Global Journal of Advanced Research
on Classical and Modern Geometries
ISSN: 2284-5569, Vol.6, (2017), Issue 1, pp.20-36
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF
2-BANACH SPACES
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
A BSTRACT. Let X and Y be two normed spaces which are also 2-Banach spaces. For the projective tensor product X ⊗γ Y, we have defined a 2-norm for which X ⊗γ Y is a 2-Banach space.
Considering the closed and bounded subspace DX , DY and DX ⊗γ Y of X, Y and X ⊗γ Y respectively, we take two pairs of mappings T1 , S1 : DX ⊗γ Y → DX and T2 , S2 : DX ⊗γ Y → DY
satisfying some specific characteristics. Using these two pairs we define a pair of self mappings
T and S on DX ⊗γ Y . Some fixed point theorems for T and S on DX ⊗γ Y are derived here with
suitable example. Moreover, taking X ⊗γ Y as a 2-Banach algebra, we establish another fixed
point theorem for a self mapping T̂ on DX ⊗γ Y with a proper example.
1. I NTRODUCTION
In 1963, Gähler [7] introduced the notion of 2-metric spaces and and their topological structures.
The concept of 2-normed linear spaces can be found in Gähler’s paper [8] in 1965. Gähler investigated a lot of results in this space and also proved that if ( X, k.k) is a linear normed space
then a 2-norm can be defined on X.
In 1969, White [19] introduced the concept of 2-Banach space and gave some important definitions and examples. Since then a lot of applications of such spaces can be found in various
aspects.
Many authors viz., Iśeki [10], Khan and Khan [11], Rhoads [15], Hadžić [9] etc. developed the
applications 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 derive
some fixed point theorems here.
P RELIMINARIES
Definition 1.1 [8] Let X be a real linear space of dimension greater than 1 and let k., .k be a
real valued function on X × X satisfying the following conditions:
(i) k x, yk = 0 if and only if x and y are linearly dependent,
(ii) k x, yk = ky, x k for all x, y ∈ X,
(iii) kαx, yk = |α|k x, yk, α being real, x, y ∈ X,
(iv) k x, y + zk 6 k x, yk + k x, zk, for all, x, y, z ∈ X
2010 Mathematics 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 k., .k is called a 2-norm on X and ( X, k., .k) is called a linear 2-normed space.
In 1988, N. Mohammad, and A.H. Siddiqui [13] introduced the concept of 2-Banach Algebra.
A 2-Banach algebra ( X, k., .k) is a real algebra (of dimension greater than 2) which is a 2Banach space (with respect to 2-norm topology) and in addition, the following condition holds:
k a, bck 6 Mk a, bkk a, ck, M > 0 ∀ a, b, c ∈ X.
Definition 1.2 A sequence { xn } in a 2-normed space ( X, k., .k) is said to be a Cauchy sequence
if limn,m→∞ k xn − xm , ak = 0 for all a in X.
Definition 1.3 A sequence { xn } in a 2-normed space X is called a convergent sequence if there
is an x in X such that limn→∞ k xn − x, ak = 0 for all a in X.
Definition 1.4 A 2-normed space in which every Cauchy sequence is convergent is called a 2Banach space.
Definition 1.5 Let X and Y be two linear 2-normed spaces. An operator T : X → Y is said
to be continuous at x ∈ X if for every sequence { xn } in X, { xn } → x as n → ∞ implies
{ T ( xn )} → T ( x ) in Y as n → ∞ .
Example [19] Let Pn denotes the set of all real polynomials of degree 6 n, on the interval
[0, 1]. Clearly, Pn is a linear vector space over the reals with respect to usual addition and scalar
multiplication,. Let { x0 , x1 , ..., x2n } be distinct fixed points in [0, 1] and g, h ∈ Pn . Define the
following 2-norm on Pn :
(
∑2n
i =0 | g ( xi ) h ( xi )|, if g and h are linearly independent
k g, hk =
0,
if g and h are linearly dependent
then ( Pn , k., .k) is a 2-Banach space.
Algebric tensor product: [6] Let X, Y be normed spaces over F with dual spaces X ∗ and Y ∗
respectively. Given x ∈ X, y ∈ Y, Let x ⊗ y be the element of BL( X ∗ , Y ∗ ; F ) (which is the set
of all bounded bilinear forms from X ∗ × Y ∗ to F), defined by
x ⊗ y ( f , g ) = f ( x ) g ( y ), ( f ∈ X ∗ , g ∈ Y ∗ )
The algebraic tensor product of X and Y, 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 spaces X and Y, the projective tensor norm γ on
X ⊗ Y is defined by
kukγ = inf{∑ k xi kkyi k : u = ∑ xi ⊗ yi }
i
i
where the infimum is taken over all (finite) representations of u.
Lemma: [6] X ⊗γ Y can be represented as a linear subspace of BL( X ∗ , Y ∗ ; F ) consisting
of all elements of the form u = ∑i xi ⊗ yi where ∑i k xi kkyi k < ∞. Moreover, kukγ =
inf{∑i k xi kkyi k} over all such representations of u.
