SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR …geometry-math-journal.ro › pdf ›...

Global Journal of Advanced Researchon Classical and Modern Geometries

ISSN: 2284-5569, Vol.6, (2017), Issue 1, pp.20-36

SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF2-BANACH SPACES

DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA

ABSTRACT. Let X andY be two normed spaces which are also 2-Banach spaces. For the pro-jective tensor productX ⊗γ Y, we have defined a 2-norm for whichX ⊗γ Y is a 2-Banach space.Considering the closed and bounded subspaceDX, DY and DX⊗γY of X, Y and X ⊗γ Y re-spectively, we take two pairs of mappingsT1, S1 : DX⊗γY → DX andT2, S2 : DX⊗γY → DY

satisfying some specific characteristics. Using these two pairs we define apair of self mappingsT andS on DX⊗γY. Some fixed point theorems forT andS on DX⊗γY are derived here withsuitable example. Moreover, takingX ⊗γ Y as a 2-Banach algebra, we establish another fixedpoint theorem for a self mappingT on DX⊗γY with a proper example.

1. INTRODUCTION

In 1963, Gahler [7] introduced the notion of2-metric spaces and and their topological structures.The concept of2-normed linear spaces can be found in Gahler’s paper [8] in 1965. Gahler in-vestigated a lot of results in this space and also proved that if(X, ‖.‖) is a linear normed spacethen a 2-norm can be defined onX.

In 1969, White [19] introduced the concept of 2-Banach space and gave some important defi-nitions and examples. Since then a lot of applications of such spaces can befound in variousaspects.

Many authors viz., Iseki [10], Khan and Khan [11], Rhoads [15], Hadzic [9] etc. developed theapplications of 2-metric spaces and 2-Banach spaces in the field of fixed point theory.

In this paper, we consider the projective tensor product space as a 2-Banach space and derivesome fixed point theorems here.

PRELIMINARIES

Definition 1.1 [8] Let X be a real linear space of dimension greater than1 and let‖., .‖ be areal valued function onX × X satisfying the following conditions:

(i) ‖x, y‖ = 0 if and only if x andy are linearly dependent,(ii) ‖x, y‖ = ‖y, x‖ for all x, y ∈ X,

(iii) ‖αx, y‖ = |α|‖x, y‖, α being real,x, y ∈ X,(iv) ‖x, y + z‖ 6 ‖x, y‖+ ‖x, z‖, for all, x, y, z ∈ X

2010Mathematics Subject Classification.46A70, 46B28, 47A80, 47H10.Key words and phrases.2-Banach space, fixed points, projective tensor product.

20

SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES 21

Then‖., .‖ is called a2-norm onX and(X, ‖., .‖) is called a linear2-normed space.

In 1988, N. Mohammad, and A.H. Siddiqui [13] introduced the concept of2-Banach Algebra.

A 2-Banach algebra(X, ‖., .‖) is a real algebra (of dimension greater than 2) which is a 2-Banach space (with respect to 2-norm topology) and in addition, the following condition holds:‖a, bc‖ 6 M‖a, b‖‖a, c‖, M > 0 ∀a, b, c ∈ X.

Definition 1.2 A sequence{xn} in a2-normed space(X, ‖., .‖) is said to be a Cauchy sequenceif limn,m→∞ ‖xn − xm, a‖ = 0 for all a in X.

Definition 1.3 A sequence{xn} in a 2-normed spaceX is called a convergent sequence if thereis anx in X such thatlimn→∞ ‖xn − x, a‖ = 0 for all a in X.

Definition 1.4 A 2-normed space in which every Cauchy sequence is convergent is calleda 2-Banach space.

Definition 1.5 Let X andY be two linear2-normed spaces. An operatorT : X → Y is saidto be continuous atx ∈ X if for every sequence{xn} in X, {xn} → x asn → ∞ implies{T(xn)} → T(x) in Y asn → ∞ .

Example [19] Let Pn denotes the set of all real polynomials of degree6 n, on the interval[0, 1]. Clearly,Pn is a linear vector space over the reals with respect to usual addition and scalarmultiplication,. Let{x0, x1, ..., x2n} be distinct fixed points in[0, 1] andg, h ∈ Pn. Define thefollowing 2-norm onPn :

‖g, h‖ =

{

∑2ni=0 |g(xi)h(xi)|, if g andh are linearly independent

0, if g andh are linearly dependent

then(Pn, ‖., .‖) is a 2-Banach space.

Algebric tensor product: [6] Let X, Y be normed spaces overF with dual spacesX∗ andY∗

respectively. Givenx ∈ X, y ∈ Y, Let x ⊗ y be the element ofBL(X∗, Y∗; F) (which is the setof all bounded bilinear forms fromX∗ × Y∗ to F), defined by

x ⊗ y( f , g) = f (x)g(y), ( f ∈ X∗, g ∈ Y∗)

The algebraic tensor product ofX andY, X ⊗Y is defined to be the linear span of{x ⊗ y : x ∈X, y ∈ Y} in BL(X∗, Y∗; F).

Projective tensor product: [6] Given normed spacesX andY, the projective tensor normγ onX ⊗ Y is defined by

‖u‖γ = inf{∑i

‖xi‖‖yi‖ : u = ∑i

xi ⊗ yi}

where the infimum is taken over all (finite) representations ofu.

Lemma: [6] X ⊗γ Y can be represented as a linear subspace ofBL(X∗, Y∗; F) consistingof all elements of the formu = ∑i xi ⊗ yi where∑i ‖xi‖‖yi‖ < ∞. Moreover,‖u‖γ =inf{∑i ‖xi‖‖yi‖} over all such representations ofu.

