PPF dependent common fixed point theorems for mappings in Banach spaces

R E S E A R C H

Open Access

PPF dependent common fixed point

theorems for mappings in Banach spaces

Anchalee Kaewcharoen

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

Centre of Excellence in

Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand

Abstract

We prove the existence of the PPF dependent common fixed point theorems and the PPF dependent coincidence points for a pair of mappings satisfying some contractive conditions in Banach spaces where the domains and ranges of the mappings are not the same. Our results extend and generalize the results in the literature.

MSC: 54H25; 55M20

Keywords: PPF dependent common fixed points; PPF dependent coincidence points; Razumikhin classes; generalized

ψ

1 Introduction

The fixed point theory in Banach spaces plays an important role and is useful in mathe-matics. It can be applied for solving various problems, for instance, variational inequalities, optimization and approximation theory. The common fixed point theorems for mappings satisfying certain contractive conditions have been continually studied for a decade (see [–] and references contained therein). Bernfeldet al.[] proved the existence of PPF (past, present and future) dependent fixed points in the Razumikhin class of functions for mappings that have different domains and ranges. Recently, Dhage [] extended the exis-tence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions. In this paper, the PPF dependent fixed point theorems and the PPF dependent coincidence points for a pair of mappings are proven in terms of more general contractive conditions in Banach spaces.

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

inR. LetE=C(I,E) be the set of all continuousE-valued functions 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 functions inEis defined by

Rc=

φ∈E:φE=φ(c)E

. (.)

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.

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.

Definition . LetS,T:E→Ebe two mappings. A pointφ∈Eis said to be a PPF dependent common fixed point or a common fixed point with PPF dependence ofSand

T ifSφ=φ(c) =Tφfor somec∈I.

Recently, Dhage [] proved the existence of PPF common fixed points for mappings satisfying the condition of Cirić type generalized contraction in a Razumikhin class.

Definition . Two mappingsS,T:E→Eare said to satisfy the condition of Cirić type generalized contraction if there exists a real numberλ∈[, ) satisfying

Sφ–Tα ≤λmax

φ–αE,φ(c) –Sφ_E,α(c) –Tα_E,



φ(c) –TαE+α(c) –SφE

(.)

for allφ,α∈Eand for somec∈I.

Theorem .(Dhage []) Suppose that S,T:E→E satisfy 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 S and T have a unique PPF dependent common fixed point inRc.

Definition . LetA:E→EandS:E→E. A pointφ∈Eis said to be a PPF depen-dent coincidence point or a coincidepen-dent point with PPF dependence ofAandSifAφ=Sφ(c) for somec∈I.

Dhage [] also assured the existence of PPF dependent coincidence points for mappings satisfying the condition of Cirić type generalized contraction (C) in a Razumikhin class.

Definition . A:E→EandS:E→Eare said to satisfy the condition of Cirić type generalized contraction (C) if there exists a real numberλ∈[, ) satisfying

Aφ–AαE≤λmax

Sφ–SαE,Sφ(c) –AφE,Sα(c) –AαE,



Sφ(c) –AαE+Sα(c) –AφE

(.)

for allφ,α∈Eand for somec∈I.

Theorem .(Dhage []) Let A:E→E and S:E→Ebe two mappings satisfying the

condition of Cirić type generalized contraction(C).Suppose that

(b) S(E)is complete; (c) Sis continuous.

IfRc is topologically closed with respect to the norm topology and is algebraically closed with respect to the difference,then A and S have a PPF dependent coincidence point inRc.

In this work, we prove the existence of PPF dependent common fixed point theorems for mappings satisfying the contractive condition which is weaker than the condition of Cirić type generalized contraction. Furthermore, we also prove the PPF dependent coincidence points for mappings satisfying the weaker contractive condition mentioned in [] without being topologically closed with respect to the norm topology of a Razumikhin class. Our results extend Theorem . and partially generalize Theorem ..

2 PPF dependent common fixed points

Let be the set of all functionsψ whereψ : [, +∞)→[, +∞) is a continuous non-decreasing function withψ(t) <tfor allt∈(, +∞) andψ() = . Ifψ∈, thenψ is called a-map. We now introduce the definition of the condition of Cirić type general-izedψ-contraction and prove our first result.

Definition . LetS,T:E→E. We say thatSandTsatisfy the condition of Cirić type generalizedψ-contraction if

Sφ–TαE≤ψ max

φ–αE,φ(c) –SφE,α(c) –TαE,



φ(c) –TαE+α(c) –SφE

(.)

for allφ,α∈Eand for somec∈I.