Let X and Y be both normed as well as 2-normed spaces. For the projective tensor product
X ⊗γ Y we define another 2-norm as follows:
for u = ∑i xi ⊗ yi , v = ∑ j p j ⊗ q j and w = ∑k gk ⊗ hk
22
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA

1


 ∑i,j k xi , p j kkyi kkq j k + kyi , q j kk xi kk p j k ,
2
ku, vk =
i f u and v are linearly independent



0, i f u and v are linearly dependent
(i) If u and v are linearly dependent then by definition, ku, vk = 0.
If ku, vk = 0 then k xi , p j kkyi kkq j k = 0 and kyi , q j kk xi kk p j k = 0 ∀i, j.
If k xi , p j k 6= 0 then kyi k = 0 or kq j k = 0 ⇒ yi = 0 or q j = 0 ∀i, j.
(Clearly then xi 6= 0 and p j 6= 0 otherwise k xi , p j k = 0)
⇒ u = 0 or v = 0 which shows that u and v are linearly dependent.
Similarly, if kyi , q j k 6= 0 then u = 0 or v = 0 and thus u and v are linearly dependent.
Now, let k xi , p j k = 0 and kyi , q j k = 0 ∀ i, j.
⇒ xi and p j are linearly dependent and yi and q j are linearly dependent ∀ i, j(being
2-norms).
This also implies that xi , x j ’s are linearly dependent ∀ i, j, and yi , y j ’s are linearly dependent ∀ i, j.
Therefore, u can be expressed as: u = α( x1 ⊗ y1 ), for some scalar α.
Similarly, pi , p j ’s are linearly dependent ∀ i, j, and qi , q j ’s are linearly dependent ∀ i, j.
So, we can take v = β( p1 ⊗ q1 ), for some scalar β. But x1 , p1 and y1 , q1 are linearly
dependent. Hence, u and v are linearly dependent.
(ii) ku, vk = kv, uk
(iii) ku, αvk = |α|ku, vk, α is a scalar
(iv) ku, v + wk 6 ku, vk + ku, wk (all these properties follow by definition.)
(Moreover, if X and Y are two 2-Banach spaces, then X ⊗γ Y is also a 2-Banach space for the
above 2-norm).
Let DX , DY and DX ⊗γ Y denote closed and bounded subsets of X, Y and X ⊗γ Y respectively.
Let T1 , S1 : DX ⊗γ Y → DX and T2 , S2 : DX ⊗γ Y → DY be such that for any u, v ∈ DX ⊗γ Y and
a ⊗ b ∈ DX ⊗γ Y with k akkbk > 1 and positive k, k/ ,
1
(kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk))
M2 N1
1
k T2 (u) − T2 (v)k 6
(k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk))
M1 N2
1
k T1 (u) − T1 (v), ak 6 2 (kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk))
N2
1
k T2 (u) − T2 (v), bk 6 2 (k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk))
N1
where,
ψ : [0, ∞) → [0, ∞) is continuous and non-decreasing, ψ(0) = 0,
max[k T1 uk, kS1 uk] 6 N1 and max[k T2 uk, kS2 uk] 6 N2 (where u ∈ DX ⊗γ Y ). (Here
DX , DY and DX ⊗γ Y are bounded in norm by N1 , N2 and N1 N2 respectively.)
max[k T1 u, ak, kS1 u, ak] 6 M1 and max[k T2 u, bk, kS2 u, bk] 6 M2 (where u ∈
DX ⊗γ Y ). (Here DX , DY and DX ⊗γ Y are bounded in 2-norm by M1 , M2 and M1 M2
respectively.)
(A) k T1 (u) − T1 (v)k 6
(B)
(C)
(D)
(a)
(b)
(c)
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
23
From the pairs ( T1 , S1 ) and ( T2 , S2 ), we define self mappings T and S on DX ⊗γ Y as earlier such
that Tu = T1 u ⊗ T2 u and Su = S1 u ⊗ S2 u ∀u ∈ DX ⊗γ Y .