Let X andY be both normed as well as 2-normed spaces. For the projective tensor productX ⊗γ Y we define another 2-norm as follows:for u = ∑i xi ⊗ yi, v = ∑j pj ⊗ qj andw = ∑k gk ⊗ hk

22 DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA

‖u, v‖ =

1

2∑i,j

(

‖xi, pj‖‖yi‖‖qj‖+ ‖yi, qj‖‖xi‖‖pj‖)

,

i f u and v are linearly independent

0, i f u and v are linearly dependent

(i) If u andv are linearly dependent then by definition,‖u, v‖ = 0.

If ‖u, v‖ = 0 then‖xi, pj‖‖yi‖‖qj‖ = 0 and‖yi, qj‖‖xi‖‖pj‖ = 0 ∀i, j.If ‖xi, pj‖ 6= 0 then‖yi‖ = 0 or ‖qj‖ = 0 ⇒ yi = 0 or qj = 0 ∀i, j.(Clearly thenxi 6= 0 andpj 6= 0 otherwise‖xi, pj‖ = 0)⇒ u = 0 or v = 0 which shows thatu andv are linearly dependent.Similarly, if ‖yi, qj‖ 6= 0 thenu = 0 or v = 0 and thusu andv are linearly dependent.Now, let‖xi, pj‖ = 0 and‖yi, qj‖ = 0 ∀ i, j.⇒ xi and pj are linearly dependent andyi and qj are linearly dependent∀ i, j(being2-norms).This also implies thatxi, xj’s are linearly dependent∀ i, j, andyi, yj’s are linearly de-pendent∀ i, j.Therefore,u can be expressed as:u = α(x1 ⊗ y1), for some scalarα.Similarly, pi, pj’s are linearly dependent∀ i, j, andqi, qj’s are linearly dependent∀ i, j.So, we can takev = β(p1 ⊗ q1), for some scalarβ. But x1, p1 andy1, q1 are linearlydependent. Hence,u andv are linearly dependent.

(ii) ‖u, v‖ = ‖v, u‖(iii) ‖u, αv‖ = |α|‖u, v‖, α is a scalar(iv) ‖u, v + w‖ 6 ‖u, v‖+ ‖u, w‖ (all these properties follow by definition.)

(Moreover, ifX andY are two 2-Banach spaces, thenX ⊗γ Y is also a 2-Banach space for theabove 2-norm).

Let DX, DY andDX⊗γY denote closed and bounded subsets ofX, Y andX ⊗γ Y respectively.Let T1, S1 : DX⊗γY → DX andT2, S2 : DX⊗γY → DY be such that for anyu, v ∈ DX⊗γY anda ⊗ b ∈ DX⊗γY with ‖a‖‖b‖ > 1 and positivek, k/,

(A) ‖T1(u)− T1(v)‖ 61

M2N1

(k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖))

(B) ‖T2(u)− T2(v)‖ 61

M1N2

(k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖))

(C) ‖T1(u)− T1(v), a‖ 61

N22

(k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖))

(D) ‖T2(u)− T2(v), b‖ 61

N21

(k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖))where,

(a) ψ : [0, ∞) → [0, ∞) is continuous and non-decreasing,ψ(0) = 0,(b) max[‖T1u‖, ‖S1u‖] 6 N1 andmax[‖T2u‖, ‖S2u‖] 6 N2 (whereu ∈ DX⊗γY). (Here

DX, DY andDX⊗γY are bounded in norm byN1, N2 andN1N2 respectively.)(c) max[‖T1u, a‖, ‖S1u, a‖] 6 M1 and max[‖T2u, b‖, ‖S2u, b‖] 6 M2 (where u ∈

DX⊗γY). (HereDX, DY and DX⊗γY are bounded in 2-norm byM1, M2 and M1M2

respectively.)


From the pairs(T1, S1) and(T2, S2), we define self mappingsT andS on DX⊗γY as earlier suchthatTu = T1u ⊗ T2u andSu = S1u ⊗ S2u ∀u ∈ DX⊗γY.

MAIN RESULTS

Theorem 2.1 For normed spacesX andY which are also 2-Banach spaces, the mappingTderived by the pair of mappings(T1, T2) satisfying (A), (B), (C) and (D) has a unique fixedpoint in DX⊗γY if

(i) p, q ∈ DX⊗γY with ‖p‖ > ‖q‖ ⇒ ‖x, p‖ > ‖x, q‖ ∀ x ∈ DX⊗γY and(ii) k + k/ 6 1

Proof. For u, v ∈ DX⊗γY and an arbitrarya ⊗ b ∈ DX⊗γY with ‖a‖‖b‖ > 1, we have,

‖Tu − Tv, a ⊗ b‖ = ‖T1u ⊗ T2u − T1v ⊗ T2v, a ⊗ b‖6 ‖(T1u − T1v)⊗ T2u, a ⊗ b‖+ ‖T1v ⊗ (T2u − T2v), a ⊗ b‖

=1

2[‖T1u − T1v, a‖.‖T2u‖‖b‖+ ‖T2u, b‖‖T1u − T1v‖‖a‖

+‖T2u − T2v, b‖.‖T1v‖‖a‖+ ‖T1v, a‖‖T2u − T2v‖‖b‖]

61

2[

1

N22

[k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)].N2N2

+1

M2N1

[k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)]M2N1

+1

N21

[k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖)].N1N1

+1

M1N2

[k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖)]M1N2]

= (k + k/)‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)− ψ(k/‖u − v, a ⊗ b‖)6 ‖u − v, a ⊗ b‖ − [ψ(k‖u − v, a ⊗ b‖) + ψ(k/‖u − v, a ⊗ b‖)] (for k + k/ 6 1)