Theorem . Suppose that S,T:E→E satisfy the condition of Cirić type generalized

ψ-contraction.Assume thatRcis topologically closed with respect to norm topology and is algebraically closed with respect to the difference,then S and T have a unique PPF depen-dent common fixed point inRc.

Proof Letφ∈Rc. SinceSφ∈E, there existsx∈Esuch thatSφ=x. Chooseφ∈Rc

such that

x=φ(c) and φ–φE=φ(c) –φ(c)E.

Sinceφ∈Rcand by assumption, we haveTφ∈E. This implies that there existsx∈E such thatTφ=x. Therefore, we can chooseφ∈Rcsuch that

x=φ(c) and φ–φE=φ(c) –φ(c)_E.

By continuing the process as before, we can construct the sequence{φn}such that

and

φn–φn+E=φn(c) –φn+(c)E

for alln∈N∪ {}. We will show that{φn}is a Cauchy sequence inE. Assume thatφN–=

φN for someN∈N. IfNis even, then we haveN= mfor somem∈N. Therefore

φm–φm+E

=φm(c) –φm+(c)_E

=Sφm–Tφm–E

≤ψ max

φm–φm–E,φm(c) –SφmE,φm–(c) –Tφm–E,



φm(c) –Tφm–E+φm–(c) –SφmE

≤ψ max

φm–φm–E,φm(c) –φm+(c)_E,φm–(c) –φm(c)_E,



φm(c) –φm(c)E+φm–(c) –φm+(c)E

≤ψ max

φm–φm–E,φm–φm+E,



φm––φm+E

≤ψ max

φm–φm–E,φm–φm+E,



φm––φmE+φm–φm+E

≤ψmaxφm–φm–E,φm–φm+E

≤ψmaxφm–φm+E

.

This implies that φm–φm+E=  and soφm=φm+. Similarly, we can prove that

φm+ =φm+. ThereforeφN =φN+. By mathematical induction, we can conclude that

φN–=φN+k for allk≥. IfN is odd, then by the same argument we also obtain that

φN–=φN+k for allk≥. Therefore{φn}is a constant sequence for alln≥N– . This

implies that{φn}is a Cauchy sequence inE. Suppose thatφn–=φnfor alln∈N. For each n∈N, we obtain that

φn–φn+E =φn(c) –φn+(c)_E

=Sφn–Tφn–E

≤ψ max

φn–φn–E,φn(c) –Sφn_E,φn–(c) –Tφn–_E,



φn(c) –Tφn–E+φn–(c) –SφnE

≤ψ max



φn(c) –φn(c)E+φn–(c) –φn+(c)E

≤ψ max

φn–φn–E,φn–φn+E,



φn––φn+E

≤ψ max

φn–φn–E,φn–φn+E,



φn––φnE+φn–φn+E

≤ψmaxφn–φn–E,φn–φn+E

.

Ifmax{φn–φn–E,φn–φn+E}=φn–φn+E, then

φn–φn+E≤ψ

φn–φn+E

<φn–φn+E.

This leads to a contradiction. Therefore

φn–φn+E≤ψ

φn–φn–E

<φn–φn–E.

Thusφn–φn+E≤ φn–φn–E. Similarly, we can prove that

φn+–φn+E≤ φn–φn+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 obtain that it is

a convergent sequence. Suppose that

lim

n→∞φn–φn+E=α

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

φn–φn+E≤ψ

φn–φn–E

for alln∈Nand the continuity ofψ, we have

α≤ψ(α) <α,

which leads to a contradiction. This implies thatα= . We next prove that the sequence

{φn}is a Cauchy sequence inE. It suffices to prove that the sequence{φn}is a Cauchy

sequence inE. Assume that{φn}is not a Cauchy sequence. It follows that there existε> 

and two sequences of even positive integers{mk}and{nk}satisfying mk> nk>kfor

eachk∈Nand

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

(.) and

φmk––φnkE<ε. (.)

We will prove thatlim_k→∞φmk–φnkE=ε. Sinceφmk–φnkE≥εfor allk∈N, we

have

lim

k→∞φmk–φnkE≥ε.

For eachk∈N, we obtain that

φmk–φnkE≤ φmk–φmk–E+φmk––φmk–E+φmk––φnkE

≤ φmk–φmk–E+φmk––φmk–E+ε.

This implies thatlim_k→∞φmk–φnkE≤ε. Therefore

lim

k→∞φmk–φnkE=ε.

