On fixed point theorems for mappings with PPF dependence

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R E S E A R C H

Open Access

On ﬁxed point theorems for mappings with

PPF dependence

Ali Farajzadeh

1

_{and Anchalee Kaewcharoen}

2,3*

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,}

Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

3_{Centre of Excellence in}

Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand Full list of author information is available at the end of the article

Abstract

In this paper, we consider two families

1and

2of mappings deﬁned on [0, +∞)

satisfy some certain properties. Using the mentioned properties for

1and

2, we

prove the analogous PPF dependent ﬁxed point theorems for mappings as in Driciet al.(Nonlinear Anal. 67:641-647, 2007) in partially ordered Banach spaces where mappings satisfy the weaker contractive conditions without assuming the topological closedness with respect to the norm topology for the Razumikhin classRc.

MSC: 54H25; 55M20

Keywords: PPF dependent ﬁxed points; Razumikhin classes; partially ordered sets; algebraic closedness with respect to diﬀerence

1 Introduction and preliminary results

The ﬁxed point theorems for mappings satisfying certain contractive conditions have been continually studied for decade (see [–] and references contained therein). Bernfeldet al.

[] proved the existence of PPF (past, present and future) dependent fixed points in the Razumikhin class for mappings that have different domains and ranges. After that Dhage [] extended the existence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions. In , Driciet al.[] proved the fixed point theorems in partially ordered metric spaces for mappings with PPF dependence. In this paper, we consider two familiesandof mappings defined on

[, +∞) satisfy some certain properties. Moreover, the PPF dependent ﬁxed point theo-rems for mappings satisfying some generalized contractive conditions in partially ordered Banach spaces are proven using the mentioned properties forand.

Suppose thatEis a real Banach space with the norm · EandIis a closed interval [a,b]

inR. LetE=C(I,E) be the set of all continuousE-valued mappings onIequipped with

the supremum norm · Edeﬁned by

φE=sup

t∈I

φ(t)_E, (.)

for allφ∈E. For a ﬁxed elementc∈I, the Razumikhin class of mappings inEis deﬁned

by

Rc=

φ∈E:φE=φ(c)_E

. (.)

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Recall that a pointφ∈Eis said to be a PPF dependent ﬁxed point or a ﬁxed point with

PPF dependence ofT:E→EifTφ=φ(c) for somec∈I.

Example . LetT:C([, ],R)→Rbe deﬁned by

Tφ= 

sup

t∈[,]

φ(t) for allφ∈C[, ],R.

ThereforeTis a contraction with a constant 

. Suppose thatφ(t) =t

_{+  for all}_t_∈_{[, ].}

SinceTφ=_(sup_t∈_[,]|φ(t)|) =  =φ(), we ﬁnd thatφ is a ﬁxed point with dependence ofT.

Deﬁnition . LetAbe a subset ofE. Then

(i) Ais said to be topologically closed with respect to the norm topology if for each sequence{xn}inAwithxn→xasn→ ∞impliesx∈A.

(ii) Ais said to be algebraically closed with respect to the diﬀerence ifx–y∈Afor all x,y∈A.

Recently, Dhage [] proved the existence of PPF ﬁxed points for mappings satisfying the condition of Cirić type generalized contraction assuming topological closedness with respect to the norm topology for a Razumikhin class.

Deﬁnition .(Dhage, []) A mappingT:E→Eis said to satisfy the condition of Cirić

type generalized contraction if there exists a real numberλ∈[, ) satisfying

Tφ–Tα ≤λmax

φ–αE,φ(c) –Tφ_E,α(c) –Tα_E,



φ(c) –TαE+α(c) –TφE

, (.)

for allφ,α∈Eand for somec∈I.

Theorem .(Dhage, []) Suppose that T:E→E satisﬁes the condition of Cirić type

generalized contraction.Assume thatRcis topologically closed with respect to the norm topology and is algebraically closed with respect to the diﬀerence,then T has a unique PPF dependent ﬁxed point inRc.

