R E S E A R C H
Open Access
Fixed points of cyclic weakly
(
ψ
,
ϕ
,
L
,
A
,
B
)-contractive mappings in
ordered
b
-metric spaces with applications
Nawab Hussain
1*, Vahid Parvaneh
2, Jamal Rezaei Roshan
3and Zoran Kadelburg
4*Correspondence:
1Department of Mathematics, King
Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract
We introduce the notion of ordered cyclic weakly (
ψ
,ϕ
,L,A,B)-contractive mappings, and we establish some fixed and common fixed point results for this class ofmappings in complete orderedb-metric spaces. Our results extend several known results from the context of ordered metric spaces to the setting of orderedb-metric spaces. They are also cyclic variants of some very recent results in orderedb-metric spaces with even weaker contractive conditions. Some examples support our results and show that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
MSC: 47H10; 54H25
Keywords: common fixed point; cyclic contraction; almost contraction; ordered
b-metric space; altering distance function
1 Introduction and preliminaries
The Banach contraction principle is a very popular tool for solving problems in nonlinear
analysis. One of the interesting generalizations of this basic principle was given by Kirket
al.[] in by introducing the following notion of cyclic representation.
Definition [] LetAandBbe non-empty subsets of a metric space (X,d) andT:A∪
B→A∪B. Then T is called a cyclic map ifT(A)⊆BandT(B)⊆A.
The following interesting theorem for a cyclic map was given in [].
Theorem Let A and B be nonempty closed subsets of a complete metric space (X,d). Suppose that T:A∪B→A∪B is a cyclic map such that
d(Tx,Ty)≤kd(x,y)
for all x∈A and y∈B,where k∈[, )is a constant.Then T has a unique fixed point u and u∈A∩B.
It should be noted that cyclic contractions (unlike Banach-type contractions) need not
be continuous, which is an important gain of this approach. Following the work of Kirket
al., several authors proved many fixed point results for cyclic mappings, satisfying various (nonlinear) contractive conditions.
Berinde initiated in [] the concept of almost contractions and obtained several
interest-ing fixed point theorems. This has been a subject of intense study since then, see,e.g., [–
]. Some authors used related notions as ‘condition (B)’ (Babuet al.[]) and ‘almost
gener-alized contractive condition’ for two maps (Ćirićet al.[]), and for four maps (Aghajaniet
al.[]). See also a note by Pacurar []. Here, we recall one of the respective definitions.
Definition [] Letf andgbe two self-mappings on a metric space (X,d). They are said
to satisfy almost generalized contractive condition, if there exist a constantδ∈(, ) and
someL≥ such that
d(fx,gy)≤δmax
d(x,y),d(x,fx),d(y,gy),d(x,gy) +d(y,fx)
+Lmind(x,fx),d(y,gy),d(x,gy),d(y,fx)
for allx,y∈X.
Khanet al.[] introduced the concept of an altering distance function as follows.
Definition [] A functionϕ: [, +∞)→[, +∞) is called an altering distance function if the following properties hold:
. ϕis continuous and non-decreasing.
. ϕ(t) = if and only ift= .
So far, many authors have studied fixed point theorems, which are based on altering distance functions.
The concept of ab-metric space was introduced by Bakhtin in [], and later used by
Czerwik in [, ]. After that, several interesting results about the existence of fixed points
for single-valued and multi-valued operators inb-metric spaces have been obtained (see,
e.g., [–]). Recently, Hussain and Shah [] obtained some results on KKM mappings
in coneb-metric spaces.
Consistent with [] and [], the following definitions and results will be needed in the sequel.
Definition [] LetXbe a (nonempty) set, and lets≥ be a given real number. A
func-tiond:X×X→R+is ab-metric if for allx,y,z∈X, the following conditions hold:
(b) d(x,y) = iffx=y,
(b) d(x,y) =d(y,x),
(b) d(x,z)≤s[d(x,y) +d(y,z)].
In this case, the pair (X,d) is called ab-metric space.