Similarly, we can prove that

lim

k→∞φmk+–φnkE=ε, _k→∞limφmk–φnk–E=ε

and

lim

k→∞φmk+–φnk–E=ε.

For eachk∈N, we obtain that

φnk–φmk+E≤φnk(c) –φmk+(c)E ≤ Sφmk–Tφnk–E

≤ψ max

φmk–φnk–E,φmk(c) –SφmkE,

φnk–(c) –Tφnk–_E, 

φmk(c) –Tφnk–E+φnk–(c) –SφnkE

≤ψ max

φmk–φnk–E,φmk(c) –φmk+(c)_E, φnk–(c) –φnk(c)E,



φmk(c) –φnk(c)E+φnk–(c) –φnk+(c)E

≤ψ max

φmk–φnk–E,φmk–φmk+E,

 

φmk–φnkE+φnk––φnk+E

≤ψ max

φmk–φnk–E,φmk–φmk+E,

φnk––φnkE,

 

φmk–φnkE+φnk––φnkE+φnk–φnk+E

.

By taking the limit of both sides, we have

ε≤ψ(ε) <ε,

which leads to a contradiction. It follows that the sequence{φn} is a Cauchy sequence

and so{φn}is a Cauchy sequence. By the completeness ofE, we have{φn}is a convergent

sequence. Suppose thatlimn→∞φn=φfor someφ∈E. SinceRcis algebraically closed

with respect to the norm topology, we haveφ∈Rc. Moreover, we also obtain that

lim

n→∞φn+=φ=n→∞limφn+.

We will prove thatφis a PPF dependent fixed point ofS. By using (.), we obtain that

Sφ–φ(c)_E≤Sφ–φn+(c)_E+φn+(c) –φ(c)_E

≤ Sφ–Tφn+E+φn+–φE

≤ψ max

φ–φn+E,φ(c) –SφE,φn+(c) –Tφn+E,



φ(c) –Tφn+E+φn+(c) –SφE

+φn+–φE

≤ψ max

φ–φn+E,φ(c) –SφE,φn+(c) –φn+(c)E,



φ(c) –φn+(c)E+φn+(c) –SφE

+φn+–φE

≤ψ max

φ–φn+E,φ(c) –Sφ_E,φn+–φn+E,

 

φ–φn+E+φn+(c) –SφE

+φn+–φE.

Lettingn→ ∞, we obtain thatSφ–φ(c)E= . ThereforeSφ=φ(c). Similarly, we can

prove thatTφ=φ(c). This implies thatφ is a PPF dependent common fixed point ofS

andT. We finally prove the uniqueness of the PPF dependent fixed point ofSandTinRc.

Letα∈Rcbe any PPF dependent common fixed point ofSandT. Therefore

φ–αE =φ(c) –α(c)E ≤ Sφ–TαE

≤ψ max



φ(c) –TαE+α(c) –SφE

≤ψ max

φ–αE,



φ(c) –α(c)E+α(c) –φ(c)E

≤ψmaxφ–αE,φ–αE

=ψφ–αE

.

It follows thatφ=α. HenceSandT have a unique PPF dependent common fixed point

inRc.

Remark . From the proof of Theorem ., we obtain the following observations: () 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

φ–αE=φ(c) –α(c)E.

() If the Razumikhin classRcis not assumed to be topologically closed, then the limit

of the sequence{φn}may be outside ofRc.

By applying Theorem ., we obtain the following corollary which is Theorem . in [].

Corollary . Suppose that S,T:E→E satisfy the condition of Cirić type generalized

contraction.Assume thatRc is topologically and algebraically closed with respect to the difference,then S and T have a unique PPF dependent fixed point inRc.

Proof Define a functionψ: [, +∞)→[, +∞) byψ(t) =λtfor allt∈[, +∞). Therefore

ψ is a continuous nondecreasing function and

ψ(t) <t for allt∈(, +∞) andψ() = .

This implies that all assumptions in Theorem . are satisfied. Hence the proof is

com-plete.

3 PPF dependent coincidence points

Definition . LetA:E→EandS:E→E. A pointφ∈E is said to be a PPF de-pendent coincidence point or a coincidence point with PPF dependence ofA andS if

Aφ=Sφ(c) for somec∈I.

Definition . LetA:E→EandS:E→E. We say thatAandSsatisfy the condition of Cirić type generalized contraction (C) if there exists a real numberλ∈[, ) satisfying

Aφ–AαE≤λmax

Sφ–SαE,Sφ(c) –Aφ_E,Sα(c) –Aα_E,



Sφ(c) –AαE+Sα(c) –AφE

(.)

