Some fixed point theorems in complex valued metric spaces

(1)

R E S E A R C H

Open Access

Some ﬁxed point theorems in complex

valued metric spaces

Kittipong Sitthikul and Satit Saejung

*

*_{Correspondence:}

[email protected]

Department of Mathematics, Khon Kaen University, Khon Kaen, 40002, Thailand

Abstract

Owning the concept of complex valued metric spaces introduced by Azamet al., we prove several ﬁxed point theorems for mappings satisfying certain point-dependent contractive conditions. The main results announced by Sintunavarat and Kumam (J. Inequal. Appl. 2012:84, 2012), Rouzkard and Imdad (Comput. Math. Appl., 2012, doi:10.1016/j.camwa.2012.02.063), and Dass and Gupta (Indian J. Pure Appl. Math. 6(12):1455-1458, 1975) are deduced from our results under weaker assumptions.

1 Introduction

The concept of a complex valued metric space which is a generalization of the classical metric space was recently introduced by Azam, Fisher and Khan (see []). To mention this, let us recall a natural relationonC, the set of complex numbers as follows: for

z,z∈C

zz ⇐⇒def Re(z)≤Re(z) and Im(z)≤Im(z),

z≺z ⇐⇒def Re(z) <Re(z) and Im(z) <Im(z).

Deﬁnition . LetXbe a nonempty set. A mappingd:X×X→Cis called acomplex

valued metriconXif the following conditions are satisﬁed:

(CM) d(x,y)for allx,y∈Xandd(x,y) = ⇐⇒x=y; (CM) d(x,y) =d(y,x)for allx,y∈X;

(CM) d(x,y)d(x,z) +d(z,y)for allx,y,z∈X.

In this case, we say that (X,d) is acomplex valued metric space.

It is obvious that this concept is a generalization of the classical metric. In fact, ifd:

X×X→Rsatisﬁes (CM)-(CM), then thisdis ametricin the classical sense; that is, the following conditions are satisﬁed:

(M) ≤d(x,y)for allx,y∈Xandd(x,y) = ⇐⇒x=y; (M) d(x,y) =d(y,x)for allx,y∈X;

(M) d(x,y)≤d(x,z) +d(z,y)for allx,y,z∈X.

The following deﬁnition is an analogue of several concepts in the classical theory of metric spaces and they are discussed in []. There are also other interesting types of generalization of metric spaces; for example, see [, ].

Deﬁnition . Suppose that (X,d) is a complex valued metric space.

(2)

• We say that a sequence{xn}is aCauchy sequenceif for every≺c∈Cthere exists an

integerNsuch thatd(xn,xm)≺cfor alln,m≥N.

• We say that{xn}converges to an elementx∈Xif for every≺c∈Cthere exists an

integerNsuch thatd(xn,x)≺cfor alln≥N. In this case, we writexn d

→x.

• We say that(X,d)iscompleteif every Cauchy sequence inXconverges to a point inX.

The following fact is summarized from Azam, Fisher and Khan’s paper []. In fact, (b) and (c) of Proposition . are their Lemmas  and .

Proposition . Let (X,d)be a complex value metric space. Suppose that d=d+id

where d,d:X×X→R,that is,d=Re(d)and d=Im(d).Then the following assertions

hold.

(a) |d|= (d_+d_)/:X×X→Ris a(classical)metric onX. (b) If{xn}is a sequence inXandx∈X,thenxn

d

→xif and only ifxn |d|

→x. (c) (X,d)is complete if and only if(X,|d|)is complete.

The following common ﬁxed point theorem was also proved by Azam, Fisher and Khan. This can be viewed as a generalization of the well-known Banach ﬁxed point theorem.

Theorem .([]) Let(X,d)be a complete complex valued metric space,and letλ,μbe nonnegative real numbers such thatλ+μ< .Suppose that S,T:X→X are mappings satisfying:

d(Sx,Ty)λd(x,y) +μd(x,Sx)d(y,Ty)

 +d(x,y) ∀x,y∈X. ()

Then S and T have a unique common ﬁxed point.

In this paper, we continue the study of fixed point theorems in complex valued metric spaces. The obtained results are generalizations of recent results proved by Sintunavarat and Kumam [], Rouzkard and Imdad []. Moreover, we improve several assumptions on the involved mappings. It should be noted that there are also some different fixed point theorems recently proved in [].

