
d(fnx,fny)
φ(s)ds≤
N
i=
d(fnxi–,fnxi)
φ(s)ds. (.)
Moreover,
d(fnxi–,fnxi)
φ(s)ds≤cn
d(xi–,xi)
since (fnx
i–,fnxi)∈E(G) for alli= , , , . . . ,Nandn∈N. From (.) and (.) we get d(fnx,fny)
φ(s)ds≤cn
N
i=
d(xi–,xi)
φ(s)ds
=cnp(x,y), (.)
wherep(x,y) =Ni=d(xi–,xi)
φ(s)ds.
Definition . Letf :X→X,y∈Xand the sequence{fny}inXbe such thatfny→x*∈X with (fny,fn+y)∈E(G) forn∈N. We say that the graphGis a (C
f)-graph if there exists a
subsequence{fnky}such that (fnky,x*)∈E(G) fork∈N.
Obviously, every (C)-graph is a (Cf)-graph for any self-mappingfonX, but the converse
may not hold as shown in the following.
Example . LetX:= [, ] with respect to the usual metricd(x,y) =|x–y|. Consider the graphGconsisting ofV(G) :=XandE(G) :={( n
n+,
n+
n+) :n∈N} ∪ {(
x
n,nx+) :n∈N,x∈
[, ]} ∪ {(xn, ) :n∈N,x∈[, ]}. Note thatGis not a (C)-graph as nn+ →. Definef :
X→Xasfx=x. ThenGis a (Cf)-graph sincefnx=xn → for eachx∈[, ].
Theorem . Let f :X→X be a sub-integral G-contraction.Assume that
(i) Xf :={x∈X: (x,fx)∈E(G)} =∅, (ii) Gis a(Cf)-graph.
Then,for any z∈Xf,f|[z]G is a Picard operator.Further,if G is weakly connected,then f
is a Picard operator.
Proof Letz∈Xf, thenfz∈[z]G. Letm>n≥, using Lemma ., we have
d(fmz
,fnz)
φ(s)ds≤
d(fnz
,fn+z)
φ(s)ds+· · ·+
d(fm–z ,fmz)
φ(s)ds
≤cn+cn++cn++· · ·+cm–p(z,fz)
≤
cn
–c
p(z,fz)→ asn→ ∞.
It follows that{fnz}is a Cauchy sequence inX. Therefore,fnz→t∈X. Letybe another
element in [z]G, then it follows from Lemma . thatfny→t, too. Next, we show thatt
is a fixed point off. Sincefnz→t∈Xand (fnz,fn+z)∈E(G) for alln∈NandGis a (Cf)-graph, then there exists a subsequence{fnkz}of{fnz}such that (fnkz,t)∈E(G) for
allk∈N. Therefore, (z,fz,fz, . . . ,fnz,t) is a path inGand so inGfromztot, thus t∈[z]G. From (.), we get
d(fnk+z,ft)
φ(s)ds≤c
d(fnkz,t)
φ(s)ds, ∀k∈N,
lettingk→ ∞, we haved(t,ft)φ(s)ds= , which implies thatd(t,ft) = . This shows that f|[z]Gis a Picard operator. Moreover, ifGis weakly connected, thenf is a Picard operator
Corollary . Let(X,d)be a complete metric space endowed with a graph G such that G is(Cf)-graph.Then the following statements are equivalent:
() Gis weakly connected.
() Every sub-integralG-contractionf onXis a Picard operator provided thatXf=∅.
Proof ()⇒(): It is immediate from Theorem ..
()⇒(): On the contrary, suppose thatGis not weakly connected, thenGis discon-nected,i.e., there existsx◦∈Xsuch that [x◦]G=∅andX\[x◦]G=∅. Lety◦∈X\[x◦]G, we
construct a self-mappingf by (as in [, Theorem .]):
fx=
⎧ ⎨ ⎩
x◦ ifx∈[x◦]G,
y◦ ifx∈X\[x◦]G.
Let (x,y)∈E(G), then [x]G:= [y]G, which impliesfx=fyhence (fx,fy)∈E(G), sinceG
con-tains all loops and further (.) is trivially satisfied (takeφ(s) = ). Butx◦andy◦are two fixed points off contradicting the fact thatf has a unique fixed point.
Theorem . Let f :X→X be a sub-integral G-contraction.Assume that f is orbitally G-continuous and Xf :={x∈X: (x,fx)∈E(G)} =∅.Then,for any z∈Xf and y∈[z]G, limn→∞fny=t∈X where t is a fixed point of f.Further,if G is weakly connected,then f is a Picard operator.
Proof Letz∈Xf, then the arguments used in the proof of Theorem . imply that{fnz} is a Cauchy sequence. Therefore,fnz→t∈X. Since (fnz,fn+z)∈E(G) for alln∈N andf is orbitallyG-continuous, thereforet=limn→∞ffnz=ft. Note that ifyis another
element from [z]G, then it follows from Lemma . thatlimn→∞fny=t. Finally, ifGis weakly connected, then [z]G:=X, which yields thatf is a Picard operator.