M AIN R ESULTS
Theorem 2.1 For normed spaces X and Y which are also 2-Banach spaces, the mapping T
derived by the pair of mappings ( T1 , T2 ) satisfying (A), (B), (C) and (D) has a unique fixed
point in DX ⊗γ Y if
(i) p, q ∈ DX ⊗γ Y with k pk > kqk ⇒ k x, pk > k x, qk ∀ x ∈ DX ⊗γ Y and
(ii) k + k/ 6 1
Proof. For u, v ∈ DX ⊗γ Y and an arbitrary a ⊗ b ∈ DX ⊗γ Y with k akkbk > 1, we have,
k Tu − Tv, a ⊗ bk = k T1 u ⊗ T2 u − T1 v ⊗ T2 v, a ⊗ bk
6 k( T1 u − T1 v) ⊗ T2 u, a ⊗ bk + k T1 v ⊗ ( T2 u − T2 v), a ⊗ bk
1
[k T1 u − T1 v, ak.k T2 ukkbk + k T2 u, bkk T1 u − T1 vkk ak
=
2
+k T2 u − T2 v, bk.k T1 vkk ak + k T1 v, akk T2 u − T2 vkkbk]
6
1 1
[ [kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk)].N2 N2
2 N22
1
+
[kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk)] M2 N1
M2 N1
1
+ 2 [k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk)].N1 N1
N1
1
[k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk)] M1 N2 ]
+
M1 N2
= (k + k/ )ku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk) − ψ(k/ ku − v, a ⊗ bk)
6 ku − v, a ⊗ bk − [ψ(kku − v, a ⊗ bk) + ψ(k/ ku − v, a ⊗ bk)] (for k + k/ 6 1)
Let x0 ∈ DX ⊗γ Y be fixed. We take xn+1 = Txn , n = 0, 1, 2, .... Now,
k xn+1 − xn , a ⊗ bk = k Txn − Txn−1 , a ⊗ bk
6 k xn − xn−1 , a ⊗ bk − ψ(kk xn − xn−1 , a ⊗ bk)
−ψ(k/ k xn − xn−1 , a ⊗ bk)
6 k x n − x n −1 , a ⊗ b k
Hence {k xn+1 − xn , a ⊗ bk} is a monotonically decreasing sequence of non-negative real numbers and so, is convergent to some real, say r.
Taking n → ∞, we get
r 6 r − (ψ(kr ) + ψ(k/ r )), (by continuity of ψ)
⇒ ψ(kr ) + ψ(k/ r ) 6 0,
this is possible only when r = 0. So,
lim k xn+1 − xn , a ⊗ bk = 0.
n→∞
24
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
Let p ⊗ q ∈ DX ⊗γ Y be such that k p ⊗ qk < 1. If
lim ( xn+1 − xn ) 6= k( p ⊗ q) for any k, then
n→∞
limn→∞ k xn+1 − xn , p ⊗ qk 6= 0. Again k a ⊗ bk > 1 and k p ⊗ qk < 1. Therefore,
k a ⊗ bk > k p ⊗ qk ⇒ ku, a ⊗ bk > ku, p ⊗ qk ∀ u ∈ DX ⊗γ Y ((by condition (i))). So,
k xn+1 − xn , a ⊗ bk > k xn+1 − xn , p ⊗ qk n = 0, 1, 2, ...
Taking limit as n → ∞, we get,
lim k xn+1 − xn , a ⊗ bk > lim k xn+1 − xn , p ⊗ qk 6= 0, a contradiction.
n→∞
n→∞
So, limn→∞ ( xn+1 − xn ) = k( p ⊗ q) for some k.
Thus, limn→∞ k xn+1 − xn , p ⊗ qk = 0 for any p ⊗ q ∈ DX ⊗γ Y with k p ⊗ qk < 1.
So, limn→∞ k xn+1 − xn , uk = 0 ∀u ∈ DX ⊗γ Y . Now, for any integer p > 0,
k xn − xn+ p , uk 6 k xn − xn+1 , uk + k xn+1 − xn+2 , uk + ... + k xn+ p−1 − xn+ p , uk
→ 0 as n → ∞
showing that { xn } is a Cauchy sequence in DX ⊗γ Y . Let it converge to some z ∈ DX ⊗γ Y . Now,
kz − Tz, uk 6 kz − xn+1 , uk + k xn+1 − Tz, uk
= kz − xn+1 , uk + k Txn − Tz, uk
6 kz − xn+1 , uk + k xn − z, uk − [ψ(kk xn − z, uk) + ψ(k/ k xn − z, uk)]
→ 0 as n → ∞
Hence, kz − Tz, uk = 0 ⇒ z = Tz.
To show the uniqueness:
Let z1 and z2 be two distinct fixed points for T in DX ⊗γ Y . Now,
kz1 − z2 , uk = k Tz1 − Tz2 , uk
6 k z1 − z2 , u k
⇒ ψ(kkz1 − z2 , uk) + ψ(k/ kz1 − z2 , uk) 6 0,
− [ψ(kkz1 − z2 , uk) + ψ(k/ kz1 − z2 , uk)]
possible only for z1 = z2 (since u is arbitrary).
Thus, T has a unique fixed point in DX ⊗γ Y .
Example 2.2 We define the following 2-norms(in the sense of White) on the normed spaces l 1
and K:
k., .k : K × K → R + ∪ {0} such that
(
| x ||b|, if x and b are linearly independent
k x, bk =
0,
if x and b are linearly dependent
and k., .k : l 1 × l 1 → R + ∪ {0} such that
(
k{ xin }kk{bin }k, if { xin }n and {bin }n are linearly independent
k{ xin }, {bin }k =
0,
if { xin }n and {bin }n are linearly dependent
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
25
Using these norms we can now define a 2-norm in the projective tensor product l 1 ⊗γ K, i.e.,
l 1 (K ) as:
for u = ∑i xi ⊗ yi , v = ∑ j p j ⊗ q j ∈ l 1 ⊗γ K
ku, vk =
=
1
k xi , p j kkyi kkq j k + kyi , q j kk xi kk p j k
∑
2 i,j
1
k xi kk p j k|yi ||q j | + k xi kk p j k|yi ||q j |
∑
2 i,j
= k xi kk p j k|yi ||q j | (for linearly independent ({ xi } and { p j }) and ({yi } and {q j }))
Let Dl 1 , DK and Dl 1 ⊗γ K be the closed subsets of the normed spaces l 1 , K and l 1 ⊗γ K, bounded
by the constants K, K and K2 respectively(K > 1).