It should be noted that the class ofb-metric spaces is effectively larger than the class of
metric spaces, since ab-metric is a metric if (and only if )s= . Here, we present an easy
example to show that in general, ab-metric need not necessarily be a metric (see also [,
Example Let (X,ρ) be a metric space andd(x,y) = (ρ(x,y))p, wherep> is a real number.
Thendis ab-metric withs= p–. Condition (b
) follows easily from the convexity of the
functionf(x) =xp(x> ).
The notions ofb-convergent andb-Cauchy sequences, as well as ofb-completeb-metric
spaces are introduced in an obvious way (see,e.g., []).
It should be noted that in general, ab-metric functiond(x,y) fors> need not be jointly
continuous in both variables. The following example (corrected from []) illustrates this fact.
Example LetX=N∪ {∞}, and letd:X×X→Rbe defined by
d(m,n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
, ifm=n,
|
m–
n|, if one ofm,nis even and the other is even or∞,
, if one ofm,nis odd and the other is odd (andm=n) or∞,
, otherwise.
Then considering all possible cases, it can be checked that for allm,n,p∈X, we have
d(m,p)≤
d(m,n) +d(n,p).
Thus, (X,d) is ab-metric space (withs= /). Letxn= nfor eachn∈N. Then
d(n,∞) =
n→ asn→ ∞,
that is,xn→ ∞, butd(xn, ) = → =d(∞, ) asn→ ∞.
Aghajani et al.[] proved the following simple lemma about the b-convergent
se-quences.
Lemma Let (X,d) be a b-metric space with s≥, and suppose that {xn} and {yn}
b-convergeto x,y,respectively.Then we have
sd(x,y)≤lim infn→∞ d(xn,yn)≤lim sup
n→∞ d(xn,yn)≤s d(x,y).
In particular,if x=y,thenlimn→∞d(xn,yn) = .Moreover,for each z∈X,we have
sd(x,z)≤lim infn→∞ d(xn,z)≤lim supn→∞ d(xn,z)≤sd(x,z).
The existence of fixed points for mappings in partially ordered metric spaces was first investigated in by Ran and Reurings [], and then by Nieto and Lopez []. After-wards, this area was a field of intensive study of many authors.
Theorem [] Let(X,,d)be a complete ordered metric space,and let A,B be closed nonempty subsets of X with X=A∪B.Let f,g:X→X be two mappings,which are(A,B) -weakly increasing(see further Definition).Assume that
(a) A∪Bis a cyclic representation ofXw.r.t.the pair(f,g),i.e.,f(A)⊂Bandg(B)⊂A; (b) there exist <δ< and an altering distance functionψsuch that for any two comparable
elementsx,y∈Xwithx∈Aandy∈B,we have
ψd(fx,gy)≤δψ
max
d(x,y),d(x,fx),d(y,gy),
d(x,gy) +d(y,fx);
(c) f orgis continuous,or
(c) the space(X,,d)is regular.
Then f and g have a common fixed point.
Here, the ordered metric space (X,,d) is called regular if for any non-decreasing
se-quence{xn}inXsuch thatxn→x∈X, asn→ ∞, one hasxnxfor alln∈N.
By an orderedb-metric space, we mean a triple (X,,d), where (X,) is a partially
or-dered set, and (X,d) is ab-metric space. Fixed points in such spaces were studied,e.g.,
by Aghajaniet al.[] and Roshanet al.[]. In the last mentioned paper, the following
common fixed point results for contractions in orderedb-metric spaces were proved.
Theorem [] Let(X,,d)be a complete ordered b-metric space,and let f,g:X→X be two weakly increasing mappings.Suppose that there exist two altering distance functions
ψ,ϕand a constant L≥such that the inequality
ψsd(fx,gy)≤ψMs(x,y)–ϕMs(x,y)+LψN(x,y)
holds for all comparable x,y∈X,where
Ms(x,y) =max
d(x,y),d(x,fx),d(y,gy),d(x,gy) +d(y,fx) s
and
N(x,y) =mind(y,gy),d(x,gy),d(y,fx).
If either[f or g is continuous],or the space(X,,d)is regular,then f and g have a common fixed point.