It is a natural question that the result of the previous theorem is still valid by omitting the topological closedness ofRc. In the next result, we will answer the question.

Proposition . The Razumikhin classRcis topologically closed with respect to the norm topology.

Proof Let{φn}be a sequence inRcconverging toφ. This implies that

lim

n→∞φn–φE=  whereφn–φE=sup

t∈I

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Therefore

lim

n→∞φnE=φE and _n→∞limφn(t)E=φ(t)E for allt∈I.

Sinceφn∈Rcfor alln∈N, we obtainφnE=φn(c)E. Therefore

lim

n→∞φn(c)E=φE.

By the uniqueness of the limit, we haveφE=φ(c)E. Henceφ∈Rcand thusRc is

topologically closed with respect to the norm topology.

Hence, using Proposition ., we can drop the topological closedness with respect to the norm topology forRcin Theorem ..

The following example shows that the algebraic closedness with respect to the diﬀerence of Razumikhin classRcmay fail.

Example . LetE=C([, ],R) andc= . If we takeφ(t) =tandα(t) =tfor allt∈[, ],

thenφ,α∈Rcwhileφ–α∈/Rc.

Proposition . If the Razumikhin classRcis algebraically closed with respect to the dif-ference,thenRcis a convex set.

Proof SinceRcis algebraically closed with respect to the diﬀerence, we haveRc–Rc⊆Rc.

Using the fact that –Rc=Rc, we obtainRc+Rc⊆Rc. SinceλRc⊆Rcfor allλ∈[, ],

we getλRc+ ( –λ)Rc⊆Rc. HenceRcis a convex set.

One can verify that Razumikhin classRcis a cone (i.e.,λφ∈Rc, for eachφ∈Rcand

λ≥). Then by applying the previous theorem Razumikhin classRcis a convex cone (also

closed).

In , Driciet al.[] proved the following the ﬁxed point theorems in partially ordered complete metric spaces for mappings with PPF dependence.

Theorem .([]) Let(E,d,≤)be a partially ordered complete metric space and T:E→

E where E=C(I,E)and I= [a,b].Assume that (i) Tis a nondecreasing mapping;

(ii) for allφ,α∈Ewithφ≤α,d(Tφ,Tα)≤kd(φ,α)where

d(φ,α) =maxs∈Id(φ(s),α(s))andk∈[, );

(iii) there exists a lower solutionφsuch thatφ(c)≤Tφ;

(iv) Tis a continuous mapping or if{φn}is a nondecreasing sequence inEconverging to

φ∈E,thenφn≤φfor alln∈N. Then T has a PPF dependent ﬁxed point in E.

It is a natural question if one can obtain the result of the aforementioned theorem for a generalized contraction (that is, condition (ii)). One of the aims of this paper is to answer the question by considering a general case of Cirić type generalized contractions.

In this paper, we consider two familiesandof mappings deﬁned on [, +∞)

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analogous PPF dependent ﬁxed point theorems for mappings as in [] in partially ordered real Banach spaces where mappings satisfy the weaker contractive conditions.

2 PPF dependent ﬁxed points in partially ordered Banach spaces

We begin this part with the consideration of the example of a partially ordered real Banach space. Recall that the setB(X,R) of all bounded linear operators from a normed spaceX

intoRis a real Banach space with a norm deﬁned by

f= sup

x∈X,x=

f(x) for allf ∈B(X,R).

We know thatB(X,R) is a partially ordered real Banach space with a partial order deﬁned as follows:

f ≤g if and only if f(x)≤g(x) for allx∈X.

From now on, let (E,≤) be a partially ordered real Banach space. In this paper, we use the following notations:

=

ψ:ψ: [, +∞)→[, +∞) is nondecreasing with

∞

n=

ψn(t) <∞for allt∈(, +∞)

and

=

ψ:ψ: [, +∞)→[, +∞) is continuous nondecreasing with

ψ(t) <tfor allt∈(, +∞).

Lemma .([]) Suppose thatψ: [, +∞)→[, +∞).Ifψis nondecreasing,then for each t∈(, +∞),lim_n→∞ψn(t) = impliesψ(t) <t.