We define T1 : Dl 1 ⊗γ K → Dl 1 by
T1 (∑ ai ⊗ xi ) =
i
1
2K5
∑ { ai
n
i
xi }, where ai = { ain }n
1
∑ k a i k.| x i | .
4 i
For arbitrary b j = {b jn }n ∈ Dl 1 , b ∈ DK with kb j k|b| > 1
and T2 : Dl 1 ⊗γ K → DK
T2 (∑i ai ⊗ xi ) =
by
k T1 (∑ ai ⊗ xi ), b j k = k
i
6
1
2K5
1
2K5
∑ { ai
i
n
xi }, b j k = k
1
2K5
∑ { ai
i
n
xi }kkb j k (for the 2-norm)
∑ kai k| xi |kbj k
i
So, taking the projective tensor norm in l 1 ⊗γ K i.e., l 1 (K ), we get,
1
1
1 3
k ∑ ai ⊗ xi kkb j k 6
K =
(= M1 )
5
5
2K
2K
2K2
i
k T1 (∑ ai ⊗ xi ), b j k 6
i
and
k T1 (∑ ai ⊗ xi )k = k
i
6
1
2K5
1
2K5
∑ { ai
i
n
xi }k
∑ kai k| xi |
i
So, taking the projective tensor norm, in l 1 ⊗γ K, we get,
k T1 (∑ ai ⊗ xi )k 6
i
1
1 2
1
k ∑ ai ⊗ xi k 6
K =
(= N1 )
5
5
2K
2K
2K3
i
26
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
Similarly,
1
k T2 (∑ ai ⊗ xi ), bk = k ∑ k ai k.| xi |, bk
4 i
i
1
= ∑ k ai k.| xi | |b| (for the 2-norm)
4 i
6
1
k ai k| xi ||b|
4∑
i
So, taking the projective tensor norm,
K3
1
(= M2 )
k T2 (∑ ai ⊗ xi ), bk 6 k ∑ ai ⊗ xi k|b| 6
4 i
4
i
and
1
k T2 (∑ ai ⊗ xi )k 6 k ∑ ai ⊗ xi k
4 i
i
6
K2
(= N2 )
4
For u = ∑i ai ⊗ xi and v = ∑i di ⊗ yi in Dl 1 ⊗γ K , we have,
k T1 u − T1 v, b j k = k
1
2K5
∑ { ai
6 k
1
2K5
∑ ai ⊗ xi − 2K5 ∑ di ⊗ yi kkbj kkbj k|b| (∵ kbj k|b| > 1)
=
=
=
i
n
xi } −
1
2K5
∑ { di
i
n
yi }kkb j k
1
i
i
1
ku − v, b j ⊗ bk
2K4


1
1 1
1
ku − v, b j ⊗ bk
 2 ku − v, b j ⊗ bk − 2 2 ku − v, b j ⊗ bk 
4

6 8


K4
K4
2
2
1
1 1
ku − v, b j ⊗ bk − ψ
ku − v, b j ⊗ bk ,
2
N22 2
1
t
where ψ(t) = , k =
2
2
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
and
k T1 u − T1 vk = k
1
2K5
∑ { ai
i
n
xi } −
1
2K5
∑ { di
i
n
yi }k
1
1
ku − vk
ku − vkkb j k|b|
4
4
=
6
K5
K5
2
2


1 1
1
 ku − v, b j ⊗ bk − 2 2 ku − v, b j ⊗ bk 
52
 (for the 2-norm)
6 4K 

K5

2
1
1 1
 2 ku − v, b j ⊗ bk − 2 2 ku − v, b j ⊗ bk 

= 


1 K3
2K3 4

=
1
1
1
ku − v, b j ⊗ bk − ψ
ku − v, b j ⊗ bk ,
M2 N1 2
2
1
t
where ψ(t) = , k =
2
2
Again,
1
1
k ai k| xi | − ∑ kdi k|yi |, bk
∑
4 i
4 i
1 6 ∑ k ai k| xi | − ∑ kdi k|yi | |b| kb j k|b|
4 i
i
k T2 u − T2 v, bk = k
Taking the projective tensor norm,
K
|kuk − kvk| |b| kb j k
4
K
6 ku − vkkb j k |b|
4
K
= ku − v, b j ⊗ bk (for the 2-norm)
4 1 1
6 1
⇒ k T2 u − T2 v, bk 6 4K
ku − v, b j ⊗ bk −
ku − v, b j ⊗ bk
(∵ K > 1)
4
2 4
1
1
ku − v, b j ⊗ bk − ψ
ku − v, b j ⊗ bk
4
4
,
=
N12
1
t
where ψ(t) = , k/ =
2
4
k T2 u − T2 v, bk 6
27
28
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
Also,
1
1
k ai k| xi | − ∑ kdi k|yi |k
∑
4 i
4 i
1 6 ∑ k ai k| xi | − ∑ kdi k|yi | kb j k |b| (∵ kb j k|b| > 1)
4 i
i
k T2 u − T2 vk = k
Taking the projective tensor norm ,
1
k T2 u − T2 vk 6 |kuk − kvk| kb j k |b|
4
1
6 ku − vkkb j k |b|
4
1
= ku − v, b j ⊗ bk 6 ku − v, b j ⊗ bk
4
1 1
1
ku − v, b j ⊗ bk −
ku − v, b j ⊗ bk
4
2 4
=
K2 1
2
4 2K
1
1
ku − v, b j ⊗ bk − ψ