2 Main result

Throughout the paper, let (X,d) be a complete complex valued metric space andS,T :

X→X.

Proposition . Let x∈X and deﬁne the sequence{xn}by xn+=Sxn,

xn+=Txn+, for all n= , , , . . . .

()

Assume that there exists a mappingλ:X×X→[, )satisfying

λ(TSx,y)≤λ(x,y) and λ(x,STy)≤λ(x,y) for all x,y∈X.

(3)

Proof Letx,y∈Xandn= , , , . . . . Then we have

λ(xn,y) =λ(TSxn–,y)≤λ(xn–,y) =λ(TSxn–,y)≤ · · · ≤λ(x,y).

Similarly, we have

λ(x,xn+) =λ(x,STxn–)≤λ(x,xn–) =λ(x,STxn–)≤ · · · ≤λ(x,x).

Lemma . Letλ,μ:X×X→[, )and x,y∈X.If S and T satisfy

d(Sx,TSx)λ(x,Sx)d(x,Sx) +μ(x,Sx)d(x,Sx)d(Sx,TSx)  +d(x,Sx) ,

d(STy,Ty)λ(Ty,y)d(Ty,y) +μ(Ty,y)d(Ty,STy)d(y,Ty)  +d(Ty,y) ,

then

d(Sx,TSx)≤λ(x,Sx)d(x,Sx)+μ(x,Sx)d(Sx,TSx), d(STy,Ty)≤λ(Ty,y)d(Ty,y)+μ(Ty,y)d(Ty,STy),

respectively.

Proof We can write

d(Sx,TSx)≤λ(x,Sx)d(x,Sx) +μ(x,Sx)d(x,Sx)d(Sx,TSx)  +d(x,Sx)

≤λ(x,Sx)d(x,Sx)+μ(x,Sx) d(x,Sx)  +d(x,Sx)

d(Sx,TSx)

≤λ(x,Sx)d(x,Sx)+μ(x,Sx)d(Sx,TSx). ()

Similarly, we get

d(STy,Ty)≤λ(Ty,y)d(Ty,y) +μ(Ty,y)d(Ty,STy)d(y,Ty)  +d(Ty,y)

≤λ(Ty,y)d(y,Ty)+μ(Ty,y) d(y,Ty)  +d(y,Ty)

d(Ty,STy)

≤λ(Ty,y)d(y,Ty)+μ(Ty,y)d(STy,Ty). ()

Lemma . Let{xn}be a sequence in X and h∈[, ).If an=|d(xn,xn+)|satisﬁes

an≤han–, for all n∈N, ()

then{xn}is a Cauchy sequence.

Proof Leth∈[, ). Then

(4)

Form,n∈Nsuch thatm>n, we have

d(xn,xm)≤an+an++· · ·+am–

≤hn +h+h+· · ·+hm–n–a

≤ hn  –ha.

Thus, we have|d(xn,xm)| → asn→ ∞, and hence{xn}is a Cauchy sequence.

Theorem . Let(X,d)be a complete complex valued metric space and S,T:X→X.If

there exist mappingsλ,μ,γ :X×X→[, )such that for all x,y∈X:

(a) λ(TSx,y)≤λ(x,y)andλ(x,STy)≤λ(x,y),

μ(TSx,y)≤μ(x,y)andμ(x,STy)≤μ(x,y),

γ(TSx,y)≤γ(x,y)andγ(x,STy)≤γ(x,y); (b) λ(x,y) +μ(x,y) +γ(x,y) < ;

(c)

d(Sx,Ty)λ(x,y)d(x,y) +μ(x,y)d(x,Sx)d(y,Ty)

 +d(x,y) +γ(x,y)

d(y,Sx)d(x,Ty)  +d(x,y) . ()

Proof Letx,y∈X. From (), we consider

d(Sx,TSx)λ(x,Sx)d(x,Sx) +μ(x,Sx)d(x,Sx)d(Sx,TSx)  +d(x,Sx)

+γ(x,Sx)d(Sx,Sx)d(x,TSx)  +d(x,Sx)

=λ(x,Sx)d(x,Sx) +μ(x,Sx)d(x,Sx)d(Sx,TSx)  +d(x,Sx) .