Hence the diﬀerence between an element ofand an element ofis continuity.

Remark .

(i) It is easily seen that ifψ: [, +∞)→[, +∞)is nondecreasing andψ(t) <tfor all t∈(, +∞), thenψ() = .

(ii) We can see that ifψ: [, +∞)→[, +∞),ψ(t) <tfor allt∈(, +∞)andψ() = , thenψis continuous at.

We now prove the PPF dependent ﬁxed point theorems for mappings satisfying the gen-eralized contractive conditions concerning withψ∈without assuming the topological

closedness with respect to the norm topology for the Razumikhin classRc.

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(ii) for allφ,α∈Ewithφ≤α,we have

Tφ–TαE≤ψ

max

φ–αE,φ(c) –TφE,α(c) –TαE,



;

(iii) there exists a lower solutionφ∈Rcsuch thatφ(c)≤Tφ; (iv) Tis a continuous mapping.

Assume that Rc is algebraically closed with respect to the diﬀerence.Then T has a PPF dependent ﬁxed point inRc.

Proof Sinceφ∈RcandTφ∈E, there existsx∈Esuch thatTφ=x. Chooseφ∈Rc

such thatx=φ(c). Sinceφis a lower solution inRcsuch thatφ(c)≤Tφ, it follows that

φ≤φ. Using the algebraic closedness with respect to the diﬀerence ofRc, this yields

φ–φE=φ(c) –φ(c)E.

By the fact thatT is nondecreasing, we obtain

φ(c) =Tφ≤Tφ.

By induction, we can construct the sequence{φn}such that

Tφ=φ(c)≤Tφ=φ(c)≤ · · · ≤Tφn=φn+(c)≤Tφn+· · ·,

φn≤φn+ and φn–φn+E=φn(c) –φn+(c)E,

for alln∈N∪ {}. Assume thatφn–=φnfor somen∈N. It follows thatφn–(c) =φn(c) = Tφn–. ThereforeThas a ﬁxed point inRc. Suppose thatφn–=φnfor alln∈N. Therefore,

for eachn∈N, we obtain

φn–φn+E =φn(c) –φn+(c)E

=Tφn–Tφn–E

≤ψ

max

φn–φn–E,φn(c) –Tφn_E,φn–(c) –Tφn–_E,



φn(c) –Tφn–E+φn–(c) –TφnE

=ψ

max

φn–φn–E,φn(c) –φn+(c)E,φn–(c) –φn(c)E,



φn(c) –φn(c)E+φn–(c) –φn+(c)E

≤ψ

max

φn–φn–E,φn–φn+E,



φn––φn+E

≤ψ

max

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

φn––φnE+φn–φn+E

≤ψmaxφn–φn–E,φn–φn+E .

Ifmax{φn–φn–E,φn–φn+E}=φn–φn+E, then

φn–φn+E≤ψ

φn–φn+E <φn–φn+E,

which leads to a contradiction. Therefore, for eachn∈N, we have

φn–φn–E .

By induction, we obtain

n_φ

–φE .

Fixε> . This implies that there existsN∈Nsuch that

n≥N

ψnφ–φE <ε.

For eachm,n∈Nwithm>n>N, we obtain

φn–φmE≤

m–

k=n

φk–φk+E≤

m–

k=n

ψkφ–φE ≤

n≥N

ψnφ–φE <ε.

This implies that {φn} is a Cauchy sequence. By the completeness of E, we have

lim_n→∞φn=φfor someφ∈Eand

lim

n→∞Tφn=n→∞limφn+(c) =φ(c).

SinceRcis algebraically closed with respect to the norm topology, we haveφ∈Rc. We

next prove thatφis a PPF dependent ﬁxed point ofT. Using the continuity ofT, we obtain lim_n→∞Tφn=Tφ. By the uniqueness of the limit, we haveTφ=φ(c).

Remark .