Let x0 ∈ DX⊗γY be fixed. We takexn+1 = Txn, n = 0, 1, 2, .... Now,

‖xn+1 − xn, a ⊗ b‖ = ‖Txn − Txn−1, a ⊗ b‖6 ‖xn − xn−1, a ⊗ b‖ − ψ(k‖xn − xn−1, a ⊗ b‖)

−ψ(k/‖xn − xn−1, a ⊗ b‖)6 ‖xn − xn−1, a ⊗ b‖

Hence{‖xn+1 − xn, a ⊗ b‖} is a monotonically decreasing sequence of non-negative real num-bers and so, is convergent to some real, sayr.Takingn → ∞, we get

r 6 r − (ψ(kr) + ψ(k/r)), (by continuity ofψ)

⇒ ψ(kr) + ψ(k/r) 6 0,

this is possible only whenr = 0. So,

limn→∞

‖xn+1 − xn, a ⊗ b‖ = 0.


Let p ⊗ q ∈ DX⊗γY be such that‖p ⊗ q‖ < 1. If

limn→∞

(xn+1 − xn) 6= k(p ⊗ q) for anyk, then

limn→∞ ‖xn+1 − xn, p ⊗ q‖ 6= 0. Again‖a ⊗ b‖ > 1 and‖p ⊗ q‖ < 1. Therefore,

‖a ⊗ b‖ > ‖p ⊗ q‖ ⇒ ‖u, a ⊗ b‖ > ‖u, p ⊗ q‖ ∀ u ∈ DX⊗γY ((by condition (i))). So,

‖xn+1 − xn, a ⊗ b‖ > ‖xn+1 − xn, p ⊗ q‖ n = 0, 1, 2, ...

Taking limit asn → ∞, we get,

limn→∞

‖xn+1 − xn, a ⊗ b‖ > limn→∞

‖xn+1 − xn, p ⊗ q‖ 6= 0, a contradiction.

So,limn→∞(xn+1 − xn) = k(p ⊗ q) for somek.Thus,limn→∞ ‖xn+1 − xn, p ⊗ q‖ = 0 for any p ⊗ q ∈ DX⊗γY with ‖p ⊗ q‖ < 1.So,limn→∞ ‖xn+1 − xn, u‖ = 0 ∀u ∈ DX⊗γY. Now, for any integerp > 0,

‖xn − xn+p, u‖ 6 ‖xn − xn+1, u‖+ ‖xn+1 − xn+2, u‖+ ... + ‖xn+p−1 − xn+p, u‖→ 0 as n → ∞

showing that{xn} is a Cauchy sequence inDX⊗γY. Let it converge to somez ∈ DX⊗γY. Now,

‖z − Tz, u‖ 6 ‖z − xn+1, u‖+ ‖xn+1 − Tz, u‖= ‖z − xn+1, u‖+ ‖Txn − Tz, u‖6 ‖z − xn+1, u‖+ ‖xn − z, u‖ − [ψ(k‖xn − z, u‖) + ψ(k/‖xn − z, u‖)]→ 0 as n → ∞

Hence,‖z − Tz, u‖ = 0 ⇒ z = Tz.To show the uniqueness:Let z1 andz2 be two distinct fixed points forT in DX⊗γY. Now,

‖z1 − z2, u‖ = ‖Tz1 − Tz2, u‖6 ‖z1 − z2, u‖

− [ψ(k‖z1 − z2, u‖) + ψ(k/‖z1 − z2, u‖)]⇒ ψ(k‖z1 − z2, u‖) + ψ(k/‖z1 − z2, u‖) 6 0,

possible only forz1 = z2 (sinceu is arbitrary).Thus,T has a unique fixed point inDX⊗γY. �

Example 2.2We define the following2-norms(in the sense of White) on the normed spacesl1

andK:‖., .‖ : K × K → R

+ ∪ {0} such that

‖x, b‖ =

{

|x||b|, if x andb are linearly independent

0, if x andb are linearly dependent

and‖., .‖ : l1 × l1 → R+ ∪ {0} such that

‖{xin}, {bin

}‖ =

{

‖{xin}‖‖{bin

}‖, if {xin}n and{bin

}n are linearly independent

0, if {xin}n and{bin

}n are linearly dependent


Using these norms we can now define a 2-norm in the projective tensor product l1 ⊗γ K, i.e.,l1(K) as:for u = ∑i xi ⊗ yi, v = ∑j pj ⊗ qj ∈ l1 ⊗γ K

‖u, v‖ =1

2∑i,j

(

‖xi, pj‖‖yi‖‖qj‖+ ‖yi, qj‖‖xi‖‖pj‖)

=1

2∑i,j

(

‖xi‖‖pj‖|yi||qj|+ ‖xi‖‖pj‖|yi||qj|)

= ‖xi‖‖pj‖|yi||qj| (for linearly independent ({xi} and{pj}) and ({yi} and{qj}))

Let Dl1 , DK andDl1⊗γK be the closed subsets of the normed spacesl1, K andl1 ⊗γ K, bounded

by the constantsK, K andK2 respectively(K > 1).We defineT1 : Dl1⊗γK → Dl1 by

T1(∑i

ai ⊗ xi) =1

2K5 ∑i

{ainxi}, whereai = {ain

}n

andT2 : Dl1⊗γK → DK by T2(∑i ai ⊗ xi) =1

4∑i ‖ai‖.|xi| .