ku − v, b j ⊗ bk
1
t
4
4
=
, where ψ(t) = , k/ =
M1 N2
2
4
Therefore, ( T1 , T2 ) satisfies the conditions (A), (B), (C), (D). Also,
1 1
k + k/ = + < 1
2 4
Moreover, for ∑i pi ⊗ xi , ∑i qi ⊗ yi and ∑ j b j ⊗ c j ∈ Dl 1 ⊗γ K ,
k ∑ pi ⊗ xi k > k ∑ qi ⊗ yi k ⇒ ∑ k pi k| xi | > ∑ kqi k|yi |
i
i
i
i
⇒ ∑ kb j k|c j | ∑ k pi k| xi | > ∑ kb j k|c j | ∑ kqi k|yi |
j
i
j
i
⇒ ∑ kb j kk pi k|c j || xi | > ∑ kb j kkqi k|c j ||yi |
i,j
i,j
⇒ k ∑ b j ⊗ c j , ∑ pi ⊗ xi k > k ∑ b j ⊗ c j , ∑ qi ⊗ yi k
j
i
j
i
So, by Theorem 2.1, the mapping T : Dl 1 ⊗γ K → Dl 1 ⊗γ K defined by
has a unique fixed point in Dl 1 ⊗γ K .
T (u) = T1 (u) ⊗ T2 (u)
Corollary 2.3 If in Theorem 2.1 T1 or T2 satisfies one of the following conditions:
k
ku − v, a ⊗ bk, kbk > 1
k T1 (u) − T1 (v), ak 6
M2
k/
ku − v, a ⊗ bk, k ak > 1
k T2 (u) − T2 (v), bk 6
M1
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
29
then T has a unique fixed point if k + k/ 6 1.
Corollary 2.4 If T1 and T2 satisfy both the above conditions, then T has a unique fixed point if
k + 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 Theorem
2.1 by the following:
1
(kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk))
M2 N1
1
(F) k T2 (u) − S2 (v)k 6
(k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk))
M1 N2
1
(G) k T1 (u) − S1 (v), ak 6 2 (kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk))
N2
1
(H) k T2 (u) − S2 (v), bk 6 2 (k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk))
N1
(E) k T1 (u) − S1 (v)k 6
Theorem 2.5 The mappings T and S derived by ( T1 , S1 ) and ( T2 , S2 ) satisfying (E), (F), (G)
and (H) have a common unique fixed point in DX ⊗γ Y , if p, q ∈ DX ⊗γ Y
(i) k pk > kqk ⇒ k x, pk > k x, qk (∀ x ∈ DX ⊗γ Y ) and
(ii) k + k/ 6 1.
Proof. For u, v ∈ DX ⊗γ Y , and a ⊗ b ∈ DX ⊗γ Y with k a ⊗ bk > 1
k Tu − Sv, a ⊗ bk = k T1 u ⊗ T2 u − S1 v ⊗ S2 v, a ⊗ bk
1
6
[k T1 u − S1 v, akk T2 ukkbk + k T2 u, bkk T1 u − S1 vkk ak
2
+k T2 u − S2 v, bkkS1 vkk ak + kS1 v, akk T2 u − S2 vkkbk]
6
1 1
[ [kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk)].N2 N2
2 N22
1
[kku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk)] M2 N1
+
M2 N1
1
+ 2 [k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk)].N1 N1
N1
1
[k/ ku − v, a ⊗ bk − ψ(k/ ku − v, a ⊗ bk)] M1 N2 ]
+
M1 N2
6 (k + k/ )ku − v, a ⊗ bk − ψ(kku − v, a ⊗ bk) − ψ(k/ ku − v, a ⊗ bk)
6 ku − v, a ⊗ bk − (ψ(kku − v, a ⊗ bk) + ψ(k/ ku − v, a ⊗ bk))
30
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
Let x0 ∈ DX ⊗γ Y be fixed. We take x1 = Sx0 , x2 = Tx1 , x3 = Sx2 , x4 = Tx3 , ... ,
x2n+1 = Sx2n , x2n+2 = Tx2n+1 . Now,
k x2n+2 − x2n+1 , a ⊗ bk = k Tx2n+1 − Sx2n , a ⊗ bk
6 k x2n+1 − x2n , a ⊗ bk − ψ(kk x2n+1 − x2n , a ⊗ bk)
−ψ(k/ k x2n+1 − x2n , a ⊗ bk)
6 k x2n+1 − x2n , a ⊗ bk
Similarly,k x2n+1 − x2n , a ⊗ bk 6 k x2n − x2n−1 , a ⊗ bk, n = 0, 1, 2, ...