(i) From the proof of Theorem ., we assume that the Razumikhin classRcis

algebraically closed with respect to diﬀerence, that is,φ–α∈Rcfor allφ,α∈Rc, in order to construct the sequence{φn}satisfying

φn–φn+E=φn(c) –φn+(c)_E for alln∈N∪ {}.

(ii) In the proof of Theorem ., if we chooseφn∈Rcto be a constant mapping for eachn∈N∪ {}, then

φn–φn+E=φn(c) –φn+(c)_E for alln∈N∪ {}.

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Example . LetE=R_{with respect to the norm}₍_x_,_y₎₌_|_x_|₊_|_y_|_and_E

=C([, ],R).

Deﬁne a mappingψ: [,∞)→[,∞) by

ψ(t) = t

 for allt∈[,∞).

We see thatψ is a nondecreasing mapping with∞_n_=ψn(t) <∞for allt∈(, +∞). Let φ∈E. Therefore, for eacht∈[, ], we obtainφ(t)∈R. Thus we can deﬁne mappings

fφ,gφ: [, ]→Rsuch that

φ(t) =fφ(t),gφ(t) for allt∈[, ].

Deﬁne a mappingT:E→Eby

Tφ=

 fφ,

 gφ

for allφ∈E,

wherefφ=supt∈[,]|fφ(t)|andgφ=supt∈[,]|gφ(t)|. Suppose thatφ,α∈Ewithφ≤α.

For eachc∈[, ], we obtain

Tφ–TαE = _fφ,

 gφ

–

 fα,

 gα

E

= fφ–

 fα

+

gφ–  gα

≤  

fφ–fα+gφ–gα

=  

sup

t∈[,]

fφ(t) –fα(t)+ sup

t∈[,]

gφ(t) –gα(t)

=  

sup

t∈[,]

fφ(t) –fα(t)+gφ(t) –gα(t)

=  

sup

t∈[,]

fφ(t) –fα(t),gφ(t) –gα(t) _E

=   sup t∈[,]

fφ(t),gφ(t) –

fα(t),gα(t) _E

=   sup t∈[,]

φ(t) –α(t)_E

=  

φ–αE

≤ψ

max



.

Suppose thatφ= . We see that all assumptions in Theorem . are now satisﬁed and 

is the PPF dependent ﬁxed point ofTinRc.

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Corollary . Suppose that c∈I and T:E→E satisﬁes the following conditions: (i) Tis a nondecreasing mapping;

Tφ–TαE≤k

max



, wherek∈[, );

Proof Deﬁne a functionψ: [, +∞)→[, +∞) byψ(t) =ktfor allt∈[, +∞). Therefore ψ is a nondecreasing mapping and

∞

n=

ψn(t) <∞ for allt∈(, +∞).

This implies that all assumptions in Theorem . are satisﬁed. Hence we obtain the desired

result.

Theorem . Suppose thatψ∈,c∈I and T:E→E satisﬁes the following conditions: (i) Tis a nondecreasing mapping;

(ii) for allφ,α∈Ewithφ≤α,Tφ–TαE≤ψ(φ–αE);

(iii) there exists a lower solutionφ∈Rcsuch thatφ(c)≤Tφ;

(iv) if{φn}is a nondecreasing sequence inEconverging toφ∈E,thenφn≤φfor all n∈N.

Proof By the analogous proof as in Theorem ., we can construct a nondecreasing se-quence{φn}inRcconverging toφ∈Rc. This implies thatφn≤φfor alln∈N. Therefore,

for eachn∈N, we have

Tφ–φ(c)_E≤Tφ–φn+(c)_E+φn+(c) –φ(c)_E

≤ Tφ–TφnE+φn+–φE

≤ψφ–φnE +φn+–φE.

Sinceψ is continuous at , we getlim_n→∞ψ(φ–φnE) =ψ() = . Taking the limit of

the above inequality, this yieldsTφ–φ(c)E=  and soφis a PPF dependent ﬁxed point

ofTinRc.