For arbitrarybj = {bjn}n ∈ Dl1 , b ∈ DK with ‖bj‖|b| > 1

‖T1(∑i

ai ⊗ xi), bj‖ = ‖ 1

2K5 ∑i

{ainxi}, bj‖ = ‖ 1

2K5 ∑i

{ainxi}‖‖bj‖ (for the 2-norm)

61

2K5 ∑i

‖ai‖|xi|‖bj‖

So, taking the projective tensor norm inl1 ⊗γ K i.e., l1(K), we get,

‖T1(∑i

ai ⊗ xi), bj‖ 61

2K5‖∑

i

ai ⊗ xi‖‖bj‖ 61

2K5K3 =

1

2K2(= M1)

and

‖T1(∑i

ai ⊗ xi)‖ = ‖ 1

2K5 ∑i

{ainxi}‖

61

2K5 ∑i

‖ai‖|xi|

So, taking the projective tensor norm, inl1 ⊗γ K, we get,

‖T1(∑i

ai ⊗ xi)‖ 61

2K5‖∑

i

ai ⊗ xi‖ 61

2K5K2 =

1

2K3(= N1)


Similarly,

‖T2(∑i

ai ⊗ xi), b‖ = ‖1

4∑

i

‖ai‖.|xi|, b‖

=

∣

∣

∣

∣

∣

1

4∑

i

‖ai‖.|xi|∣

∣

∣

∣

∣

|b| (for the 2-norm)

61

4∑

i

‖ai‖|xi||b|

So, taking the projective tensor norm,

‖T2(∑i

ai ⊗ xi), b‖ 61

4‖∑

i

ai ⊗ xi‖|b| 6K3

4(= M2)

and

‖T2(∑i

ai ⊗ xi)‖ 61

4‖∑

i

ai ⊗ xi‖

6K2

4(= N2)

For u = ∑i ai ⊗ xi andv = ∑i di ⊗ yi in Dl1⊗γK, we have,

‖T1u − T1v, bj‖ = ‖ 1

2K5 ∑i

{ainxi} −

1

2K5 ∑i

{dinyi}‖‖bj‖

6 ‖ 1

2K5 ∑i

ai ⊗ xi −1

2K5 ∑i

di ⊗ yi‖‖bj‖‖bj‖|b| (∵ ‖bj‖|b| > 1)

=1

2K4‖u − v, bj ⊗ b‖

=

1

4‖u − v, bj ⊗ b‖

K4

2

6 8

1

2‖u − v, bj ⊗ b‖ − 1

2

[

1

2‖u − v, bj ⊗ b‖

]

K4

2

=1

N22

[

1

2‖u − v, bj ⊗ b‖ − ψ

(

1

2‖u − v, bj ⊗ b‖

)]

,

whereψ(t) =t

2, k =

1

2


and

‖T1u − T1v‖ = ‖ 1

2K5 ∑i

{ainxi} −

1

2K5 ∑i

{dinyi}‖

=

1

4‖u − v‖

K5

2

6

1

4‖u − v‖‖bj‖|b|

K5

2

6 4K5

1

2‖u − v, bj ⊗ b‖ − 1

2

[

1

2‖u − v, bj ⊗ b‖

]

K5

2

(for the 2-norm)

=

1

2‖u − v, bj ⊗ b‖ − 1

2

[

1

2‖u − v, bj ⊗ b‖

]

1

2K3

K3

4

=1

M2N1

[

1

2‖u − v, bj ⊗ b‖ − ψ

(

1

2‖u − v, bj ⊗ b‖

)]

,

whereψ(t) =t

2, k =

1

2

Again,

‖T2u − T2v, b‖ = ‖1

4∑

i

‖ai‖|xi| −1

4∑

i

‖di‖|yi|, b‖

61

4

∣

∣

∣

∣

∣

∑i

‖ai‖|xi| − ∑i

‖di‖|yi|∣

∣

∣

∣

∣

|b| ‖bj‖|b|

Taking the projective tensor norm,

‖T2u − T2v, b‖ 6K

4|‖u‖ − ‖v‖| |b| ‖bj‖

6K

4‖u − v‖‖bj‖ |b|

=K

4‖u − v, bj ⊗ b‖ (for the 2-norm)

⇒ ‖T2u − T2v, b‖ 6 4K6

(

1

4‖u − v, bj ⊗ b‖ − 1

2

[

1

4‖u − v, bj ⊗ b‖

])

(∵ K > 1)

=

1

4‖u − v, bj ⊗ b‖ − ψ

(

1

4‖u − v, bj ⊗ b‖

)

N21

,

whereψ(t) =t

2, k/ =

1

4


Also,

‖T2u − T2v‖ = ‖1

4∑

i

‖ai‖|xi| −1

4∑

i

‖di‖|yi|‖

61

4

∣

∣

∣

∣

∣

∑i

‖ai‖|xi| − ∑i

‖di‖|yi|∣

∣

∣

∣

∣

‖bj‖ |b| (∵ ‖bj‖|b| > 1)

Taking the projective tensor norm ,

‖T2u − T2v‖ 61

4|‖u‖ − ‖v‖| ‖bj‖ |b|

61

4‖u − v‖‖bj‖ |b|

=1

4‖u − v, bj ⊗ b‖ 6 ‖u − v, bj ⊗ b‖

=

1

4‖u − v, bj ⊗ b‖ − 1

2

[

1

4‖u − v, bj ⊗ b‖

]

K2

4

1

2K2

=

1

4‖u − v, bj ⊗ b‖ − ψ

(

1

4‖u − v, bj ⊗ b‖

)