In this paper, we introduce the notion of ordered cyclic weakly (ψ,ϕ,L,A,B
)-contrac-tions and then derive fixed point and common fixed point theorems for these cyclic
con-tractions in the setup of complete orderedb-metric spaces. Our results extend some fixed
point theorems from the framework of ordered metric spaces, in particular Theorem . On the other hand, they are cyclic variants of Theorem with even weaker contractive conditions.
2 Common fixed point results
In this section, we introduce the notion of ordered cyclic weakly (ψ,ϕ,L,A,B)-contractive
pair of self-mappings and prove our main results.
Definition Let (X,,d) be an orderedb-metric space, letf,g:X→Xbe two mappings,
and letAandBbe nonempty closed subsets ofX. The pair (f,g) is called an ordered cyclic
weakly (ψ,ϕ,L,A,B)-contraction if
() X=A∪Bis a cyclic representation ofXw.r.t. the pair(f,g); that is,fA⊆Band
gB⊆A;
() there exist two altering distance functionsψ,ϕand a constantL≥, such that for arbitrary comparable elementsx,y∈Xwithx∈Aandy∈B, we have
ψsd(fx,gy)≤ψMs(x,y)–ϕMs(x,y)+LψN(x,y), (.)
where
Ms(x,y) =max
d(x,y),d(x,fx),d(y,gy),d(x,gy) +d(y,fx) s
(.)
and
N(x,y) =mind(y,gy),d(x,gy),d(y,fx). (.)
Definition [] Let (X,) be a partially ordered set, and letAandBbe closed subsets
ofXwithX=A∪B. Letf,g:X→Xbe two mappings. The pair (f,g) is said to be (A,B
)-weakly increasing iffxgfxfor allx∈Aandgyfgyfor ally∈B.
Theorem Let(X,,d)be a complete ordered b-metric space,and let A and B be closed subsets of X.Let f,g:X→X be two(A,B)-weakly increasing mappings with respect to. Suppose that
(a) the pair(f,g)is an ordered cyclic weakly(ψ,ϕ,L,A,B)-contraction; (b) f orgis continuous.
Then f and g have a common fixed point u∈A∩B.
Proof Let us divide the proof into two parts.
First part. We prove thatu∈A∩Bis a fixed point off if and only ifuis a fixed point
ofg. Suppose thatuis a fixed point off. Asuuandu∈A∩B, by (.), we have
ψsd(u,gu)=ψsd(fu,gu)
≤ψ
max
d(u,fu),d(u,gu), s
d(u,gu) +d(u,fu)
–ϕ
max
d(u,fu),d(u,gu), s
d(u,gu) +d(u,fu)
+Lmind(u,gu),d(u,fu)
=ψd(u,gu)–ϕd(u,gu)
It follows thatϕ(d(u,gu)) = . Therefore,d(u,gu) = , and hencegu=u. Similarly, we can
show that ifuis a fixed point ofg, thenuis a fixed point off.
Second part (construction of a sequence by iterative technique).
Letx∈A, and letx=fx. SincefA⊆B, we havex∈B. Also, letx=gx. SincegB⊆A,
we havex∈A. Continuing this process, we can construct a sequence{xn}in X such that
xn+=fxn,xn+=gxn+,xn∈Aandxn+∈B. Sincefandgare (A,B)-weakly increasing,
we have
x=fxgfx=x=gxfgx=x · · ·
xn+=fxngfxn=xn+ · · ·.
Ifxn=xn+, for somen∈N, thenxn=fxn. Thus,xnis a fixed point off. By the first
part of proof, we conclude thatxnis also a fixed point ofg. Similarly, ifxn+=xn+, for
somen∈N, thenxn+=gxn+. Thus,xn+is a fixed point ofg. By the first part of proof,
we conclude thatxn+is also a fixed point off. Therefore, we assume thatxn=xn+for all
n∈N. Now, we complete the proof in the following steps.
Step . We will prove that
lim
n→∞d(xn,xn+) = .
