Nonlinear operators between intuitionistic fuzzy ...

Chaos, Solitons and Fractals 42 (2009) 1010–1015

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Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet derivative M. Mursaleen, S.A. Mohiuddine * Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

a r t i c l e

i n f o

Article history: Accepted 26 February 2009

a b s t r a c t In this paper, we define and study intuitionistic fuzzy continuity, intuitionistic fuzzy boundedness and Fréchet differentiation of nonlinear operators between intuitionistic fuzzy normed spaces (IFNS). We also display here some interesting examples by using classical sequence spaces lp and co . Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction and preliminaries We often come across some situations when many mathematical problems can not be solved by using classical techniques of nonlinear functional analysis. There are many challenging problems of nonlinear phenomenons in nature and some of these problems can be reduced to the operator equations. Fréchet derivative of nonlinear operators plays a fundamental role in solving such operator equations. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, e.g. population dynamics [2], chaos control [9,10], computer programming [11], nonlinear dynamical systems [12], statistical convergence [17], etc. Recently, the fuzzy topology [1,8,13,14,24] proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The most fascinating application of fuzzy topology in quantum particle physics arises in string and ð1Þ -theory of ElNaschie [3–7]. The usual uncertainty principle of W. Heisenberg leads to generalized uncertainty principle, which has been motivated by string theory and non-commutative geometry. In strong quantum gravity regime space-time points are determined in a fuzzy manner. Thus impossibility of determining position of articles gives space-time a fuzzy structure [3,16,18]. Because of this fuzzy structure, position space representation of quantum mechanics breaks down and therefore a generalized normed space of quasi-position eigenfunction is required [18] and it is seen that in quantum gravity regime the very basic notion of spacetime itself induces dispersion on the wave packet profile. This fact has origin on the quantum fluctuation of space-time which can be described as fuzzy space-time. There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm [19–21] seems to be more suitable in such cases, that is, we can deal with such situations by modelling the inexactness through the intuitionistic fuzzy norm. In this paper, we define and study the Fréchet derivative of nonlinear operators between intuitionistic fuzzy normed spaces which provides a useful functional tool to solve the operator equations involving such operators. For example, local and global bifurcation analysis of solutions of operator equations are established on this important notion. In fact, we develop classical techniques of nonlinear functional analysis of operator equations along with Fréchet derivative. We recall some notations and basic definitions used in this paper.

* Corresponding author. E-mail addresses: [email protected] (M. Mursaleen), [email protected] (S.A. Mohiuddine). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.02.041

M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 1010–1015

1011

Definition 1.1. [22] A binary operation : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-norm if it satisfies the following conditions: (a) is associative and commutative, (b) is continuous, (c) a 1 = a for all a 2 ½0; 1, (d) a b 6 c d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Definition 1.2. [22] A binary operation } : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-conorm if it satisfies the following conditions: 0 0 0 0 (a ) } is associative and commutative, (b ) } is continuous, (c ) a } 0 = a for all a 2 ½0; 1, (d ) a } b 6 c } d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Using the notions of continuous t-norm and t-conorm, Saadati and Park [20] have recently introduced the concept of intuitionistic fuzzy normed space as follows: Definition 1.3. The five-tuple ðX; l; m; ; }Þ is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, is a continuous t -norm, } is a continuous t-conorm, and l; m are fuzzy sets on X ð0; 1Þ satisfying the following conditions. For every x; y 2 X and s; t > 0 (i) lðx; tÞ þ mðx; tÞ 6 1, (ii) lðx; tÞ > 0, (iii) lðx; tÞ ¼ 1 if and only if x ¼ 0, (iv) lðax; tÞ ¼ lðx; jat jÞ for each a–0; ðv Þlðx; tÞ lðy; sÞ 6 lðx þ y; t þ sÞ, (vi) lðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (vii) limt!1 lðx; tÞ ¼ 1 and limt!0 lðx; tÞ ¼ 0, (viii) mðx; tÞ < 1, (ix) mðx; tÞ ¼ 0 if and only if x ¼ 0, (x) mðax; tÞ ¼ mðx; jat jÞ for each a–0, (xi) lðx; tÞ}lðy; sÞ P mðx þ y; t þ sÞ, (xii) mðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (xiii) limt!1 mðx; tÞ ¼ 0 and limt!0 mðx; tÞ ¼ 1. In this case ðl; mÞ is called an intuitionistic fuzzy norm. Remark 1.1. Let ðX; l; m; ; }Þ be an IFNS with the condition

lðx; tÞ > 0 and mðx; tÞ < 1 implies x ¼ 0 for all t 2 R:

ð1:1:1Þ

Let kxka ¼ infft > 0 : lðx; tÞ P a and mðx; tÞ 6 1 ÿ ag, for all a 2 ð0; 1Þ. Then fk:ka : a 2 ð0; 1Þg is an ascending family of norms on X. These norms are called a-norms on X corresponding to intuitionistic fuzzy norm ðl; mÞ. 2. Intuitionistic fuzzy continuity In this section, we define strong and weak intuitionistic fuzzy limits as well as strong and weak intuitionistic fuzzy continuity and boundedness of the mapping between IFNSs. Definition 2.1. Let ðX; l1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two IFNSs and f : X ! Y be a mapping. Then, (i) L is said to be strong intuitionistic fuzzy limit of f at some x0 2 X iff for every that

> 0 there exists some d ¼ dðÞ > 0 such

l2 ðf ðxÞ ÿ L; Þ P l1 ðx ÿ x0 ; dÞ and m2 ðf ðxÞ ÿ L; Þ 6 m1 ðx ÿ x0 ; dÞ: In this case we write ðsifÞ ÿ limx!x0 f ðxÞ ¼ L, which also means that

lim

l1 ðxÿx0 ;tÞ!1

l2 ðf ðxÞ ÿ L; tÞ ¼ 1ðsifÞ and

lim

m1 ðxÿx0 ;tÞ!0

m2 ðf ðxÞ ÿ L; tÞ ¼ 0ðsifÞ

or

l2 ðf ðxÞ ÿ L; tÞ ¼ 1ðsifÞ as l1 ðx ÿ x0 ; tÞ ! 1; and m2 ðf ðxÞ ÿ L; tÞ ¼ 0ðsifÞ as m1 ðx ÿ x0 ; tÞ ! 0;

for all t > 0. (ii) L is said to be weak intuitionistic fuzzy limit of f at some x0 2 X iff for given d ¼ dð; aÞ > 0 such that

l1 ðx ÿ x0 ; dÞ P a implies l2 ðf ðxÞ ÿ L; Þ P a; and

m1 ðx ÿ x0 ; dÞ 6 1 ÿ a implies m2 ðf ðxÞ ÿ L; Þ 6 1 ÿ a: In this case we write ðwifÞ ÿ limx!x0 f ðxÞ ¼ L, which also means that

lim

l1 ðxÿx0 ;tÞ!1

l2 ðf ðxÞ ÿ L; tÞ ¼ 1ðwifÞ and

lim

m1 ðxÿx0 ;tÞ!0

m2 ðf ðxÞ ÿ L; tÞ ¼ 0ðsifÞ

or

l2 ðf ðxÞ ÿ L; tÞ ¼ 1ðwifÞ as l1 ðx ÿ x0 ; tÞ ! 1; and m2 ðf ðxÞ ÿ L; tÞ ¼ 0ðwifÞ as m1 ðx ÿ x0 ; tÞ ! 0;

for all t > 0.

>0

and a 2 ð0; 1Þ there exists some

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Proposition 2.1. ðsifÞ ÿ lim implies ðwifÞ ÿ lim but the converse need not hold. Further, ðsifÞ ÿ lim ¼ ðwifÞ ÿ lim whenever ðsifÞ ÿ lim exists. Proof. It is easy to prove. Now we show that ðwifÞ ÿ lim does not imply ðsifÞ ÿ lim in general.

h

Example 2.1. Let X ¼ Y ¼ R and

l1 ðx; tÞ ¼

(

t tþjxj

if t > 0;

0

if t 6 0;

m1 ðx; tÞ ¼

(

jxj tþjxj

if t > 0;

1

if t 6 0;

l2 ðx; tÞ ¼

1 if t > jxj; 0

if t 6 jxj;

and

m2 ðx; tÞ ¼

1 if t 6 jxj; 0

if t > jxj:

Let the function f from ðR; l1 ; m1 ; ; }Þ onto ðR; l2 ; m2 ; ; }Þ be defined by f ðxÞ ¼ x. Then ðwifÞ ÿ limx!0 f ðxÞ ¼ 0. However, ðsifÞ ÿ limx!0 f ðxÞ does not exist. Because, for jxj ¼ , there is no d > 0 satisfying the conditions

l2 ðx; Þ ¼ 0 P l1 ðx; dÞ ¼

d d ¼ d þ jxj d þ

m2 ðx; Þ ¼ 0 6 m1 ðx; dÞ ¼

and

dþ

:

Now, we define the strong and weak intuitionistic fuzzy continuity of mappings between IFNSs. Definition 2.2. Let ðX; l; m; ; }Þ be an IFNS and U X. Then U is said to be intuitionistic fuzzy open subset of X if for each x 2 U there exist some t > 0 and a 2 ð0; 1Þ such that Bðx; a; tÞ # U, where Bðx; a; tÞ :¼ fy : lðx ÿ y; tÞ > 1 ÿ a and mðxÿ y; tÞ < ag. Definition 2.3. Let ðX; l1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two IFNSs and f : X ! Y be a mapping. Then f is said to be (i) weakly intuitionistic fuzzy continuous at x0 2 X if for given that

> 0 and a 2 ð0; 1Þ, there exists some d ¼ dð; aÞ > 0 such

l1 ðx ÿ x0 ; dÞ P a implies l2 ðf ðxÞ ÿ f ðx0 Þ; Þ P a; and

m1 ðx ÿ x0 ; dÞ 6 1 ÿ a implies m2 ðf ðxÞ ÿ f ðx0 Þ; Þ 6 1 ÿ a; for all x 2 X. (ii) strongly intuitionistic fuzzy continuous at x0 2 X if for given

> 0 there exists some d ¼ dðÞ > 0 such that l2 ðf ðxÞ ÿ f ðx0 Þ; Þ P l1 ðx ÿ x0 ; dÞ and m2 ðf ðxÞ ÿ f ðx0 Þ; Þ 6 m1 ðx ÿ x0 ; dÞ;

for all x 2 X. (iii) Let f be linear. Then f is called weakly intuitionistic fuzzy bounded (for short, WIFB) on X if for given a 2 ð0; 1Þ, there exists some ma > 0 such that

l1 x;

t ma

t ma

P a implies

6 1 ÿ a implies

l2 ðf ðxÞ; tÞ P a;

and

m1 x;

m2 ðf ðxÞ; tÞ 6 1 ÿ a;

for all x 2 X and t > 0. Let F 0 ðX; YÞ denote the set of all WIFB linear operators. (iv) Let f be linear. We say that f is strongly intuitionistic fuzzy bounded (for short, SIFB) on X if for given a 2 ð0; 1Þ, there exists some M > 0 such that

l2 ðf ðxÞ; tÞ P l1 x; for all x 2 X

and

t M

and

m2 ðf ðxÞ; tÞ 6 m1 x;

t ; M

t > 0. Let FðX; YÞ denote the set of all SIFB linear operators.

As in classical theory, the following is easy to prove. Theorem 2.1. Let ðX; l1 ; m1 ; ; }Þ and ðX; l2 ; m2 ; ; }Þ be two IFNSs and f : X ! Y be a linear mapping. Then f is strongly (weakly) intuitionistic fuzzy continuous if and only if it is strongly (weakly) intuitionistic fuzzy bounded.