M1N2

, whereψ(t) =t

2, k/ =

1

4

Therefore,(T1, T2) satisfies the conditions (A), (B), (C), (D). Also,

k + k/ =1

2+

1

4< 1

Moreover, for∑i pi ⊗ xi, ∑i qi ⊗ yi and ∑j bj ⊗ cj ∈ Dl1⊗γK,

‖∑i

pi ⊗ xi‖ > ‖∑i

qi ⊗ yi‖ ⇒ ∑i

‖pi‖|xi| > ∑i

‖qi‖|yi|

⇒ ∑j

‖bj‖|cj|∑i

‖pi‖|xi| > ∑j

‖bj‖|cj|∑i

‖qi‖|yi|

⇒ ∑i,j

‖bj‖‖pi‖|cj||xi| > ∑i,j

‖bj‖‖qi‖|cj||yi|

⇒ ‖∑j

bj ⊗ cj, ∑i

pi ⊗ xi‖ > ‖∑j

bj ⊗ cj, ∑i

qi ⊗ yi‖

So, by Theorem 2.1, the mappingT : Dl1⊗γK → Dl1⊗γK defined by

T(u) = T1(u)⊗ T2(u)

has a unique fixed point inDl1⊗γK.

Corollary 2.3 If in Theorem 2.1T1 or T2 satisfies one of the following conditions:

‖T1(u)− T1(v), a‖ 6k

M2

‖u − v, a ⊗ b‖, ‖b‖ > 1

‖T2(u)− T2(v), b‖ 6k/

M1

‖u − v, a ⊗ b‖, ‖a‖ > 1


thenT has a unique fixed point ifk + k/ 6 1.

Corollary 2.4 If T1 andT2 satisfy both the above conditions, thenT has a unique fixed point ifk + k/

< 1.

Now, we discuss common fixed points for the pair of mappings(T, S) derived by the pairs(T1, S1) and(T2, S2). For this, we replace the conditions (A), (B), (C) and (D) of the Theorem2.1 by the following:

(E) ‖T1(u)− S1(v)‖ 61

M2N1

(k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖))

(F) ‖T2(u)− S2(v)‖ 61

M1N2

(k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖))

(G) ‖T1(u)− S1(v), a‖ 61

N22

(k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖))

(H) ‖T2(u)− S2(v), b‖ 61

N21

(k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖))

Theorem 2.5The mappingsT andS derived by(T1, S1) and(T2, S2) satisfying (E), (F), (G)and (H) have a common unique fixed point inDX⊗γY, if p, q ∈ DX⊗γY

(i) ‖p‖ > ‖q‖ ⇒ ‖x, p‖ > ‖x, q‖ (∀x ∈ DX⊗γY) and(ii) k + k/ 6 1.

Proof. For u, v ∈ DX⊗γY, anda ⊗ b ∈ DX⊗γY with ‖a ⊗ b‖ > 1

‖Tu − Sv, a ⊗ b‖ = ‖T1u ⊗ T2u − S1v ⊗ S2v, a ⊗ b‖

61

2[‖T1u − S1v, a‖‖T2u‖‖b‖+ ‖T2u, b‖‖T1u − S1v‖‖a‖

+‖T2u − S2v, b‖‖S1v‖‖a‖+ ‖S1v, a‖‖T2u − S2v‖‖b‖]

61

2[

1

N22

[k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)].N2N2

+1

M2N1

[k‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)]M2N1

+1

N21

[k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖)].N1N1

+1

M1N2

[k/‖u − v, a ⊗ b‖ − ψ(k/‖u − v, a ⊗ b‖)]M1N2]

6 (k + k/)‖u − v, a ⊗ b‖ − ψ(k‖u − v, a ⊗ b‖)− ψ(k/‖u − v, a ⊗ b‖)6 ‖u − v, a ⊗ b‖ − (ψ(k‖u − v, a ⊗ b‖) + ψ(k/‖u − v, a ⊗ b‖))


Let x0 ∈ DX⊗γY be fixed. We takex1 = Sx0, x2 = Tx1, x3 = Sx2, x4 = Tx3, ... ,x2n+1 = Sx2n, x2n+2 = Tx2n+1. Now,

‖x2n+2 − x2n+1, a ⊗ b‖ = ‖Tx2n+1 − Sx2n, a ⊗ b‖6 ‖x2n+1 − x2n, a ⊗ b‖ − ψ(k‖x2n+1 − x2n, a ⊗ b‖)

−ψ(k/‖x2n+1 − x2n, a ⊗ b‖)6 ‖x2n+1 − x2n, a ⊗ b‖ (1)

Similarly,‖x2n+1 − x2n, a ⊗ b‖ 6 ‖x2n − x2n−1, a ⊗ b‖, n = 0, 1, 2, ...

Hence{‖xn+1 − xn, a ⊗ b‖} is a monotonically decreasing sequence of non-negative real num-bers and so, is convergent to some real, sayr.Takingn → ∞ in (1), we getr = 0. Now, as in Theorem 2.1, we can show that

limn→∞

‖xn+1 − xn, u‖ = 0 ∀u ∈ DX⊗γY

Now, it can be easily shown that{xn} is a Cauchy sequence inDX⊗γY. Let it converge to somez ∈ DX⊗γY. Now, proceeding as in Theorem 2.1, we can show thatz = Sz and alsoz = Tz.Uniqueness can also be shown in a similar manner.So,z is the common unique fixed point forT andS in DX⊗γY. �

Next we consider algebraic tensor product space as a 2-Banach algebra.