(1)
Hence {k xn+1 − xn , a ⊗ bk} is a monotonically decreasing sequence of non-negative real numbers and so, is convergent to some real, say r.
Taking n → ∞ in (1), we get r = 0. Now, as in Theorem 2.1, we can show that
lim k xn+1 − xn , uk = 0 ∀u ∈ DX ⊗γ Y
n→∞
Now, it can be easily shown that { xn } is a Cauchy sequence in DX ⊗γ Y . Let it converge to some
z ∈ DX ⊗γ Y . Now, proceeding as in Theorem 2.1, we can show that z = Sz and also z = Tz.
Uniqueness can also be shown in a similar manner.
So, z is the common unique fixed point for T and S in DX ⊗γ Y .
Next we consider algebraic tensor product space as a 2-Banach algebra.
Let X and Y be two unital Banach algebras which are also 2-Banach algebras. We show that
X ⊗ Y with projective tensor norm is a 2-normed algebra (with the 2-norm already defined). For
k
u = ∑in=1 xi ⊗ yi , v = ∑m
j=1 p j ⊗ q j and w = ∑r =1 er ⊗ f r in X ⊗ γ Y we have,
min(n,k)
kuw, vk = k
∑
i,r =1
m
x i er ⊗ y i f r , ∑ p j ⊗ q j k
j =1
1
2
min(m,n,k)
1
6
2
min(m,n,k)
=
∑
i,r,j=1
∑
i,r,j=1
(k xi er , p j kkyi f r kkq j k + kyi f r , q j kk xi er kk p j k)
(K k xi , p j kker , p j kkyi f r kkq j k + K / kyi , q j kk f r , q j kk xi er kk p j k)
(for some constant K and K / )
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
31
Let ǫ = min j=1,2,...,m (k p j k, kq j k), p j 6= 0, q j 6= 0. We choose N(> 1) be such that max(K, K / ) 6
Nǫ. Now,
kuw, vk 6
=
Nǫ
2
N
2
min(m,n,k)
∑
(k xi , p j kker , p j kkyi f r kkq j k + kyi , q j kk f r , q j kk xi er kk p j k)
i,r,j=1
min(m,n,k)
∑
i,r,j=1
[ min (k p j k, kq j k)(k xi , p j kker , p j kkyi f r kkq j k)
j=1,2,...,m
+ min (k p j k, kq j k)(kyi , q j kk f r , q j kk xi er kk p j k)]
j=1,2,...,m
6
N
2
min(m,n,k)
∑
i,r,j=1
(k xi , p j kkyi kkq j k.ker , p j kk f r kkq j k + kyi , q j kk xi kk p j k.k f r , q j kker kk p j k
+ k xi , p j kkyi kkq j k.k f r , q j kker kk p j k + kyi , q j kk xi kk p j k.ker , p j kk f r kkq j k)
min(m,n,k)
=M
∑
i,r,j=1
1
1
(k xi , p j kkyi kkq j k + kyi , q j kk xi kk p j k) (ker , p j kk f r kkq j k + k f r , q j kker kk p j k)
2
2
= Mku, vkkw, vk, where M = 2N
showing that X ⊗ 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 where T1 : DX ⊗γ Y → DX ⊗γ Y and T2 : DX ⊗γ Y → DX ⊗γ Y
are such that for any u, v, a, b ∈ DX ⊗γ Y ∪ {e} with k ak > 1 and kbk > 1,
(I) k T1 (u) − T1 (v), ak 6 √
1
p
MM2
p
1
kku − v, abk − ψ(kku − v, abk)
k/ ku − v, abk − ψ(k/ ku − v, abk),
MM1
(for constants M1 , M2 > 0 and M is defined as above)
where,
(a) ψ : [0, ∞) → [0, ∞) is continuous and non-decreasing, ψ(0) = 0
M2
M1
and k T2 u, bk 6
. [Here DX ⊗γ Y is a closed and bounded subspace
(b) k T1 u, ak 6
M
M
of X ⊗γ Y.]
(J) k T2 (u) − T2 (v), bk 6 √
From the pair ( T1 , T2 ) we define a self mapping T̂ : DX ⊗γ Y → DX ⊗γ Y such that T̂u = T1 uT2 u
for u ∈ DX ⊗γ Y .