Asxnandxn+are comparable andxn∈Aandxn+∈B, by (.), we have
ψd(xn+,xn+)
≤ψsd(xn+,xn+)
=ψsd(fxn,gxn+)
≤ψMs(xn,xn+)
–ϕMs(xn,xn+)
+LψN(xn,xn+)
,
where
Ms(xn,xn+) = max
d(xn,xn+),d(xn,fxn),d(xn+,gxn+),
d(fxn,xn+) +d(xn,gxn+)
s
=max
d(xn,xn+),d(xn+,xn+),
d(xn,xn+)
s
≤max
d(xn,xn+),d(xn+,xn+),
s[d(xn,xn+) +d(xn+,xn+)]
s
=maxd(xn,xn+),d(xn+,xn+)
,
and
N(xn,xn+) =min
d(xn+,gxn+),d(xn+,fxn),d(xn,gxn+)
=mind(xn+,xn+),d(xn+,xn+),d(xn,xn+)
Hence, we have
ψd(xn+,xn+)
≤ψmaxd(xn,xn+),d(xn+,xn+)
–ϕmaxd(xn,xn+),d(xn+,xn+)
. (.)
If
maxd(xn,xn+),d(xn+,xn+)
=d(xn+,xn+),
then (.) becomes
ψd(xn+,xn+)
≤ψd(xn+,xn+)
–ϕd(xn+,xn+)
<ψd(xn+,xn+)
,
which gives a contradiction. So,
maxd(xn,xn+),d(xn+,xn+)
=d(xn,xn+),
and hence, (.) becomes
ψd(xn+,xn+)
≤ψd(xn,xn+)
–ϕd(xn,xn+)
<ψd(xn,xn+)
. (.)
Similarly, we can show that
ψd(xn+,xn)
<ψd(xn,xn–)
. (.)
By (.) and (.), we get that{d(xn,xn+) :n∈N}is a non-increasing sequence of positive
numbers. Hence, there isr≥ such that
lim
n→∞d(xn,xn+) =r.
Lettingn→ ∞in (.), we get
ψ(r)≤ψ(r) –ϕ(r),
which implies thatϕ(r) = , and hencer= . So, we have
lim
n→∞d(xn,xn+) = . (.)
Step . We will prove that{xn}is ab-Cauchy sequence. Because of (.), it is sufficient
to show that{xn}is ab-Cauchy sequence. Suppose on the contrary,i.e., that{xn}is not
ab-Cauchy sequence. Then there existsε> , for which we can find two subsequences
{xmi}and{xni}of{xn}such thatniis the smallest index, for which
This means that
d(xmi,xni–) <ε. (.)
From (.) and using the triangular inequality, we get
ε≤d(xmi,xni)≤sd(xmi,xmi+) +sd(xmi+,xni).
Using (.) and taking the upper limit asi→ ∞, we get
ε
s ≤lim supi→∞ d(xmi+,xni). (.)
On the other hand, we have
d(xmi,xni–)≤sd(xmi,xni–) +sd(xni–,xni–).
Using (.), (.) and taking the upper limit asi→ ∞, we get
lim sup i→∞
d(xmi,xni–)≤εs. (.)
Again, using the triangular inequality, we have
d(xmi,xni)≤sd(xmi,xni–) +sd(xni–,xni)
≤sd(xmi,xni–) +s d(x
ni–,xni–) +sd(xni–,xni)
and
d(xmi+,xni–)≤sd(xmi+,xmi) +sd(xmi,xni–).
Taking the upper limit asi→ ∞in the above inequalities, and using (.), (.) and (.),
we get
lim sup
i→∞ d(xmi,xni)≤εs (.)
and
lim sup
i→∞ d(xmi+,xni–)≤εs
. (.)
Sincexmiandxni–are comparable andxmi∈Aandxni–∈B, using (.) we have
ψsd(xmi+,xni)
=ψsd(fxmi,gxni–)
≤ψMs(xmi,xni–)
–ϕMs(xmi,xni–)
+LψN(xmi,xni–)
where
Ms(xmi,xni–) =max
d(xmi,xni–),d(xmi,xmi+),d(xni–,xni),
d(xmi,xni) +d(xmi+,xni–)
s
(.)
and
N(xmi,xni–) =min
d(xni–,xni),d(xmi,xni),d(xni–,xmi+)
. (.)
