R E S E A R C H
Open Access
A new type fixed point theorem for a
contraction on partially ordered generalized
complete metric spaces with applications
Madjid Eshaghi Gordji
1, Maryam Ramezani
1, Farhad Sajadian
1, Yeol Je Cho
2*and Choonkil Park
3*Correspondence: [email protected] 2Department of Mathematics
Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract
In this paper, we prove a fixed point theorem for a contraction in generalized complete metric spaces endowed with partial order. As an application, we use the fixed point theorem to prove the Hyers-Ulam stability of the Cauchy functional equation in Banach spaces endowed with a partial order.
MSC: 54H25; 47H10; 39B52
Keywords: fixed point; generalized complete metric space; partial order; Hyers-Ulam stability
1 Introduction
In , Ulam gave a wide ranging talk in front of the mathematics club of University of Wisconsin in which he discussed a number of important unsolved problems (see []). One of the problems was the question concerning the stability of homomorphisms:
LetGbe a group andGbe a metric group with a metricd(·,·). Given> , does there
existδ> such that, if a mappingh:G→Gsatisfies the inequalityd(h(xy),h(x)h(y)) <δ
for allx,y∈G, then there exists a homomorphismH:G→Gwithd(h(x),H(x)) <for
allx∈G?
In , Hyers [] affirmatively answered the question of Ulam for the case whereG
andGare Banach spaces. Taking this fact into account, the additive Cauchy functional
equationf(x+y) =f(x) +f(y) is said to satisfy theHyers-Ulam stability.
On the other hand, Banach’s contraction principle is one of the pivotal results of analy-sis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics. Many kinds of gen-eralizations of the above principle have been a heavily investigated branch of research. In particular, Diaz and Margolis [] presented the following definition and fixed point theo-rem in a ‘generalized complete metric space’.
Definition . LetX be an abstract (nonempty) set and assume that, in the Cartesian productX×X, a distance functiond(x,y) (≤d(x,y)≤ ∞for allx,y∈X) is defined and satisfies the following conditions:
(D) d(x,y) = if and only ifx=y; (D) d(x,y) =d(y,x)(symmetry);
(D) d(x,y)≤d(x,z) +d(z,y)(triangle inequality);
(D) everyd-Cauchy sequence inXisd-convergent,i.e.,limm,n→∞d(xn,xm) = for a
sequence{xn}inXimplies the existence of an elementx∈Xwith
limn→∞d(x,xn) = (the pointxis unique by (D) and (D)).
Then we call (X,d) ageneralized complete metric space.
Theorem . Suppose that(X,d)is a generalized complete metric space and the function T:X→X is a contraction,that is,T satisfies the following condition:
(CI) There exists a constantqwith <q< such that,wheneverd(x,y) <∞,
d(Tx,Ty)≤qd(x,y).
Let x∈X and consider a sequence{Tlx}of successive approximations with initial
ele-ment x.Then the following alternative holds:either
(A) for alll≥,one hasd(Tl(x
),Tl+(x)) =∞
or
(B) the sequence{Tlx
}isd-convergent to a fixed point of T.
Recently, Nieto and Rodriguez-Lopez [] proved a fixed point theorem in partially or-dered sets as follows.
Theorem . Let(X,≤)be a partially ordered set.Suppose that there exists a metric d in
X such that(X,d)is a complete metric space.Let f:X−→X be a continuous and nonde-creasing mapping such that there exists k∈[, )with
df(x),f(y)≤kd(x,y)
for all x,y∈X.If there exists x∈X with x≤f(x),then f has a fixed point.
In , Cˇadariu and Radu [] applied the fixed point method to investigate the Jensen functional equation (see also [–]) and presented a short and simple proof (different from thedirect methodinitiated by Hyers in ) for the Hyers-Ulam stability of the Jensen functional equation [] for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable [] for the stability of some nonlinear equations []. Recently, Brzdek [], Brzdek and Cieplinski [, ] reported some inter-esting results in this direction (see also [–]).
In this paper, we prove a fixed point theorem for self-mappings on a partially ordered setXwhich has a generalized metricd. Moreover, we give a generalization of the Hyers-Ulam stability of the conditional Cauchy equation as an important result of our fixed point theorem.
2 Main results
We start our work by the following fixed point theorem in generalized complete metric spaces.
k∈[, )with
df(x),f(y)≤kd(x,y)
for all x,y∈X with x≥y.If there exists x≤f(x),then the following alternative holds:
either
(A) for alll≥,one has
dfl(x),fl+(x)
=∞
or
(B) the sequence of{fn(x
)}isd-convergent to a fixed point off.
Proof Consider the sequence{d(fl(x
),fl+(x))}of real numbers. Then we consider two
cases as follows:
(a) If, for alll≥,d(fl(x),fl+(x)) = +∞, then (A) holds;
(b) If, for some integerl,d(fl(x),fl+(x)) < +∞, thenN=N(x)denotes the smallest
nonnegative integer such thatd(fN(x
),fN+(x)) < +∞.