From Lemma ., we have

_d₍_Sx_,_TSx₎_≤_λ(_x_,_Sx₎_d₍_x_,_Sx₎₊_μ(_x_,_Sx₎_d₍_Sx_,_TSx₎_. _()

Similarly, we get

d(STy,Ty)λ(Ty,y)d(Ty,y) +μ(Ty,y)d(Ty,STy)d(y,Ty)  +d(Ty,y)

+γ(Ty,y)d(y,STy)d(Ty,Ty)  +d(Ty,y)

=λ(Ty,y)d(Ty,y) +μ(Ty,y)d(Ty,STy)d(y,Ty)  +d(Ty,y) .

From Lemma ., we have

(5)

Letx∈Xand the sequence{xn}be deﬁned by (). We show that{xn}is a Cauchy sequence.

From Proposition ., (), () and for allk= , , , . . . , we have

d(xk+,xk)=d(STxk–,Txk–)

≤λ(Txk–,xk–)d(Txk–,xk–)

+μ(Txk–,xk–)d(Txk–,STxk–)

=λ(xk,xk–)d(xk–,xk)+μ(xk,xk–)d(xk,xk+)

≤λ(x,xk–)d(xk–,xk)+μ(x,xk–)d(xk+,xk)

≤λ(x,x)d(xk–,xk)+μ(x,x)d(xk+,xk),

which implies that

d(xk+,xk)≤

λ(x,x)

 –μ(x,x)

d(xk–,xk). ()

Similarly, we get

d(xk+,xk+)=d(TSxk,Sxk)

≤λ(xk,Sxk)d(xk,Sxk)+μ(xk,Sxk)d(Sxk,TSxk)

=λ(xk,xk+)d(xk,xk+)+μ(xk,xk+)d(xk+,xk+)

≤λ(x,xk+)d(xk,xk+)+μ(x,xk+)d(xk+,xk+)

≤λ(x,x)d(xk,xk+)+μ(x,x)d(xk+,xk+),

which implies that

d(xk+,xk+)≤

λ(x,x)

 –μ(x,x)

d(xk,xk+). ()

Leth=_–λ_μ(x₍_x,x_,_x)₎ < . Then we have

d(xn+,xn)≤hd(xn–,xn), for alln∈N. ()

From Lemma ., we have{xn}is a Cauchy sequence in (X,d). By the completeness ofX,

there existsz∈Xsuch thatxn→zasn→ ∞.

Next, we show thatzis a ﬁxed point ofS. By () and Proposition ., we have

d(z,Sz)d(z,Txn+) +d(Txn+,Sz)

=d(z,xn+) +d(Sz,Txn+)

d(z,xn+) +λ(z,xn+)d(z,xn+) +μ(z,xn+)

d(z,Sz)d(xn+,Txn+)

 +d(z,xn+)

+γ(z,xn+)

d(xn+,Sz)d(z,Txn+)

(6)

d(z,xn+) +λ(z,x)d(z,xn+) +μ(z,x)

d(z,Sz)d(xn+,xn+)

 +d(z,xn+)

+γ(z,x)

d(xn+,Sz)d(z,xn+)

 +d(z,xn+)

.

Thus,d(z,Sz) =  and hencez=Sz.

We also show thatzis a ﬁxed point ofT. By (), we have

d(z,Tz)d(z,Sxn) +d(Sxn,Tz)

d(z,xn+) +λ(xn,z)d(xn,z) +μ(xn,z)

d(xn,Sxn)d(z,Tz)

 +d(xn,z)

+γ(xn,z)

d(z,Sxn)d(xn,Ty)

 +d(xn,z)

d(z,xn+) +λ(x,z)d(xn,z) +μ(x,z)

d(xn,xn+)d(z,Tz)

 +d(xn,z)

+γ(x,z)

d(z,xn+)d(xn,Ty)

 +d(xn,z)

.

Thus,d(z,Tz) =  and hencez=Tz. Therefore,zis a common ﬁxed point ofSandT. Finally, we show the uniqueness. Suppose that there isz*_∈_X_{such that}_z*₌_Sz*₌_Tz*_.