Taking the upper limit in (.) and using (.) and (.)-(.), we get
lim sup i→∞
Ms(xmi,xni–) =max
lim sup
i→∞
d(xmi,xni–), , ,
lim supi→∞d(xmi,xni) +lim supi→∞d(xmi+,xni–)
s
≤max
εs,εs+εs
s
=εs.
Hence, we have
lim sup
i→∞ Ms(xmi,xni–)≤εs, (.)
and, from (.),
lim sup i→∞
N(xmi,xni–) = . (.)
Now, taking the upper limit asi→ ∞in (.) and using (.), (.) and (.), we
have
ψ(εs) =ψ
sε
s
≤ψ
slim sup
i→∞ d(xmi+,xni)
≤ψ
lim sup
i→∞
Ms(xmi,xni–)
–ϕ
lim inf
i→∞ Ms(xmi,xni–)
≤ψ(εs) –ϕ
lim inf
i→∞ Ms(xmi,xni–)
,
which implies that ϕ(lim infi→∞Ms(xmi,xni–)) = . By (.), it follows that
lim infi→∞d(xmi,xni) = , which is in contradiction with (.). Hence{xn}is ab-Cauchy
sequence inX.
Step (existence of a common fixed point).
As{xn}is ab-Cauchy sequence inXwhich is ab-completeb-metric space, there exists
u∈Xsuch thatxn→uasn→ ∞, and
lim
Now, without loss of generality, we may assume thatf is continuous. Using the triangular inequality, we get
d(u,fu)≤sd(u,fxn) +sd(fxn,fu).
Lettingn→ ∞, we get
d(u,fu)≤s lim
n→∞d(u,fxn) +snlim→∞d(fxn,fu) = .
Hence, we havefu=u. Thus,uis a fixed point off and, sinceAandBare closed subsets
ofX,u∈A∩B. By the first part of proof, we conclude thatuis also a fixed point ofg.
The assumption of continuity of one of the mappingsforgin Theorem can be replaced
by another condition, which is often used in similar situations. Namely, we shall use the
notion of a regular orderedb-metric space, which is defined analogously to the case of the
standard metric (see the paragraph following Theorem ).
Theorem Let the hypotheses of Theorembe satisfied,except that condition(b)is re-placed by the assumption
(b) the space(X,,d)is regular.
Then f and g have a common fixed point in X.
Proof Repeating the proof of Theorem , we construct an increasing sequence{xn} in
X such thatxn→ufor someu∈X. AsAandBare closed subsets ofX, we haveu∈
A∩B. Using the assumption (b) onX, we havexnufor alln∈N. Now, we show that
fu=gu=u. By (.), we have
ψsd(xn+,gu)
=ψsd(fxn,gu)
≤ψMs(xn,u)
–ϕMs(xn,u)
+LψN(xn,u)
, (.)
where
Ms(xn,u) =max
d(xn,u),d(xn,fxn),d(u,gu),
d(xn,gu) +d(fxn,u) s
=max
d(xn,u),d(xn,xn+),d(u,gu),
d(xn,gu) +d(xn+,u)
s
(.)
and
N(xn,u) =min
d(u,gu),d(u,fxn),d(xn,gu)
=mind(u,gu),d(u,xn+),d(xn,gu)
. (.)
Lettingn→ ∞in (.) and (.) and using Lemma , we get
lim sup i→∞
Ms(xn,u)≤max
d(u,gu),sd(u,gu) s
andN(xn,u)→. Now, taking the upper limit asn→ ∞in (.) and using Lemma and (.), we get
ψsd(u,gu)=ψ
s
sd(u,gu)
≤ψ
slim sup
n→∞ d(xn+,gu)
≤ψ
lim sup
n→∞ Ms(xn,u)
–ϕ
lim inf
n→∞ Ms(xn,u)
≤ψsd(u,gu)–ϕ
lim inf
n→∞ Ms(xn,u)
.