We introduce the following contractive condition which is weaker than the condition of Cirić type generalized contraction (C).

Definition . LetA:E→EandS:E→E. We say thatAandSsatisfy the condition of Cirić type generalizedψ-contraction (C) if

Aφ–AαE≤ψ max

Sφ–SαE,Sφ(c) –Aφ_E,Sα(c) –Aα_E,



Sφ(c) –AαE+Sα(c) –AφE

(.)

for allφ,α∈Eand for somec∈I.

We now prove the existence of PPF dependent coincidence points for mappings satis-fying the weaker contractive condition assumed in [] without being topologically closed of the Razumikhin class.

Theorem . Let A:E→E and S:E→Ebe two mappings satisfying the condition of

Cirić type generalizedψ-contraction(C).Suppose that

(a) A(E)⊆S(E)(c); (b) S(Rc)is complete;

(c) Sis continuous.

IfRcis algebraically closed with respect to the difference,then A and S have a PPF depen-dent coincidence point inRc.

Proof Letφ∈Rc. SinceAφ∈E, there existsx∈Esuch thatAφ=x. BecauseA(E)⊆

S(E)(c), we can chooseφ∈Rcsuch that

x=Sφ(c) =α(c) and α–αE=α(c) –α(c)_E.

SinceAφ∈Eand by assumption, we haveAφ=x for somex∈E. BecauseA(E)⊆

S(E)(c), we can chooseφ∈Rcsuch that

x=Sφ(c) =α(c) and α–αE=α(c) –α(c)E.

By continuing the process as before, we can construct the sequence{αn}such that

Aφn=Sφn+(c), Sφn+=αn+

and

αn–αn+E=αn(c) –αn+(c)_E

for alln∈N∪ {}. We will show that{αn}is a Cauchy sequence inE. IfαN =αN+for someN∈N, then we have

αN+–αN+E =αN+(c) –αN+(c)E

≤ψ max

SφN–SφN+E,SφN(c) –AφN_E,

SφN+(c) –AφN+_E, 

SφN(c) –AφN+E+SφN+(c) –AφNE

≤ψ max

αN–αN+E,αN(c) –αN+(c)_E,

αN+(c) –αN+(c)_E,



αN(c) –αN+(c)E+αN+(c) –αN+(c)E

≤ψ max

αN–αN+E,αN–αN+E,

αN+–αN+E,



αN–αN+E

≤ψ max

αN–αN+E,αN+–αN+E,

 

αN–αN+E

≤ψ max

αN–αN+E,αN+–αN+E,

 

αN–αN+E+αN+–αN+E

≤ψmaxαN–αN+E,αN+–αN+E

≤ψαN+–αN+E

.

ThereforeαN+=αN+. By mathematical induction, we obtain that

αN=αN+k for allk∈N.

This implies that{αn}is a constant sequence forn≥N. Thus{αn}is a Cauchy sequence

inE. Suppose thatαn=αn+for alln∈N. For eachn∈N, we have

αn+–αn+E =αn+(c) –αn+(c)_E

=Aφn–Aφn+E

≤ψ max

Sφn–Sφn+E,Sφn(c) –Aφn_E,

Sφn+(c) –Aφn+_E, 

Sφn(c) –Aφn+E+Sφn+(c) –AφnE

≤ψ max

αn–αn+E,αn(c) –αn+(c)_E,

αn+(c) –αn+(c)_E,



αn(c) –αn+(c)E+αn+(c) –αn+(c)E

≤ψ max

αn–αn+E,αn–αn+E,

αn+–αn+E,



αn–αn+E

≤ψ max

αn–αn+E,αn+–αn+E,

 

αn–αn+E

≤ψ max

αn–αn+E,αn+–αn+E,

 

αn–αn+E+αn+–αn+E

≤ψmaxαn–αn+E,αn+–αn+E

.

Ifmax{αn–αn+E,αn+–αn+E}=αn+–αn+E, then

αn+–αn+E≤ψ

αn+–αn+E

<αn+–αn+E.

This leads to a contradiction. Therefore

αn+–αn+E≤ψ

αn–αn+E

<αn–αn+E.