Let X andY be two unital Banach algebras which are also 2-Banach algebras. We show thatX ⊗Y with projective tensor norm is a 2-normed algebra (with the 2-norm alreadydefined). Foru = ∑

ni=1 xi ⊗ yi, v = ∑

mj=1 pj ⊗ qj andw = ∑

kr=1 er ⊗ fr in X ⊗γ Y we have,

‖uw, v‖ = ‖min(n,k)

∑i,r=1

xier ⊗ yi fr,m

∑j=1

pj ⊗ qj‖

=1

2

min(m,n,k)

∑i,r,j=1

(‖xier, pj‖‖yi fr‖‖qj‖+ ‖yi fr, qj‖‖xier‖‖pj‖)

61

2

min(m,n,k)

∑i,r,j=1

(K‖xi, pj‖‖er, pj‖‖yi fr‖‖qj‖+ K/‖yi, qj‖‖ fr, qj‖‖xier‖‖pj‖)

(for some constantK andK/)


Let ǫ = minj=1,2,...,m(‖pj‖, ‖qj‖), pj 6= 0, qj 6= 0. We chooseN(> 1) be such thatmax(K, K/) 6Nǫ. Now,

‖uw, v‖ 6Nǫ

2

min(m,n,k)

∑i,r,j=1

(‖xi, pj‖‖er, pj‖‖yi fr‖‖qj‖+ ‖yi, qj‖‖ fr, qj‖‖xier‖‖pj‖)

=N

2

min(m,n,k)

∑i,r,j=1

[ minj=1,2,...,m

(‖pj‖, ‖qj‖)(‖xi, pj‖‖er, pj‖‖yi fr‖‖qj‖)

+ minj=1,2,...,m

(‖pj‖, ‖qj‖)(‖yi, qj‖‖ fr, qj‖‖xier‖‖pj‖)]

6N

2

min(m,n,k)

∑i,r,j=1

(‖xi, pj‖‖yi‖‖qj‖.‖er, pj‖‖ fr‖‖qj‖+ ‖yi, qj‖‖xi‖‖pj‖.‖ fr, qj‖‖er‖‖pj‖

+ ‖xi, pj‖‖yi‖‖qj‖.‖ fr, qj‖‖er‖‖pj‖+ ‖yi, qj‖‖xi‖‖pj‖.‖er, pj‖‖ fr‖‖qj‖)

= Mmin(m,n,k)

∑i,r,j=1

1

2(‖xi, pj‖‖yi‖‖qj‖+ ‖yi, qj‖‖xi‖‖pj‖)

1

2(‖er, pj‖‖ fr‖‖qj‖+ ‖ fr, qj‖‖er‖‖pj‖)

= M‖u, v‖‖w, v‖, whereM = 2N

showing thatX ⊗ Y is a 2-normed algebra. Taking the completion with respect to the 2-norm,we can make it a 2-Banach algebra.Let (T1, T2) be a pair of mappings whereT1 : DX⊗γY → DX⊗γY andT2 : DX⊗γY → DX⊗γY

are such that for anyu, v, a, b ∈ DX⊗γY ∪ {e} with ‖a‖ > 1 and‖b‖ > 1,

(I) ‖T1(u)− T1(v), a‖ 61√

MM2

√

k‖u − v, ab‖ − ψ(k‖u − v, ab‖)

(J) ‖T2(u)− T2(v), b‖ 61√

MM1

√

k/‖u − v, ab‖ − ψ(k/‖u − v, ab‖),(for constantsM1, M2 > 0 andM is defined as above)where,

(a) ψ : [0, ∞) → [0, ∞) is continuous and non-decreasing,ψ(0) = 0

(b) ‖T1u, a‖ 6M1

Mand‖T2u, b‖ 6

M2

M. [HereDX⊗γY is a closed and bounded subspace

of X ⊗γ Y.]

From the pair(T1, T2) we define a self mappingT : DX⊗γY → DX⊗γY such thatTu = T1uT2ufor u ∈ DX⊗γY.

Theorem 2.6From the pair of mappings(T1, T2) satisfying (I) and (J) the mappingT definedabove has a unique fixed point inDX⊗γY if

(i) For p, q ∈ DX⊗γY, ‖p‖ > ‖q‖ ⇒ ‖u, p‖ > ‖u, q‖ ∀u ∈ DX⊗γY,(ii) k + k/ 6 1


Proof. For u, v, a, b ∈ DX⊗γY with ‖a‖ > 1, ‖b‖ > 1

‖Tu − Tv, ab‖ = ‖T1uT2u − T1vT2v, ab‖6 ‖(T1u − T1v)T2u, ab‖+ ‖T1v(T2u − T2v), ab‖6 M‖T1u − T1v, ab‖‖T2u, ab‖+ M‖T1v, ab‖‖T2u − T2v, ab‖6 M3‖T1u − T1v, a‖‖T1u − T1v, b‖‖T2u, a‖‖T2u, b‖

+ M3‖T1v, a‖‖T1v, b‖‖T2u − T2v, a‖‖T2u − T2v, b‖

6 M3

[

1√MM2

√

k‖u − v, ab‖ − ψ(k‖u − v, ab‖)(

M2

M

)]2

+ M3

[

1√MM1

√

k/‖u − v, ab‖ − ψ(k/‖u − v, ab‖)(

M1

M

)]2

= (k + k/)‖u − v, ab‖ − ψ(k‖u − v, ab‖)− ψ(k/‖u − v, ab‖)6 ‖u − v, ab‖ − (ψ(k‖u − v, ab‖) + ψ(k/‖u − v, ab‖))