It follows thatϕ(lim infn→∞Ms(xn,u)) = , and hence, by (.), thatd(u,gu) = . Thus,u
is a fixed point ofg. On the other hand, similar to the first part of the proof of Theorem ,
we can show thatfu=u. Hence,uis a common fixed point off andg.
3 Consequences and examples
As consequences, we have the following results.
By puttingA=B=Xin Theorems and , we obtain improvements of the main results
(Theorems and ) of Roshanet al.[],i.e., of Theorem of the present paper (note that
we havesinstead ofsin the contractive condition).
Takingϕ= ( –δ)ψ, <δ< in Theorems and , we get the following.
Corollary Let(X,,d)be a complete ordered b-metric space,and let A and B be closed subsets of X.Let f,g:X→X be two(A,B)-weakly increasing mappings with respect to. Suppose that
(a) X=A∪Bis a cyclic representation ofXw.r.t.the pair(f,g);
(b) there exist <δ< ,L≥and an altering distance functionψ such that for any com-parable elementsx,y∈Xwithx∈Aandy∈B,we have
ψsd(fx,gy)≤δψMs(x,y)+LψN(x,y), (.)
whereMs(x,y)andN(x,y)are given by(.)and(.),respectively; (c) f orgis continuous,or
(c) the space(X,,d)is regular.
Then f and g have a common fixed point u∈A∩B.
Takings= andL= in Corollary , we obtain Theorems . and . of Shatanawi and
Postolache [] (Theorem in this paper).
Takingψ(t) =tfort∈[, +∞) in Corollary , we get the following.
Corollary Let(X,,d)be a complete ordered b-metric space.Let A and B be nonempty closed subsets of X,and let f,g:X→X be two(A,B)-weakly increasing mappings with respect tosuch that f(A)⊆B and g(B)⊆A.Suppose that there existδ∈(, )and L≥ such that
d(fx,gy)≤ δ smax
d(x,y),d(x,fx),d(y,gy),d(x,gy) +d(fx,y) s
+ L
smin
for all comparable elements x,y∈X with x∈A and y∈B.If either f or g is continuous,or the space(X,,d)is regular,then f and g have a common fixed point.
Puttingf =gin Theorems and , the following corollary is obtained which extends
and improves Theorems and in [].
Corollary Let(X,,d)be a complete ordered b-metric space,and let A and B be closed subsets of X.Let f :X→X be a mapping such that f is non-decreasing with respect to. Assume the following:
(a) A∪Bis a cyclic representation ofXw.r.t.f,that is,fA⊆B,fB⊆A; (b) there exist two altering distance functionsψ,ϕ,andL≥such that
ψsd(fx,fy)≤ψMs(x,y)–ϕMs(x,y)+LψN(x,y) (.)
for all comparablex,y∈Xwithx∈Aandy∈B,where
Ms(x,y) =max
d(x,y),d(x,fx),d(y,fy),d(x,fy) +d(y,fx) s
and
N(x,y) =mind(x,fx),d(x,fy),d(y,fx).
(c) f is continuous,or
(c) the space(X,,d)is regular.
If there exists x∈X such that xfx,then f has a fixed point.
Again, takingϕ= ( –δ)ψ, <δ< in Corollary , we get the following.
Corollary Let(X,,d)be a complete ordered b-metric space,let and A and B be closed subsets of X.Let f :X→X be a non-decreasing map with respect to.Suppose that
(a) X=A∪Bis a cyclic representation ofXw.r.t.f;
(b) there exist <δ< ,L≥and an altering distance functionψ such that for any com-parable elementsx,y∈Xwithx∈Aandy∈B,we have
ψsd(fx,fy)≤δψMs(x,y)+LψN(x,y), (.)
whereMs(x,y)andN(x,y)are given in Corollary; (c) f is continuous,or
(c) the space(X,,d)is regular.
Then f has a fixed point u∈A∩B.
Remark (Common) fixed points of the given mappings in Theorems and and Corol-laries and need not be unique (see further Example ). However, it is easy to show that they must be unique in the case that the respective sets of (common) fixed points are well
ordered (recall that a subsetWof a partially ordered set is said to be well ordered if every
We illustrate our results with the following two examples.