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Theorem . Suppose thatψ∈,c∈I and T:E→E satisﬁes the following

condi-tions:

(i) Tis a nondecreasing mapping; (ii) for allφ,α∈Ewithφ≤α,we have

Tφ–TαE≤ψ

max



;

Proof Sinceφ∈RcandTφ∈E, there existsx∈Esuch thatTφ=x. As in the proof

of Theorem ., we can construct the sequence{φn}such that

Tφ=φ(c)≤Tφ=φ(c)≤ · · · ≤Tφn=φn+(c)≤Tφn+· · ·,

φn≤φn+ and φn–φn+E=φn(c) –φn+(c)_E,

for alln∈N∪ {}. Assume thatφn–=φnfor somen∈N. It follows thatφn–(c) =φn(c) = Tφn–. ThereforeThas a ﬁxed point inRc. Suppose thatφn–=φnfor alln∈N. For each n∈N, we obtain

φn–φn+E =φn(c) –φn+(c)E

=Tφn–Tφn–E

≤ψ

max

φn–φn–E,φn(c) –Tφn_E,φn–(c) –Tφn–_E,



φn(c) –Tφn–E+φn–(c) –TφnE

=ψ

max

φn–φn–E,φn(c) –φn+(c)E,φn–(c) –φn(c)E,



φn(c) –φn(c)E+φn–(c) –φn+(c)E

≤ψ

max



φn––φn+E

≤ψ

max



φn––φnE+φn–φn+E

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Ifmax{φn–φn–E,φn–φn+E}=φn–φn+E, then

φn–φn+E <φn–φn+E.

This leads to a contradiction. Therefore

φn–φn–E

<φn–φn–E.

It follows that φn–φn+E ≤ φn––φnE for alln∈N. Since the sequence{φn–

φn+E}is a nonincreasing sequence of nonnegative real numbers, we see that it is a

con-vergent sequence. Suppose that

lim

n→∞φn–φn+E=α,

for some nonnegative real numberα. We will prove thatα= . Suppose thatα> . Since

φn–φn–E ,

for alln∈Nand the continuity ofψ, we haveα≤ψ(α) <αwhich leads to a contradic-tion. This implies thatα= . We next prove that the sequence{φn}is a Cauchy sequence

inE. Assume that{φn}is not a Cauchy sequence. It follows that there existε>  and two

sequences of positive integers{mk}and{nk}satisfyingmk>nk>kfor eachk∈Nand

φmk–φnkE≥ε. (.)

Let{mk}be the sequence of the least positive integers exceeding{nk}which satisﬁes (.)

and

φmk––φnkE<ε. (.)

We will prove thatlim_k→∞φmk –φnkE=ε. Sinceφmk –φnkE≥εfor allk∈N, we

have

lim

k→∞φmk–φnkE≥ε.

For eachk∈N, we obtain

φmk–φnkE≤ φmk–φmk–E+φmk––φnkE

≤ φmk–φmk–E+ε.

This implies thatlim_k→∞φmk–φnkE≤ε. Therefore

lim

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Similarly, we can prove that

lim

k→∞φmk+–φnkE=ε, _k→∞limφmk–φnk–E=ε,

and

lim

k→∞φmk+–φnk–E=ε.

SinceRcis algebraically closed with respect to the diﬀerence, for eachk∈N, we obtain

φnk–φmk+E =φnk(c) –φmk+(c)E

=Tφmk–Tφnk–E

≤ψ

max

φmk–φnk–E,φmk(c) –TφmkE, φnk–(c) –Tφnk–E,



φmk(c) –Tφnk–E+φnk–(c) –TφnkE

=ψ

max

φmk–φnk–E,φmk(c) –φmk+(c)E, φnk–(c) –φnk(c)E,



φmk(c) –φnk(c)E+φnk–(c) –φnk+(c)E

≤ψ

max

φmk–φnk–E,φmk–φmk+E,φnk––φnkE,

 

φmk–φnkE+φnk––φnk+E

≤ψ

max

φmk–φnk–E,φmk–φmk+E,φnk––φnkE,

 

φmk–φnkE+φnk––φnkE+φnk–φnk+E

.