We see thatd(fN+(x
),fN+(x)) <d(fN(x),fN+(x)) < +∞and, by induction, we have
dfN+l(x),fN+l+(x)
< +∞
for alll≥.
Note thatf is nondecreasing. Then we have
x≤f(x)≤f(x)≤ · · · ≤fn(x)≤ · · ·
and we can writed(fn(x),fn+(x))≤kd(fn(x),fn–(x)) for alln≥.
Now, by induction, we show that
dfn+(x),fN(x)
≤kndf(x),x
()
for alln>N(x). Forn= , sincex≤f(x), we have
df(x),f(x)
≤kdf(x),x
.
Supposing that () holds for somenand using thatfn(x
)≤fn+(x), we obtain
dfn+(x),fn+(x)
≤kd(fn+(x),fn(x)≤kknd
f(x),x
=kn+df(x),x
.
Thus it follows that{fn(x
)}is a Cauchy sequence inX. Indeed, letm>n>N(x). Then
we have
dfm(x),fn(x)
≤dfm(x),fm–(x)
+· · ·+dfn+(x),fn(x)
≤km–+km–+· · ·+kndf(x),x
=k
n–km
–k d
f(x),x
On the other hand, sinceXis a complete generalized metric space, there existsy∈X such thatlimn→∞fn(x
) =y.
Finally, we prove thaty∈Xis a fixed point off, that is,f(y) =y. Let> be a positive real number. Using the continuity off aty, for/, there existsδ> such thatd(z,y) <δimplies thatd(f(z),f(y)) </. Now, by the convergence of{fn(x)}toyandη=min{/,δ}> ,
there existsn∈Nsuch thatn>N(x) and, for alln≥n,d(fn+(x),y) <η. Therefore,
for alln>n, we have
df(y),y≤df(y),ffn(x)
+dfn+(x),y
</ +η≤,
and hencef(y) =y. This completes the proof.
Theorem . In Theorem.,we can replace the following condition with the continuity of f:
If{xn}is a nondecreasing sequence and xn→x in X,then xn≤x for all n∈N.
Then f has a fixed point.
Proof In Theorem ., we just showed thatyis a fixed point off. Let> be given. Since fn(x
)→yand{fn(x)}is a nondecreasing sequence, we havefn(x)≤y. For any/ > ,
there existsn∈Nsuch thatn≥land, for alln≥n,d(fn(x),y) </. Therefore, we
have
df(y),y≤df(y),fn+(x
)
+dfn+(x
),y
≤kdy,fn(x
)
+dfn+(x
),y
<k/ +/
<.
This shows thatf(y) =y. This completes the proof.
Remark . In Theorem ., sincefn(x
)→yand{fn(x)}is nondecreasing, we have
x≤yand
d(x,y)≤d
x,f(x)
+df(x),f(y)
.
Therefore, we have
d(x,y)≤d
x,f(x)
+kd(x,y).
Thus it follows that
d(x,y)≤
–kd
x,f(x)
.
Theorem . If,for all x,y∈X,there exists z which is comparable to x and y and d(z,x) <
Proof If x∈X is another fixed point of f, then we prove that d(x,y) = , where y=
limn→∞fn(x
). Since there existsz∈Xwhich is comparable toxandy,fn(z) is
compa-rable tofn(x) =xandfn(y) =yfor alln∈N∪ {}and
d(x,y)≤dfn(x),fn(z)+dfn(z) +fn(y)≤knd(x,z) +knd(z,y)→
whenevern→ ∞and so we haved(x,z) = . This completes the proof.
3 Application
In this section, we suppose that (E, · ,≤) is a partially ordered normed space with the
following conditions:
(a) for allx,y∈E,x≤y ⇒rx≤ryfor allr∈R+;
(b) for allx,y∈E, there existsz∈Esuch thatzis comparable toxandy.
Also, we suppose that (E, · ,≤) is a partially ordered Banach space with the condition
(i) and satisfies the following:
(c) for allx,y∈E, there existsz∈Esuch thatzis an upper bound of{x,y};
(d) if{xn}is a nondecreasing sequence inEandxn→x, thenx≥xnfor alln∈N.
As a simple example, we can show thatRsatisfies the conditions (a), (b), (c) and (d). Also, in this section, we consider × ∞= .
Now, we prove the main result of this section as follows.
Theorem . Suppose that f :E→Eis a mapping satisfying
f(x)≤f(x) (.)
and
f(x+y+z–w) –f(x) –f(y) –f(z) +f(w)≤φ(x,z) +φ(y,w) (.)
for all x,y,z,w∈E,where x is comparable to z,y is comparable to w,whereφ:E×E→
[,∞)is a function satisfyingφ(, ) = and the following condition:
φ(x,y)≤Lφ
x ,
y
(.)
for all x,y∈E,where x is comparable to y and L∈(, )is a constant.Then there exists a
unique additive mapping T:E→Esuch that
T(x) –f(x)≤
–Lφ(x,x) (.)
for all x∈E.