Example Consider theb-metric space (X,d) given in Example , ordered by natural
ordering and a mappingf :X→Xgiven as
fn= ⎧ ⎨ ⎩
n, ifn∈N,
∞, ifn=∞.
IfA={n:n∈N} ∪ {∞}andB={n:n∈N} ∪ {∞}, thenA∪Bis a cyclic representation of
Xwith respect tof. Takeψ: [, +∞)→[, +∞) given asψ(t) =√t,δ= /√ (< ) and
L≥ arbitrary. In order to check the contractive condition (.), consider the following
cases.
Ifx,y∈N, then
ψsd(fx,fy)=ψ
d(x, y) = · x–
y
≤
√ψ
d(x,y)≤δψMs(x,y)
+LψN(x,y)
and (.) holds. Ifx=∞andyis an even integer, then
ψsd(fx,fy)=ψ
d(∞, y) = ·· y ≤
√ψ
d(x,y)≤δψMs(x,y)
+LψN(x,y).
Finally, ifx=∞andyis an odd integer, thend(x,y) = and (.) trivially holds.
Hence, all the conditions of Corollary are satisfied. Obviously,f has a (unique) fixed
point∞, belonging toA∩B.
We now present an example showing that there are situations where our results can be used to conclude about the existence of (common) fixed points, while some other known results cannot be applied.
Example LetX={, , , , }be equipped with the following partial order:
:=(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ).
Define ab-metricd:X×X→R+by
d(x,y) = ⎧ ⎨ ⎩
, ifx=y,
(x+y), ifx=y.
It is easy to see that (X,d) is a b-complete b-metric space with s= /. Set A=
{, , , , }andB={, }, and define self-mapsf andgby
f =
, g=
It is easy to see thatf andgare (A,B)-weakly increasing mappings with respect to, and
thatf andgare continuous. Also,A∪B=X,f(A)⊆Bandg(B)⊆A.
Defineψ : [,∞)→[,∞) byψ(t) =√t. One can easily check that the pair (f,g)
sat-isfies the requirements of Corollary , with anyδandL≥, as the left-hand side of the
contractive condition (.) is equal to for all comparablex,ysuch thatx∈Aandy∈B.
Hence,f andghave a common fixed point. Indeed, and are two common fixed points
off andg. (Note that the ordered set ({, },) is not well ordered).
However, takex= ∈Aandy= ∈B(which are not comparable). Then
ψsd(f,g)=s( + )= s>
> δ+L· =δψMs(, )+LψN(, ),
where <δ< andL≥ are arbitrary, since
Ms(, ) =max
d(, ),d(, ),d(, ),d(, ) +d(, ) s
=
and
N(, ) =mind(, ),d(, ),d(, )= .
Hence, this result cannot be applied in the context ofb-metric spaces without order.
4 Application to existence of solutions of integral equations
Integral equations like (.) have been studied in many papers (see, e.g., [, ] and
the references therein). In this section, we look for a nonnegative solution to (.) in X=C([,T],R).
Consider the integral equation
u(t) =
T
G(t,s)fs,u(s)ds for allt∈[,T], (.)
whereT> ,f : [,T]×R→RandG: [,T]×[,T]→[,∞) are continuous functions.
LetX=C([,T]) be the set of real continuous functions on [,T]. We endowXwith the
b-metric
D(u,v) = max t∈[,T]
u(t) –v(t) for allu,v∈X.
Clearly, (X,D) is a completeb-metric space (with the parameters= ). We endowXwith
the partial ordergiven by
xy ⇐⇒ x(t)≤y(t) for allt∈[,T].
Clearly, the space (X,,D) is regular.
Letα,β∈Xandα,β∈Rsuch that
Assume that for allt∈[,T], we have
α(t)≤
T
G(t,s)fs,β(s)ds (.)
and
β(t)≥
T
G(t,s)fs,α(s)ds. (.)
Let for alls∈[,T],f(s,·) be a decreasing function, that is,
x,y∈R, x≥y ⇒ f(s,x)≤f(s,y). (.)
Assume thatγ > is such that
γ
max t∈[,T]
T
G(t,s)ds
< . (.)