Theorem 2.6 From the pair of mappings ( T1 , T2 ) satisfying (I) and (J) the mapping T̂ defined
above has a unique fixed point in DX ⊗γ Y if
(i) For p, q ∈ DX ⊗γ Y , k pk > kqk ⇒ ku, pk > ku, qk ∀u ∈ DX ⊗γ Y ,
(ii) k + k/ 6 1
32
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
Proof. For u, v, a, b ∈ DX ⊗γ Y with k ak > 1, kbk > 1
k T̂u − T̂v, abk = k T1 uT2 u − T1 vT2 v, abk
6 k( T1 u − T1 v) T2 u, abk + k T1 v( T2 u − T2 v), abk
6 Mk T1 u − T1 v, abkk T2 u, abk + Mk T1 v, abkk T2 u − T2 v, abk
6 M3 k T1 u − T1 v, akk T1 u − T1 v, bkk T2 u, akk T2 u, bk
+ M3 k T1 v, akk T1 v, bkk T2 u − T2 v, akk T2 u − T2 v, bk
q
M2 2
1
kku − v, abk − ψ(kku − v, abk)
6 M3 √
M
MM2
q
1
M1 2
3
/
/
+M √
k ku − v, abk − ψ(k ku − v, abk)
M
MM1
= (k + k/ )ku − v, abk − ψ(kku − v, abk) − ψ(k/ ku − v, abk)
6 ku − v, abk − (ψ(kku − v, abk) + ψ(k/ ku − v, abk))
Let x0 ∈ DX ⊗γ Y be fixed. We take xn+1 = T̂xn , n = 0, 1, 2, ... Now, for arbitrary a ∈ DX ⊗γ Y
with k ak > 1,
k xn+1 − xn , ak = k xn+1 − xn , aek = k T̂xn − T̂xn−1 , aek
6 k xn − xn−1 , aek − ψ(kk xn − xn−1 , aek)
−ψ(k/ k xn − xn−1 , aek)
6 k x n − x n −1 , a k
Hence {k xn+1 − xn , ak} is a monotonically decreasing sequence of non-negative real numbers
and so, is convergent to some real, say r.
Taking n → ∞, we get
r 6 r − (ψ(kr ) + ψ(k/ r )), (by continuity of ψ)
⇒ ψ(kr ) + ψ(k/ r ) 6 0,
this is possible only when r = 0. So, limn→∞ k xn+1 − xn k = 0. Now, let p ∈ DX ⊗γ Y be such
that k pk < 1. So, k pk < k ak.
By condition (i), k xn+1 − xn , pk < k xn+1 − xn , ak, n = 0, 1, 2, ... Taking limit as n → ∞ on
both sides, we get,
lim k xn+1 − xn , pk = 0.
n→∞
Hence, limn→∞ k xn+1 − xn , uk = 0 ∀u ∈ DX ⊗γ Y So, { xn } is a Cauchy sequence in DX ⊗γ Y .
DX ⊗γ Y being closed, this sequence converges to some z ∈ DX ⊗γ Y . Now, as in Theorem 2.1, it
can be shown that
kz − T̂z, uk → 0 as n → ∞,
which implies that z = T̂z. Uniqueness can be shown in a similar manner. Thus, T̂ has a unique
fixed point in DX ⊗γ Y .
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
33
Example 2.7 Let Dl 1 ⊗γ K be the closed and bounded (in 2-norm by some constant K) subspace
of the 2-Banach algebra l 1 ⊗γ K with the 2-norm defined earlier. We define the self mappings
1
T1 and T2 on Dl 1 ⊗γ K by T1 (∑i ai ⊗ xi ) =
∑ ai ⊗ xi and
2d i
1
T2 (∑i ai ⊗ xi ) =
∑ ai ⊗ xi , where ai = { ain }n ∈ l 1 , and d is a positive constant such that
4d i
√
d > 2K, and d > 2.