Let x0 ∈ DX⊗γY be fixed. We takexn+1 = Txn, n = 0, 1, 2, ... Now, for arbitrarya ∈ DX⊗γY

with ‖a‖ > 1,

‖xn+1 − xn, a‖ = ‖xn+1 − xn, ae‖ = ‖Txn − Txn−1, ae‖6 ‖xn − xn−1, ae‖ − ψ(k‖xn − xn−1, ae‖)

−ψ(k/‖xn − xn−1, ae‖)6 ‖xn − xn−1, a‖

Hence{‖xn+1 − xn, a‖} is a monotonically decreasing sequence of non-negative real numbersand so, is convergent to some real, sayr.Takingn → ∞, we get

r 6 r − (ψ(kr) + ψ(k/r)), (by continuity ofψ)

⇒ ψ(kr) + ψ(k/r) 6 0,

this is possible only whenr = 0. So,limn→∞ ‖xn+1 − xn‖ = 0. Now, let p ∈ DX⊗γY be suchthat‖p‖ < 1. So,‖p‖ < ‖a‖.By condition (i),‖xn+1 − xn, p‖ < ‖xn+1 − xn, a‖, n = 0, 1, 2, ... Taking limit asn → ∞ onboth sides, we get,

limn→∞

‖xn+1 − xn, p‖ = 0.

Hence,limn→∞ ‖xn+1 − xn, u‖ = 0 ∀u ∈ DX⊗γY So,{xn} is a Cauchy sequence inDX⊗γY.DX⊗γY being closed, this sequence converges to somez ∈ DX⊗γY. Now, as in Theorem 2.1, itcan be shown that

‖z − Tz, u‖ → 0 as n → ∞,

which implies thatz = Tz. Uniqueness can be shown in a similar manner. Thus,T has a uniquefixed point inDX⊗γY. �


Example 2.7Let Dl1⊗γK be the closed and bounded (in 2-norm by some constantK) subspace

of the2-Banach algebral1 ⊗γ K with the 2-norm defined earlier. We define the self mappings

T1 andT2 on Dl1⊗γK by T1(∑i ai ⊗ xi) =1

2d∑i ai ⊗ xi and

T2(∑i ai ⊗ xi) =1

4d∑i ai ⊗ xi , whereai = {ain

}n ∈ l1, andd is a positive constant such that

d >√

2K, andd > 2.For p = ∑k bk ⊗ yk, q = ∑j cj ⊗ y/

j ∈ l1 ⊗γ K with ‖p‖ > 1, ‖q‖ > 1,

‖T1(∑i

ai ⊗ xi), ∑k

bk ⊗ yk‖ = ‖ 1

2d ∑i

ai ⊗ xi, ∑k

bk ⊗ yk‖

=1

2d

1

2∑i,k

(‖ai, bk‖‖xi‖‖yk‖+ ‖xi, yk‖‖ai‖‖bk‖)

=1

2d ∑i,k

‖ai‖‖bk‖‖xi‖‖yk‖ (for the 2-norms inl1 andK )

61

2d ∑i,k

‖ai‖‖bk‖|xi||yk|(∑j

‖cj‖|y/j |) (∵ ‖q‖ > 1)

=1

2d ∑i,k,j

‖ai‖‖bk‖‖cj‖|xi||yk||y/j |

=1

2d ∑i,k,j

‖ai‖‖bkcj‖|xi||yky/j |

=1

2d‖∑

i

ai ⊗ xi, ∑k,j

bkcj ⊗ yky/j ‖

=1

2d‖∑

i

ai ⊗ xi, pq‖

6K

2d6

1

2d

d2

2(K 6

d2

2)

=

1

2M

(=M1

M), M =

2

d, M1 =

1

2

Similarly,

‖T2(∑i

ai ⊗ xi), ∑k

bk ⊗ yk‖ 61

4d‖∑

i

ai ⊗ xi, ∑k,j

bkcj ⊗ yky/j ‖

=1

4d‖∑

i

ai ⊗ xi, pq‖ 6K

4d

=

1

4M

(=M2

M)


For arbitraryu, v ∈ Dl1⊗γK, we have,

‖T1u − T1v, p‖ = ‖ 1

2du − 1

2dv, ∑

k

bk ⊗ yk‖ 6

1

4d‖u − v, pq‖

1

2

=

√

‖ 1

4d(u − v), pq‖

√

‖ 1

4d(u − v), pq‖

1

2

6

√

‖ 1

4d(u − v), pq‖

√

1

4d‖u, pq‖+ 1

4d‖v, pq‖

1

2

6

√

‖ 1

4d(u − v), pq‖

√

K

2d1

2

6

√

‖ 1

4d(u − v), pq‖

√

d

4

1

2

6

√

1

4d‖(u − v), pq‖

1√

d

2

√

1

2

6

√

2

d

√

1

4‖(u − v), pq‖

√

2

d

, M =1

d

2

=2

d

6 4

√

1

4‖(u − v), pq‖

√M

, M =1

d

2

=2

d

6

√

1

4‖(u − v), pq‖√

MM2

‖T1u − T1v, p‖ 61√

MM2

√

[

1

2‖u − v, pq‖ − ψ

(

1

2‖u − v, pq‖

)]

, whereψ(t) =t

2, k =

1

2

Similarly,

‖T2u − T2v, p‖ 61√

MM1

√

[

1

2‖u − v, pq‖ − ψ

(

1

2‖u − v, pq‖

)]

, whereψ(t) =t

2, k/ =

1

2


Therefore,T1 andT2 satisfy the conditions (I) and (J) respectively. Also

k + k/ =1

2+

1

2= 1

Now, proceeding as in Example 2.2, we can show that forp, q ∈ l1 ⊗γ K

‖p‖ > ‖q‖ ⇒ ‖u, p‖ > ‖u, q‖ ∀u ∈ Dl1⊗γK

So, the mappingT : Dl1⊗γK → Dl1⊗γK defined byT(u) = T1uT2u has a unique fixed pointin Dl1⊗γK.