Then

dz,z*=dSz,Tz*

λz,z*dz,z*+μz,z*d(z,Sz)d(z

*_,_Tz*₎

 +d(z,z*₎ +γ

z,z*d(z

*_,_Sz₎_d₍_z_,_Tz*₎

 +d(z,z*₎

=λz,z*dz,z*+γz,z*d(z

*_,_z₎_d₍_z_,_z*₎

 +d(z,z*₎ .

Therefore, we have

dz,z*≤λz,z*dz,z*+γz,z*dz,z* d(z,z

*₎

 +d(z,z*₎

≤λz,z*dz,z*+γz,z*dz,z*

≤λz,z*+γz,z*dz,z*.

Sinceλ(z,z*_{) +}_γ₍_z_,_z*_{) < , we have}_|_d₍_z_,_z*₎_|_{= . Thus}_z₌_z*_.

By settingS=T in Theorem ., we deduce the following corollary.

Corollary . Let(X,d)be a complete complex valued metric space and T:X→X.If there exist mappingsλ,μ,γ :X×X→[, )such that for all x,y∈X:

(a) λ(Tx,y)≤λ(x,y)andλ(x,Ty)≤λ(x,y),

μ(Tx,y)≤μ(x,y)andμ(x,Ty)≤μ(x,y),

(7)

(c)

d(Tx,Ty)λ(x,y)d(x,y) +μ(x,y)d(x,Tx)d(y,Ty)

 +d(x,y) +γ(x,y)

d(y,Tx)d(x,Ty)  +d(x,y) . ()

Then T has a unique common ﬁxed point.

By choosingγ =  in Theorem ., we deduce the following corollary.

Corollary . Let(X,d)be a complete complex valued metric space and S,T:X→X.If

there exist mappingsλ,μ:X×X→[, )such that for all x,y∈X:

μ(TSx,y)≤μ(x,y)andμ(x,STy)≤μ(x,y); (b) λ(x,y) +μ(x,y) < ;

(c)

d(Sx,Ty)λ(x,y)d(x,y) +μ(x,y)d(x,Sx)d(y,Ty)

 +d(x,y) . ()

By choosingμ=  in Theorem ., we deduce the following corollary.

Corollary . Let(X,d)be a complete complex valued metric space and S,T:X→X.If

there exist mappingsλ,γ :X×X→[, )such that for all x,y∈X:

γ(TSx,y)≤γ(x,y)andγ(x,STy)≤γ(x,y); (b) λ(x,y) +γ(x,y) < ;

(c)

d(Sx,Ty)λ(x,y)d(x,y) +γ(x,y)d(y,Sx)d(x,Ty)

 +d(x,y) . ()

The following result is closely related to Corollary . withγ = . The real valued metric space version of this result is an extension of Dass and Gupta’s result [].

Theorem . Let(X,d)be a complete complex valued metric space and T:X→X.If there exist mappingsλ,μ:X×X→[, )such that for all x,y∈X:

(a) λ(Tx,y)≤λ(x,y)andλ(x,Ty)≤λ(x,y),

μ(Tx,y)≤μ(x,y)andμ(x,Ty)≤μ(x,y); (b) λ(x,y) +μ(x,y) < ;

(c)

d(Tx,Ty)λ(x,y)d(x,y) +μ(x,y)d(y,Ty)[ +d(x,Tx)]

 +d(x,y) . ()

(8)

Proof Letx∈Xand the sequence{xn}be deﬁned by

xn+=Txn, wheren= , , , . . . . ()

We show that{xn}is a Cauchy sequence. From (), we have

d(xn+,xn+) = d(Txn,Txn+)

λ(xn,xn+)d(xn,xn+) +μ(xn,xn+)

d(xn+,Txn+)[ +d(xn,Txn)]

 +d(xn,xn+)

=λ(xn,xn+)d(xn,xn+) +μ(xn,xn+)

d(xn+,xn+)[ +d(xn,xn+)]

 +d(xn,xn+)

=λ(xn,xn+)d(xn,xn+) +μ(xn,xn+)d(xn+,xn+).

It follows from (a) that

d(xn+,xn+)λ(xn,xn+)d(xn,xn+) +μ(xn,xn+)d(xn+,xn+)

λ(x,xn+)d(xn,xn+) +μ(x,xn+)d(xn+,xn+)

λ(x,x)d(xn,xn+) +μ(x,x)d(xn+,xn+).