Remark . Theorem . generalizes claims & of [, Theorem .].
Theorem . Let f :X→X be a sub-integral G-contraction.Assume that f is orbitally continuous and if there exists some z∈ X such that fz∈ [z]G, then, for y ∈[z]G, limn→∞fny=t∈X,where t is a fixed point of f.Further,if G is weakly connected,then f is a Picard operator.
Proof Let z ∈X be such that fz∈[z]G, then using the same arguments as in the
proof of Theorem .,{fnz
} is Cauchy and thus limn→∞fnz=t∈X. Moreover, t=
limn→∞ffnz=ft, sincef is orbitally continuous. Note that ifyis another element from
[z]G, then it follows from Lemma . thatlimn→∞fny=t. IfGis weakly connected, then
[z]G:=X. This yields thatf is a Picard operator.
Remark . Theorem . generalizes claims & of [, Theorem .] and thus generalizes and extends the results of Nieto and Rodrýguez-López [, Theorems . and .], Petrusel and Rus [, Theorem .] and Ran and Reurings [, Theorem .].
() Gis weakly connected.
() Every sub-integralG-contractionf onXis a Picard operator provided thatf is orbitally continuous.
Proof () ⇒() is obvious from Theorem .. Note that the example constructed in
Corollary . is orbitally continuous. Hence, ()⇒().
Remark . Corollary . generalizes claims & of [, Corollary .].
Kirket al.[] introduced cyclic representations and cyclic contractions and they have been further investigated by many authors (see,e.g., [–]). LetXbe a nonempty set,m be a positive integer and{Ai}m
i=be nonempty closed subsets ofXandf :
m i=Ai→
m i=Ai
be an operator. ThenX:=mi=Aiis known as a cyclic representation ofXw.r.t.f if
f(A)⊂A, . . . , f(Am–)⊂Am, f(Am)⊂A (.)
and the operatorf is known as a cyclic operator.
Theorem . Let(X,d)be a complete metric space.Let m be a positive integer,{Ai}m i=be nonempty closed subsets of X,Y:=mi=Aiand f :Y→Y.Assume that
(i) mi=Aiis a cyclic representation ofYw.r.t.f;
(ii) there existsφ∈such thatd(fx,fy)φ(s)ds≤c(d(x,y))φ(s)dswhenever,x∈Ai,
y∈Ai+,whereAm+=A.
Then f has a unique fixed point t∈mi=Aiand fny→t for any y∈ m
i=Ai.
Proof We note that (Y,d) is a complete metric space. Let us consider a graphGconsisting ofV(G) :=YandE(G) :=∪ {(x,y)∈Y×Y:x∈Ai,y∈Ai+;i= , . . . ,m}. By (i) and (ii) it follows thatfis a sub-integralG-contraction. Now letfnx→x*inYsuch that (fnx,fn+x)∈ E(G) for alln≥. Then in view of (.), the sequence{fnx}has infinitely many terms in eachAiso that one can easily extract a subsequence of{fnx}converging tox*in eachAi.
SinceAi’s are closed, thenx*∈ m
i=Ai. Now it is easy to form a subsequence{fnkx}in some
Aj,j∈ {, . . . ,m}such that (fnkx,x*)∈E(G) fork≥, it vindicatesGis a weakly connected
(Cf)-graph and thus conclusion follows from Theorem ..
Remark . Takingφ(s) = , Theorem . subsumes the main result of [].
Theorem . Let f :X→X be an integral G-contraction. Assume that the following assertions hold:
(i) there existsz∈Xf such that
d(fx,fy)
φ(s)ds≤c
d(x,y)
φ(s)ds, ∀x,y∈O(z)⊂[z]G, (.)
whereO(z) ={z,fz,fz, . . .}.
(ii) Gis a(Cf)-graph.
Then,for any z∈Xf,f|[z]G is a Picard operator.Furthermore,if G is weakly connected
Proof Letz∈Xf, thenfz∈[z]G. Now, it follows from Proposition .(ii) thatO(z)⊂
[z]G. Moreover, from Lemma . we get
lim n→∞d
fnz,fn+z
= . (.)
We claim that{fnz
}is Cauchy sequence. Otherwise, there exists some> in such a way that, for eachk∈N, there aremk,nk∈Nwithnk>mk>ksatisfying
dfmkz,fnkz≥. (.)
We may choose sequences{mk},{nk}such that corresponding tomk, natural numbernk
is smallest satisfying (.). Therefore,
≤dfnkz
,fmkz
≤dfnkz
,fnk–z
+dfnk–z
,fmkz
<dfnkz
,fnk–z
+.
On lettingk→ ∞and using (.), we get
lim k→∞d
fnkz
,fmkz
=. (.)