Proof It is clear thatf() = . Puttingz:=xandy=w:= in (.), we get
for allx∈E. Hence we have
f(x) –f(x)
≤
φ(x,x)≤φ(x,x) (.)
for allx∈E. ConsiderX:={g:g:E→E}and introduce the generalized metricdonX
by
d(h,g) :=infC∈R+:h(x) –g(x)≤Cφ(x,x),∀x∈E
for allh,g∈X. It is easy to show that (X,d) is a complete generalized metric space. Now, we put the partial order≤onXas follows: for allh,g∈X,
h≤g ⇐⇒ h(x)≤g(x)
for allx∈E. Now, we define a mappingJ:X→Xby
J(h)(x) := h(x)
for allx∈E. For anyg,h∈Xwithg≤h, it follows that, for allx∈E,
d(g,h) <C ⇒ g(x) –h(x)≤Cφ(x,x)
⇒ g(x)
–
h(x)
≤Cφ(x, x)
⇒ J(g)(x) –J(h)(x)≤LCφ(x,x).
It follows that
dJ(g),J(h)≤Ld(g,h).
It is easy to show thatJis a nondecreasing mapping.
Now, we show thatJis a continuous function. To this end, let{hn}be a sequence in (X,d) which converges toh∈Xand let> be given. Then there existN∈NandC∈R+with
C≤such that
hn(x) –h(x)≤Cφ(x,x)
for allx∈Eandn≥Nand so
hn(x) –h(x)≤Cφ(x, x)
for allx∈Eandn≥N. By inequality (.) and the definition ofJ, we get
for allx∈Eandn≥N. Hence
dJ(hn),J(h)
≤LC<
for alln≥N. It follows thatJis continuous. On the other hand, by (.), we havef ≤J(f) and, by applying inequality (.), we see thatd(J(f),f)≤. Applying Theorem ., it follows thatJhas a fixed pointT∈Xsuch thatlimn→∞d(Jn(f),T) = . It follows that
T(x) = lim
n→∞ f(nx)
n (.)
for allx∈E. For anyx∈E, it follows from (.) that the sequence{f(
nx)
n } is a
nonde-creasing sequence inEand so, by (.), we havef(
nx)
n ≤T(x) for alln≥. In particular,
f(x)≤T(x). This shows thatf≤T. Now, we can see that
dJ(f),J(T)≤Ld(f,T)
and hence
d(f,T)≤ –L.
This implies inequality (.).
On the other hand, by using inequality (.), we have
φnx, ny≤Lnφ(x,y) (.)
for allx,y∈Eandn∈N, wherexis comparable toy. Letx,y∈Ebe arbitrary elements.
Then there exists z∈E such that zis comparable to xandy. This implies that nzis
comparable to nxand nyfor alln∈N. It follows from (.) that
fn(x+y)–fnx–fny
=fnx+ ny+ nz– nz–fnx–fny–fnz+fnz
≤φnx, nz+φny, nz
for alln∈N. By using (.) and (.), it follows thatTis a Cauchy mapping.
To prove the uniqueness property ofT, suppose thatT is another additive function
satisfying (.). It is clear thatJ(T) =T. Then, for anyx∈E, there existsg(x)∈Esuch
thatg(x) is an upper bound of{T(x),T(x)}. This shows thatg:E→Eis a mapping which
is comparable toT andT. Hence we have
d(T,T)≤d
T,Jn(g)+dJn(g),T
=dJn(T),Jn(g)+dJn(g),Jn(T)
≤Lnd(T,g) +Lnd(g,T
)
Corollary . Let∈[,∞)and f :E→Ebe a function such that f() = and
f(x)≤f(x)
and
f(x+y+z–w) –f(x) –f(y) –f(z) +f(w)≤
for all x,y,z,w∈E,where x is comparable to z and y is comparable to w.Then there exists
a unique additive mapping T:E→Esuch that
T(x) –f(x)≤
for all x∈E.
Proof Setφ(x,y) = for allx,y∈Ewithx,y= ,φ(, ) = , and letL=in Theorem ..
Then we get the desired result.
Corollary . Let p∈(, )and∈[,∞).Suppose that f :E→Eis a mapping such
that
f(x)≤f(x)
and
f(x+y+z–w) –f(x) –f(y) –f(z) +f(w)≤xp+yp+zp+wp
for all x,y,z,w∈E,where x is comparable to z and y is comparable to w.Then there exists
a unique additive mapping T:E→Esuch that
T(x) –f(x)≤
–p
–p– x p
for all x∈E.
Proof Setφ(x,y) =(xp+yp) for allx,y∈E
, and letL= p–in Theorem .. Then we
get the desired result.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran.2Department of Mathematics
Acknowledgements
YJ Cho and C Park were supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170) and (NRF-2012R1A1A2004299), respectively.
Received: 20 May 2013 Accepted: 29 October 2013 Published:20 Jan 2014
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