Therefore,

d(xn+,xn+)≤λ(x,x)d(xn,xn+)+μ(x,x)d(xn+,xn+),

and hence

d(xn+,xn+)≤

λ(x,x)

 –μ(x,x)

d(xn,xn+), for alln= , , , . . . . ()

Leth=_–λ_μ(x₍_x,x_,_x)₎< . Then

d(xn+,xn+)≤hd(xn,xn+), for alln= , , , . . . .

From Lemma ., we have{xn}is a Cauchy sequence in (X,d). By the completeness ofX,

there existsz∈Xsuch thatxn→zasn→ ∞. Next, we show thatzis a ﬁxed point ofT.

Then

d(z,Tz)d(z,Txn) +d(Txn,Tz)

d(z,xn+) +λ(xn,z)d(xn,z) +μ(xn,z)

d(z,Tz)[ +d(xn,Txn)]

 +d(z,xn)

d(z,xn+) +λ(x,z)d(xn,z) +μ(x,z)

d(z,Tz)[ +d(xn,xn+)]

 +d(z,xn)

.

(9)

Finally, we show the uniqueness. Suppose that there isz*_∈_X_{such that}_z*₌_Tz*_{. Then}

dz,z*=dTz,Tz*

λz,z*dz,z*+μz,z*d(z

*_,_Tz*_{)[ +}_d₍_z_,_Tz_)]

 +d(z,z*₎

=λz,z*dz,z*.

Sinceλ(z,z*)∈[, ), we haved(z,z*) = , that is,z=z*. This completes the proof.

3 Deduced results

3.1 Sintunavarat and Kumam’s results

We deduce the main result of [] as follows.

Theorem .([, Theorem .]) Let(X,d)be a complete complex valued metric space and S,T:X→X.If there exist mappings,:X→[, )such that for all x,y∈X:

(i) (Sx)≤(x)and(Sx)≤(x); (ii) (Tx)≤(x)and(Tx)≤(x); (iii) (+)(x) < ;

(iv)

d(Sx,Ty)(x)d(x,y) +(x)d(x,Sx)d(y,Ty)  +d(x,y) .

Proof Deﬁneλ,μ:X×X→[, ) by

λ(x,y) =(x) and μ(x,y) =(x), for allx,y∈X. ()

Then for allx,y∈X,

(a) λ(TSx,y) =(TSx)≤(Sx)≤(x) =λ(x,y)andλ(x,STy) =(x) =λ(x,y);

μ(TSx,y) =(TSx)≤(Sx)≤(x) =μ(x,y)andμ(x,STy) =(x) =μ(x,y); (b) λ(x,y) +μ(x,y) =(x) +(x) < ;

(c)

d(Sx,Ty)(x)d(x,y) +(x)d(x,Sx)d(y,Ty)  +d(x,y)

=λ(x,y)d(x,y) +μ(x,y)d(x,Sx)d(y,Ty)  +d(x,y) .

By Corollary .,SandT have a unique common ﬁxed point.

Remark  It is worth mentioning that (i) and (ii) of Theorem . above can be weakened by the condition

(10)

3.2 Rouzkard and Imdad’s results

The following corollary is easily obtained from our Theorem ..

Corollary . Let(X,d)be a complete complex valued metric space and S,T:X→X.If there exist mappingsλ,μ,γ :X→[, )such that for all x,y∈X:

(a) λ(TSx)≤λ(x),μ(TSx)≤μ(x)andγ(TSx)≤γ(x); (b) λ(x) +μ(x) +γ(y) < ;

(c)

d(Sx,Ty)λ(x)d(x,y) +μ(x)d(x,Sx)d(y,Ty)  +d(x,y) +γ(x)

d(y,Sx)d(x,Ty)

 +d(x,y) . ()

Proof Deﬁneλ,μ,γ:X×X→[, ) by

λ(x,y) =λ(x), μ(x,y) =μ(x) and γ(x,y) =γ(x), for allx,y∈X. ()

Then for allx,y∈X,

(a) λ(TSx,y) =λ(TSx)≤λ(x) =λ(x,y)andλ(x,STy) =λ(x) =λ(x,y);

μ(TSx,y) =μ(TSx)≤μ(x) =μ(x,y)andμ(x,STy) =μ(x) =μ(x,y);