Moreover, using (.) and (.), it follows from
dfnk–z,fmk–z≤dfnk–z,fnkz+dfnkz,fmkz+dfmkz,fmk–z
and
dfnkz,fmkz≤dfnkz,fnk–z+dfnk–z,fmk–z+dfmk–z,fmkz
that
lim n→∞d
fnk–z,fmk–z=. (.)
Sincefnk–z
,fmk–z∈O(z), it follows from assertion (ii) that
d(fnkz,fmkz)
φ(s)ds≤c
d(fnk–z,fmk–z)
φ(s)ds. (.)
Lettingk→ ∞and using (.), (.), we get (–c)φ(s)ds≤. As <c< , this implies that= . Therefore,{fnz}is a Cauchy sequence inX. The rest of the proof runs on the
same lines as the proof of Theorem ..
Remark . Theorem . generalizes [, Theorem .].
Example . LetX:= [, ] be equipped with the usual metric d. Definef :X→X, φ: [, +∞)→[, +∞) by
fx=
⎧ ⎨ ⎩
x
+px ifx=
n,
ifx=n and φ(s) =
⎧ ⎨ ⎩
ss–( –logs) ifs> ,
ifs=
for alln∈Nandp≥ is any fixed positive integer. Consider the graphGsuch thatV(G) := XandE(G) :=∪{(,x) :x∈X}∪{(n+ ,n) :n∈N}. We observe that (.) holds. Moreover,
τ
φ(s)ds=τ
τ, so that (.) is equivalent to
d(fx,fy)d(fx,fy)≤cd(x,y)d(x,y) for (x,y)∈E(G). (.)
Next we show that (.) is satisfied forc=+p< . Case i. Let (x,x)∈E(G), then (.) is trivially satisfied.
Case ii. Let (,x)∈E(G);x=n forn∈N, (.) is trivially satisfied. Let (,x)∈E(G);x=n forn∈N
d(fx,fy)d(fx,fy)= n+p–
|
n+p–|
=
(n+p)(n+p) (.)
and
d(x,y)d(x,y)= n–
|
n–|
=
(n)n. (.)
From inequality (.) we need to show that
(n+p)(n+p) ≤
( +p)(n)n (.)
or, equivalently,
n n+p
n
(n+p)p ≤
+p.
Since
(n+p)p≤(n+p)<(+p) for alln∈Nandnn+p< , thus inequality (.) is satisfied.
Case iii. Let (
n+,
n)∈E(G) forn∈N, then we have
d(fx,fy)d(fx,fy)= n+ +p–
n+p
|
n++p– n+p|
=
(n+ +p)(n+p)
(n++p)(n+p)
(.)
and
d(x,y)d(x,y)= n–
n+
|
n– n+|
=
n(n+ )
n(n+)
, (.)
so we need to show that
(n+ +p)(n+p)
(n++p)(n+p)
≤
( +p)
n(n+ )
n(n+)
On rearranging we have
[n(n+ )]n(n+)
[(n+ +p)(n+p)](n++p)(n+p) ≤ (p+ )
or
n n+p
n(n+) n+ n+ +p
n(n+)
[(n+ +p)(n+p)](p+np+p) ≤
+p.
By analyzing L.H.S., we see thatn+np < andnn+++p< for alln∈Nand
[(n++p)(n+p)](p+np+p) <
[(n++p)(n+p)]< (n+p) ≤
(+p) for alln∈N, which infers that inequality (.) is indeed true. Therefore,f is an integralG-contraction with contraction constantc=+p. Note thatG is a weakly connected (C)-graph and (.) also holds forO(). Thus all the conditions of Theorem . are satisfied andf is a Picard operator with fixed point . Note thatf is not a BanachG-contraction since, for (n+,n)∈E(G),
d(fn+,fn) d(n+,n) =
|
n++p–
n+p| |
n–
n+|
→ asn→ ∞.
By settingp= in the above example, we have
d
f ,f
d(f ,f )
=
=d
,
d( , )
.
Therefore one cannot apply Theorem . [].
Conclusion
The notion of an integralG-contraction not only generalizes/extends the notion of a Ba-nachG-contraction, but it also improves the integral inequality (.). Whereas, the no-tion of a sub-integral G-contraction generalizes the notion of a BanachG-contraction, but it partially generalizes the integral inequality (.). Therefore, Theorem . general-izes/extends some results of Jachymski [] and provides partial improvement to the main result of Branciari []. A very natural question is bound to be posed: Are the conclusions of Theorems ., ., . still valid for integralG-contractions? In Theorem ., we have provided a partial answer to this question by imposing condition (.). But it re-mains open to investigate an affirmative answer without the crucial condition of (.). Furthermore, Example . invokes the generality of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally in this article.
Author details
1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad,
Pakistan. 2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.
Acknowledgements
Authors are grateful to referees for their suggestions.
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doi:10.1186/1687-1812-2013-149