2. CONCLUDING REMARKS:

In 1973 [2], Gahler, et al. introduced the concept of 2-inner product spaces.For a linear space X of dimension greater than 1 let(., .|.) be a real-valued function onX × X ×X which satisfies the following conditions:

(i) (x, x|z) > 0; (x, x|z) = 0 if and only if x andz are linearly dependent;(ii) (x, x|z) = (z, z|x) ;

(iii) (x, y|z) = (y, x|z) ;(iv) (αx, y|z) = α(x, y|z) for any real numberα;(v) (x + x/, y|z) = (x, y|z) + (x/, y|z), x, x/, y, z ∈ X .

(., .|.) is called a 2-inner product and(X, (., .|.)) a 2-inner product space.

Different results on 2-inner product spaces and linear 2-normed spaces can be found in [1], [20].

Here we can raise the following problem:

Can we establish some analogous fixed point theorems for self mappings on 2-inner productspaces and also their tensor products?

REFERENCES

[1] Y. J. Cho and S. S. Kim,Gateaux derivatives and 2-inner product spaces, Glasnik Mat. 27(1992), 271-282[2] C. Dimininnie, S. Gahler and A. White,2-inner product spaces, Demon-stratio Math., 6(1973), 525-536.[3] D. Das, N. Goswami,Fixed Points of Mapping Satisfying a Weakly Contractive Type Condition, Journal of Math.

Res. with Appl., Vol 36(2016), No. 1 pp. 70-78[4] D. Das, N. Goswami,Fixed Points of Mapping Satisfying a Weakly Contractive Type Condition, Journal of Math.

Res. with Appl., Vol 36(2016), No. 1 pp. 70-78[5] D. Das, N. Goswami and Vandana,Some Fixed Point Theorems in 2-Banach Spaces and 2-normed Tensor Prod-

uct Spaces, New Trends in Mathematical Sciences(to appear)[6] F. F. Bonsal and J. Duncan,Complete Normed algebras, Springer-Verlag, Berlin Heidelberg New York, 1973[7] S. Gahler,2-metrische Raiume und ihre topologische Strucktur, Math. Nachr. 26 (1963), 115-148.[8] S. Gahler,Lineare 2-normierte Raume, Math. Nachr. 28 (1965), 1-43.[9] O. Hadzic, On Common Fixed Point Theorems in 2-metric Spaces, Review of Research Faculty of Science-

University of Novi Sad, Vol 12(1982)[10] K. Iseki, Fixed point theorems in 2-metric space, Math.Seminar.Notes, Kobe Univ.,3(1975), 133 - 136.

MR0405395[11] M.S. Khan and M.D. Khan,Involutions with Fixed Points in2-Banach SpacesInternat. J. Math. and Math. Sci.

VOL. 16 NO. 3 (1993) 429-434[12] S. S. Kim, Y. J. Cho, A. White,Linear Operators on Linear 2-Normed Spaces, Glasnik Matematicki, vol 27(47)

(1992), 63-70[13] N. Mahammad, and A.H. Siddiqui,On 2-Banach Algebras, International Centre for theoretical Physics,

Preprint, 1988.[14] A. Raymond Ryan,Introduction to Tensor Product of Banach Spaces, London, Springer -Verlag,2002.


[15] B.E. Rhoades,Contractive type mappings on a 2-metric space, Math.Nachr.,91(1979), 151 - 155. MR0563606[16] S. Radenovic, Z. Kadelburg, D. Jandrli’c, A. Jandrli’c,Some Results on Weakly Contractive maps, Bull. of the

Iranian Math. Soc., 38(3) (2012), 625-645[17] H. Huang, S. Radenovic, Common fixed point theorems of generalized Lipschitz mappings in cone b-metric

spaces over Banach algebras and applications, J. Nonlinear Sci. Appl. 8 (2015), 787-799[18] V. C. Rajic, S. Radenovic, S. Chauhan,Common Fixed Point of Generalized Weakly Contractive Maps in Partial

Metric Spaces, Acta Mathematica Scientia, 34B (4) (2014), 1345-1356[19] A. White,2-Banach spaces, Math.Nachr., 42(1969), 43-60. MR0257716 (41 2365) Zbl 0185.20003[20] A. White, Y. J. Cho and S. S. KimCharacterizations of 2-inner product spaces, Math. Japon., 47(1988), 239-

244.

DEPARTMENT OFMATHEMATICS, GAUHATI UNIVERSITY, GUWAHATI -781014, ASSAM, INDIA

E-mail address: [email protected]

DEPARTMENT OFMATHEMATICS, GAUHATI UNIVERSITY, GUWAHATI -781014, ASSAM, INDIA ,E-mail address: nila−[email protected]

DEPARTMENT OFMATHEMATICS, APPLIEDMATHEMATICS AND HUMANITIES DEPARTMENT, SARDAR VAL -LABHBHAI NATIONAL INSTITUTE OF TECHNOL- OGY, ICHCHHANATH MAHADEV DUMAS ROAD, SURAT 395007, GUJARAT, INDIA , [.2CM] L. 1627 AWADH PURI COLONY BENIGANJ, PHASE -III, O PPOSITE- INDUS-TRIAL TRAINING INSTITUTE (I.T.I.), AYODHYA MAIN ROAD FAIZABAD 224 001, UTTAR PRADESH, INDIA ,

E-mail address: [email protected], vishnu [email protected]

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