For p = ∑k bk ⊗ yk , q = ∑ j c j ⊗ y/j ∈ l 1 ⊗γ K with k pk > 1, kqk > 1,
k T1 (∑ ai ⊗ xi ), ∑ bk ⊗ yk k = k
i
k
1
2d
∑ a i ⊗ x i , ∑ bk ⊗ y k k
i
k
=
1 1
(k ai , bk kk xi kkyk k + k xi , yk kk ai kkbk k)
2d 2 ∑
i,k
=
1
2d
1
6
2d
∑ kai kkbk kk xi kkyk k (for the 2-norms in l 1 and K )
i,k
∑ kai kkbk k| xi ||yk |(∑ kc j k|y/j |) (∵ kqk > 1)
j
i,k
=
1
∑ kai kkbk kkc j k| xi ||yk ||y/j |
2d i,k,j
=
1
∑ kai kkbk c j k| xi ||yk y/j |
2d i,k,j
=
1
k ai ⊗ xi , ∑ bk c j ⊗ yk y/j k
2d ∑
i
k,j
=
1
k ai ⊗ xi , pqk
2d ∑
i
1 d2
d2
K
6
(K 6 )
2d
2d 2
2
1
M
2
1
= 2 (= 1 ), M = , M1 =
M
M
d
2
6
Similarly,
k T2 (∑ ai ⊗ xi ), ∑ bk ⊗ yk k 6
i
k
=
1
k ∑ ai ⊗ xi , ∑ bk c j ⊗ yk y/j k
4d i
k,j
1
K
k ∑ ai ⊗ xi , pqk 6
4d i
4d
1
M2
)
= 4 (=
M
M
34
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
For arbitrary u, v ∈ Dl 1 ⊗γ K , we have,
1
ku − v, pqk
1
1
k T1 u − T1 v, pk = k u − v, ∑ bk ⊗ yk k 6 4d
1
2d
2d k
2
r
r
1
1
k (u − v), pqk k (u − v), pqk
4d
4d
=
1
2r
r
1
1
1
k (u − v), pqk
ku, pqk + kv, pqk
4d
4d
4d
6
1
r
r
r
r2
1
K
1
d
k (u − v), pqk
k (u − v), pqk
4d
2d
4d
4
6
6
1
1
2
2
r
1
k(u − v), pqk
4d
r
6
1
1
r
d 2
2
6
r
2
d
r
k T1 u − T1 v, pk 6
r
1
k(u − v), pqk
2
1
4 r
=
, M=
d
d
2
2
d
1
k(u − v), pqk
2
1
=
, M=
6 4 4 √
d
d
M
2
r
1
k(u − v), pqk
4 √
6
MM2
s
1
t
1
1
1
√
ku − v, pqk − ψ
ku − v, pqk , where ψ(t) = , k =
2
2
2
2
MM2
Similarly,
k T2 u − T2 v, pk 6 √
1
MM1
s
1
ku − v, pqk − ψ
2
1
ku − v, pqk
2
, where ψ(t) =
t /
1
,k =
2
2
SOME FIXED POINT THEOREMS IN THE PROJECTIVE TENSOR PRODUCT OF 2-BANACH SPACES
35
Therefore, T1 and T2 satisfy the conditions (I) and (J) respectively. Also
1 1
k + k/ = + = 1
2 2
Now, proceeding as in Example 2.2, we can show that for p, q ∈ l 1 ⊗γ K
k pk > kqk ⇒ ku, pk > ku, qk ∀u ∈ Dl1 ⊗γ K
So, the mapping T̂ : Dl 1 ⊗γ K → Dl 1 ⊗γ K defined by T̂ (u) = T1 uT2 u has a unique fixed point
in Dl 1 ⊗γ K .
2. C ONCLUDING R EMARKS :
In 1973 [2], Gähler, 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 on X × X ×
X which satisfies the following conditions:
(i) ( x, x |z) > 0; ( x, x |z) = 0 if and only if x and z 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 number α;
(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 product
spaces and also their tensor products?
R EFERENCES
[1] Y. J. Cho and S. S. Kim, Gâteaux derivatives and 2-inner product spaces, Glasnik Mat. 27(1992), 271-282
[2] C. Dimininnie, S. Gähler 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 Product 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. Gähler, 2-metrische Räiume und ihre topologische Strucktur, Math. Nachr. 26 (1963), 115-148.
[8] S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28 (1965), 1-43.
[9] O. Hadžić, On Common Fixed Point Theorems in 2-metric Spaces, Review of Research Faculty of ScienceUniversity of Novi Sad, Vol 12(1982)
[10] K. Iśeki, 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 in 2-Banach Spaces Internat. 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 Matematicǩi, 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.
36
DIPANKAR DAS, NILAKSHI GOSWAMI, AND VISHNU NARAYAN MISHRA
[15] B.E. Rhoades, Contractive type mappings on a 2-metric space, Math.Nachr.,91(1979), 151 - 155. MR0563606
[16] S. Radenović, 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. Radenović, 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. Ć. Rajić, S. Radenović, 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. Kim Characterizations of 2-inner product spaces, Math. Japon., 47(1988), 239244.
D EPARTMENT OF M ATHEMATICS , G AUHATI U NIVERSITY, G UWAHATI -781014, A SSAM , I NDIA
E-mail address: [email protected]
D EPARTMENT OF M ATHEMATICS , G AUHATI U NIVERSITY, G UWAHATI -781014, A SSAM , I NDIA ,
E-mail address: nila− [email protected]
D EPARTMENT OF M ATHEMATICS , A PPLIED M ATHEMATICS AND H UMANITIES D EPARTMENT, S ARDAR VAL NATIONAL I NSTITUTE OF T ECHNOL - OGY, I CHCHHANATH M AHADEV D UMAS ROAD , S URAT 395
007, G UJARAT, I NDIA , [.2 CM ] L. 1627 AWADH P URI C OLONY B ENIGANJ , P HASE -III, O PPOSITE - I NDUS TRIAL T RAINING I NSTITUTE (I.T.I.), AYODHYA M AIN ROAD FAIZABAD 224 001, U TTAR P RADESH , I NDIA ,
E-mail address: [email protected], vishnu [email protected]
LABHBHAI