Define a mappingT :X→Xby
Tu(t) =
T
G(t,s)fs,u(s)ds for allt∈[,T].
Suppose that for alls∈[,T] and for all comparablex,y∈Xwith (x(s)≤βandy(s)≥α)
or (x(s)≥αandy(s)≤β),
≤fs,x(s)–fs,y(s)
≤
γmaxx(s) –y(s),x(s) –Tx(s),y(s) –Ty(s),
|x(s) –Ty(s)|+|y(s) –Tx(s)|
. (.)
Theorem Under the assumptions(.)-(.),the integral equation(.)has a solution in the set{u∈C([,T]) :αu≤β}.
Proof Define closed subsets ofX,AandAby
A={u∈X:uβ} and A={u∈X:uα}.
Consider the mappingT :X→Xdefined above. We will prove that
T(A)⊆A and T(A)⊆A. (.)
Suppose thatu∈A, that is,
Applying, condition (.), sinceG(t,s)≥ for allt,s∈[,T], we obtain that
G(t,s)fs,u(s)≥G(t,s)fs,β(s) for allt,s∈[,T].
The above inequality with condition (.) implies that
T
G(t,s)fs,u(s)ds≥
T
G(t,s)fs,β(s)ds≥α(t)
for allt∈[,T]. Thus, we haveTu∈A.
Similarly, letu∈A, that is,
u(s)≥α(s) for alls∈[,T].
Using condition (.), sinceG(t,s)≥ for allt,s∈[,T], we obtain that
G(t,s)fs,u(s)≤G(t,s)fs,α(s) for allt,s∈[,T].
The above inequality with condition (.) implies that
T
G(t,s)fs,u(s)ds≤
T
G(t,s)fs,α(s)ds≤β(t)
for allt∈[,T]. Hence, we haveTu∈A. Thus, (.) holds.
Now, let (u,v)∈A×A, that is, for allt∈[,T],
u(t)≤β(t), v(t)≥α(t).
This implies from condition (.) that for allt∈[,T],
u(t)≤β, v(t)≥α.
Also, ifxy, then by (.), we have
Ty(t) –Tx(t) =
T
G(t,s)fs,y(s)–fs,x(s)ds≥
for allt∈[,T]. That is,TxTy. Hence,T is increasing.
Now, by the conditions (.) and (.), we have for allt∈[,T] and for all
comparablex∈Aandy∈A,
Tx(t) –Ty(t)
=
T
G(t,s)fs,x(s)–fs,y(s)ds
≤
T
G(t,s)fs,x(s)–fs,y(s)ds
≤
T
G(t,s)
|x(s) –Ty(s)|+|y(s) –Tx(s)|
ds
≤ T
G(t,s)
γmax
max s∈[,T]
x(s) –y(s), max s∈[,T]
x(s) –Tx(s),
max s∈[,T]
y(s) –Ty(s)
,
maxs∈[,T]x(s) –Ty(s)+maxs∈[,T]y(s) –Tx(s)
ds
=γ
T
G(t,s)ds
max
D(x,y),D(x,Tx),D(y,Ty),D(x,Ty) +D(y,Tx) s
,
which implies that
D(Tx,Ty)≤ δ
smax
D(x,y),D(x,Tx),D(y,Ty),D(x,Ty) +D(y,Tx) s
withδ= γ(maxt∈[,T]
T
G(t,s)ds)< .
Now, all the conditions of Corollary (withT =g=f andL= ) hold, andT has a fixed
pointzin
A∩A=
u∈C[,T]:α(t)≤u(t)≤β(t), for allt∈[,T].
That is,z∈A∩Ais the solution to (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Young Researchers
and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran. 3Department of Mathematics, Qaemshahr
Branch, Islamic Azad University, Qaemshahr, Iran.4Faculty of Mathematics, University of Belgrade, Beograd, Serbia.
Acknowledgements
The authors are highly indebted to the referees of this paper who helped us to improve it in several places. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support. The fourth author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
Received: 18 June 2013 Accepted: 23 September 2013 Published:07 Nov 2013 References
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