By taking the limit of both sides, we have

ε≤ψ(ε) <ε.

This leads to a contradiction. It follows that the sequence{φn}is a Cauchy sequence. By

the completeness ofE, we havelimn→∞φn=φfor someφ∈Eand

lim

n→∞Tφn=n→∞limφn+(c) =φ(c).

SinceRcis algebraically closed with respect to the norm topology, we haveφ∈Rc. We

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Example . Assume thatE=RandE=C([, ],R). Deﬁne a mappingψ : [,∞)→

[,∞) by

ψ(t) = t

 for allt∈[,∞).

We see thatψ is a continuous nondecreasing mapping withψ(t) <t. Deﬁne a mapping

T:E→Eby

Tφ= φ

 

for allφ∈E.

Suppose thatφ,α∈Ewithψ≤αandc∈[, ]. Therefore

Tφ–TαE =

 

φ

 

–α

 

≤ 

φ–αE ≤ψ

max



.

Suppose thatφ= . We ﬁnd that all assumptions in Theorem . are now satisﬁed and 

is the PPF dependent ﬁxed point ofTinRc.

For the next result, we drop the continuity ofT.

Theorem . Suppose thatψ ∈,c∈I,and that T :E→E satisﬁes the following

conditions:

(i) Tis a nondecreasing mapping; (ii) for allφ,α∈Ewithφ≤α,we have

Tφ–TαE≤ψ

max



;

(iv) if{φn}is a nondecreasing sequence inEconverging toφ∈E,thenφn≤φfor all n∈N.

Proof As in the proof of Theorem ., we can construct a nondecreasing sequence{φn}

converging toφ∈Rcand this yields

lim

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Using (iv), we haveφn≤φfor alln∈N. Therefore, for eachn∈N, we obtain

Tφ–φ(c)_E≤Tφ–φn+(c)_E+φn+(c) –φ(c)_E

≤ Tφ–TφnE+φn+–φE

≤ψ

max

φ–φnE,φ(c) –Tφ_E,φn(c) –Tφn_E,



φ(c) –TφnE+φn(c) –TφE

+φn+–φE

=ψ

max

φ–φnE,φ(c) –Tφ_E,φn(c) –φn+(c)_E,



φ(c) –φn+(c)E+φn(c) –TφE

+φn+–φE

≤ψ

max

φ–φnE,φ(c) –TφE,φn–φn+E

 

φ–φn+E+φn(c) –Tφ_E

+φn+–φE.

Lettingn→ ∞, we obtainTφ–φ(c)E≤ψ(φ(c) –TφE). Ifφ(c)=Tφ, then

Tφ–φ(c)_E≤ψφ(c) –Tφ_E <φ(c) –Tφ_E.

This leads to a contradiction. ThereforeTφ=φ(c). This implies thatφis a PPF dependent

ﬁxed point ofT.

By applying Theorem . and Theorem ., we obtain the following corollary.

Corollary . Suppose that c∈I and that T:E→E satisﬁes the following conditions: (i) Tis a nondecreasing mapping;

Tφ–TαE≤k

max



, wherek∈[, );

(iv) Tis a continuous mapping or if{φn}is a nondecreasing sequence inEconverging to

φ∈E,thenφn≤φfor alln∈N.

Proof Deﬁne a functionψ: [, +∞)→[, +∞) byψ(t) =ktfor allt∈[, +∞). Therefore ψ is a continuous nondecreasing mapping and

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This implies that all assumptions in Theorem . or Theorem . are satisﬁed. Hence the

proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, Razi University, Kermanshah, 67149, Iran.}2_{Department of Mathematics,}

Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand.3_{Centre of Excellence in Mathematics, CHE, Si} Ayutthaya Rd., Bangkok, 10400, Thailand.

Acknowledgements

The second author would like to express her deep thanks to the Centre of Excellence in Mathematics, the Commission of Higher Education and Naresuan University, Thailand for the support.

Received: 10 October 2013 Accepted: 10 September 2014 Published:26 Sep 2014

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