γ(TSx,y) =γ(TSx)≤γ(x) =γ(x,y)andγ(x,STy) =γ(x) =γ(x,y); (b) λ(x,y) +μ(x,y) +γ(x,y) =λ(x) +μ(x) +γ(x) < ;

(c)

d(Sx,Ty)λ(x)d(x,y) +μ(x)d(x,Sx)d(y,Ty)  +d(x,y) +γ(x)

d(y,Sx)d(x,Ty)  +d(x,y)

=λ(x,y)d(x,y) +μ(x,y)d(x,Sx)d(y,Ty)

 +d(x,y) +γ(x,y)

d(y,Sx)d(x,Ty)  +d(x,y) .

By Theorem .,SandThave a unique common ﬁxed point.

Lettingλ(·) =λ,μ(·) =μandγ(·) =γ in Corollary . gives the following result proved by Rouzkard and Imdad in [].

Corollary .([]) If S and T are self-mappings deﬁned on a complete complex valued

metric space(X,d)satisfying the condition

d(Sx,Ty)λd(x,y) +μd(x,Sx)d(y,Ty)  +d(x,y) +γ

d(y,Sx)d(x,Ty)  +d(x,y)

for all x,y∈X,whereλ,μ,γ are nonnegative reals withλ+μ+γ< ,then S and T have a unique common ﬁxed point.

3.3 Dass and Gupta’s results

(11)

Theorem .([]) Let(X,d)be a real valued metric space.Let T:X→X be such that

(i)

d(Tx,Ty)≤λd(x,y) +μd(y,Ty)[ +d(x,Tx)]  +d(x,y)

for allx,y∈X,λ> ,μ> ,λ+μ< ,and

(ii) for somex∈X,the sequence of iterates{Tn(x)}has a subsequence{Tnk(x)}with

z=lim_k_→∞Tnk₍_x

).

Then z is a unique ﬁxed point of T.

Proof Deﬁneλ,μ:X×X→[, ) by

λ(x,y) =λ and μ(x,y) =μ, for allx,y∈X.

Then the conditions (a), (b) and (c) of Theorem . are satisﬁed. Hence, we have{xn}is a

Cauchy sequence in (X,d). By (ii), the whole sequencexn→zasn→ ∞. It follows again

from the proof of Theorem . thatzis a unique ﬁxed point ofTas desired.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly to this research work. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors thank the referee for their comments and suggestions on the manuscript. The ﬁrst author is supported by Khon Kaen University-Integrated Multidisciplinary Research Cluster (Sciences and Technologies). The research work of the second author was also supported by the Higher Education Research Promotion and National Research University Project of Thailand, Oﬃce of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

Received: 18 July 2012 Accepted: 10 October 2012 Published: 24 October 2012

References

1. Azam, A, Fisher, B, Khan, M: Common ﬁxed point theorems in complex valued metric spaces. Numer. Funct. Anal. Optim.32(3), 243-253 (2011)

2. Ding, HS, Kadelburg, Z, Karapinar, E, Radenovi´c, S: Common ﬁxed points of weak contractions in cone metric spaces. Abstr. Appl. Anal.2012, Article ID 793862 (2012). doi:10.1155/2012/793862

3. Ding, HS, Kadelburg, Z, Nashine, HK: Common ﬁxed point theorems for weakly increasing mappings on ordered orbitally complete metric spaces. Fixed Point Theory Appl.2012, 85 (2012). doi:10.1186/1687-1812-2012-85 4. Sintunavarat, W, Kumam, P: Generalized common ﬁxed point theorems in complex valued metric spaces and

applications. J. Inequal. Appl.2012, 84 (2012). doi:10.1186/1029-242X-2012-84

5. Rouzkard, F, Imdad, M: Some common ﬁxed point theorems on complex valued metric spaces. Comput. Math. Appl. (2012). doi:10.1016/j.camwa.2012.02.063

6. Bhatt, S, Chaukiyal, S, Dimri, RC: Common ﬁxed point of mappings satisfying rational inequality in complex valued metric space. Int. J. Pure Appl. Math.73(2), 159-164 (2011)

7. Dass, BK, Gupta, S: An extension of Banach contraction principle through rational expression. Indian J. Pure Appl. Math.6(12), 1455-1458 (1975)

doi:10.1186/1687-1812-2012-189