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

αn+E}is a nonincreasing sequence of real numbers, we obtain that it is a convergent

sequence. Suppose that

lim

n→∞αn–αn+E=α

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

αn+–αn+E≤ψ

αn–αn+E

for alln∈Nand the continuity ofψ, we have

α≤ψ(α) <α,

which leads to a contradiction. This implies thatα= . We will prove that{αn}is a Cauchy

sequence inE. It suffices to prove that the sequence{αn}is a Cauchy sequence inE. Assume that{αn}is not a Cauchy. It follows that there existε>  and two sequences of

even positive integers{mk}and{nk}satisfying mk> nk>kfor eachk∈Nand

αmk–αnkE≥ε. (.)

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

(.) and

We will prove thatlim_k→∞αmk–αnkE=ε. Sinceαmk–αnkE≥εfor allk∈N, we

have

lim

k→∞αmk–αnkE≥ε.

For eachk∈N, we obtain that

αmk–αnkE≤ αmk–αmk–E+αmk––αmk–E+αmk––αnkE

≤ αmk–αmk–E+αmk––αmk–E+ε.

This implies thatlim_k→∞αmk–αnkE≤ε. Therefore

lim

k→∞αmk–αnkE=ε.

Similarly, we can prove that

lim

k→∞αmk+–αnkE=ε, _k→∞lim αmk–αnk–E=ε

and

lim

k→∞αmk+–αnk–E=ε.

Since

αnk–αmk+E≤αnk(c) –αmk+(c)E

=Aφnk––AφmkE

≤ψ max

Sφnk––SφmkE,Sφnk–(c) –Aφnk–E,

Sφmk(c) –Aφmk_E, 

Sφnk–(c) –AφmkE+Sφmk(c) –Aφnk–E

≤ψ max

αnk––αmkE,αnk–(c) –αnk(c)_E, αmk(c) –αmk+(c)E,



αnk–(c) –αmk+(c)E+αmk(c) –αnk(c)E

≤ψ max

αnk––αmkE,αnk––αnkE,

αmk–αmk+E,



αnk––αmk+E+αmk–αnkE

,

by taking the limit of both sides, we have

which leads to a contradiction. It follows that the sequence{αn} is a Cauchy sequence

and so{αn}is a Cauchy sequence inE. Therefore{Sφn}is a Cauchy sequence inS(Rc).

By the completeness of S(Rc), we have{Sφn} is a convergent sequence. Suppose that limn→∞Sφn=φ∗ for someφ∗∈S(Rc). Thereforeφ∗=Sφ for someφ∈Rc. Moreover,

we also have

lim

n→∞Aφn=n→∞limSφn+(c) =Sφ(c).

We will prove thatφis a PPF dependent coincidence fixed point ofAandS. By using (.), we obtain that

Aφ–Sφ(c)_E≤ Aφ–AφnE+Aφn–Sφ(c)_E ≤ Aφ–AφnE+Sφn+(c) –Sφ(c)_E

≤ψ max

Sφ–SφnE,Sφ(c) –AφE,Sφn(c) –AφnE,



Sφ(c) –AφnE+Sφn(c) –AφE

+Sφn+(c) –Sφ(c)_E.

By takingn→ ∞, we obtain thatAφ=Sφ(c). Henceφ is a PPF dependent coincidence

point ofAandS.

By applying Theorem ., we obtain the following corollary.

Corollary . Let A:E→E and S:E→Ebe two mappings satisfying the condition of

Cirić type generalized contraction(C).Suppose that

(a) A(E)⊆S(E)(c); (b) S(Rc)is complete;

(c) Sis continuous.

IfRcis algebraically closed with respect to the difference,then A and S have a PPF depen-dent coincidence point inRc.

Proof Define a functionψ: [, +∞)→[, +∞) byψ(t) =λtfor allt∈[, +∞). Therefore

A:E→EandS:E→Esatisfy the condition of Cirić type generalizedψ-contraction (C). This implies that all assumptions in Theorem . are now satisfied. Hence the proof

is complete.

Questions

(i) Are the results in Theorem . and Theorem . still true when the norm closedness forRcis replaced by weak closedness or weak∗closedness (for dual Banach spaces)?

(ii) Is there some way to improve the results to more than two mappings or a family of mappings as in the case of nonexpansive mappings (see, for example, [] and references contained therein)?

Competing interests

Acknowledgements

This research is supported by the Centre of Excellence in Mathematics, the Commission of Higher Education, Thailand. I would like to express my deep thanks to the referees for their suggestions and comments which led to the improvement of the manuscript.

Received: 19 October 2012 Accepted: 22 May 2013 Published: 6 June 2013

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Cite this article as:Kaewcharoen:PPF dependent common fixed point theorems for mappings in Banach spaces.