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Definition 2.4. Let ðX; l; m; ; }Þ be an IFNS. Then, a sequence ðxk Þ is said to be (i) weakly intuitionistic fuzzy convergent to x 2 X if and only if, for every > 0 and a 2 ð0; 1Þ, there exists some k0 ¼ k0 ða; Þ wif such that lðxk ÿ x; Þ P 1 ÿ a and mðxk ÿ x; Þ 6 a for all k P k0 . In this case we write xk ! x. (ii) strongly intuitionistic fuzzy convergent to x 2 X if and only if, for every a 2 ð0; 1Þ, there exists some k0 ¼ k0 ðaÞ such that sif lðxk ÿ x; tÞ P 1 ÿ a and mðxk ÿ x; tÞ 6 a for all t > 0. In this case we write xk ! x. Similarly, we can define a sif(wif)-Cauchy sequence, sif(wif)-closure of a subset and a sif(wif)-complete IFNS. Theorem 2.2. If a sequence ðxk Þ is sif-convergent then it is wif-convergent to the same limit, but not conversely. It is straightforward that sif-convergence implies wif-convergence. For converse, we construct the following example. Example 2.2. Let X ¼ co , the Banach space of all sequences x ¼ ðxk Þ convergent to zero with the sup-norm kxk1 ¼ sup jxk j k and consider the intuitionistic fuzzy norm

( tÿkxk

tþkxk1

if t > kxk1

0

if t 6 kxk1

1

lðx; tÞ ¼

and

mðx; tÞ ¼

(

2kxk1 tþkxk1

if t < kxk1

1

ift 6 kxk1 :

on X. We can find a-norms of intuitionistic fuzzy norm ðl; mÞ since it satisfies (1.1.1) condition. Thus

lðx; tÞ P a ()

t ÿ kxk1 1þa P a () kxk1 6 t; t þ kxk1 1ÿa

and

mðx; tÞ 6 1 ÿ a ()

2kxk1 1þa 6 1 ÿ a () kxk1 6 t: t þ kxk1 1ÿa

This shows that

kxka ¼ infft > 0 : lðx; tÞ P a and

mðx; tÞ 6 1 ÿ ag ¼

1þa kxk1 : 1ÿa

Now, we show that the sequence x ¼ ðxk Þ ¼ ð1k Þ1 k¼1 is wif-convergent but not sif-convergent. Since each k:ka is equivalent to k:k1 , obviously ðxk Þ is wif-convergent to 0. However, this convergence is not uniform in a. Indeed for given > 0

kxka ¼

1þa 1þa kxk1 < () < 1; 1ÿa ð1 ÿ aÞ

since kxk1 ¼ 1 for x ¼ ð1k Þ1 k¼1 . But this is not possible, since

1þa ð1ÿaÞ

! 1 as a ! 1.

3. Intuitionistic fuzzy Fréchet derivative (IFFD) In this section, we define strong and weak IFFD and give some analogues of classical results. Definition 3.1. Let ðX; l1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two IFNSs, U # X be an intuitionistic fuzzy open subset and f : U ! Y be a mapping, probably nonlinear. Then f is called strong (weak) intuitionistic fuzzy Fréchet differentiable at x0 2 U if there exists a strongly (weakly) intuitionistic fuzzy bounded linear operator T from ðX; l1 ; m1 ; ; }Þ to ðY; l2 ; m2 ; ; }Þ such that for every t > 0

lim

l2

lim

m2

l1 ðh;tÞ!1

f ðx0 þ hÞ ÿ f ðx0 Þ ÿ Th ; t ¼ 1ðsifðwifÞÞ; 1 ÿ l1 ðh; tÞ

and

m1 ðh;tÞ!0

f ðx0 þ hÞ ÿ f ðx0 Þ ÿ Th ; t ¼ 0ðsifðwifÞÞ: m1 ðh; tÞ

In this case, the operator T is called strong (weak) intuitionistic fuzzy, or briefly, sif(wif)-Fréchet derivative of f at x0 and is denoted by dsifðwifÞ f ½x0 . f is said to be sif(wif)-Fréchet differentiable on U if it is sif(wif)-Fréchet differentiable at every point of U. In this case, x ! dsifðwifÞ f ½x is a function from U to FðX; YÞðF 0 ðX; YÞÞ, denote by dsifðwifÞ f . Theorem 3.1. A strongly (weakly) intuitionistic fuzzy bounded linear operator A is sif(wif)-Fréchet differentiable at every point x0 and dsifðwifÞ A½x0 ¼ A. Proof. This is obvious since

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Aðx0 þ hÞ ÿ Aðx0 Þ ÿ Ah ; t ¼ l2 ð0; tÞ ¼ 1; l2 1 ÿ l1 ðh; tÞ and

m2

Aðx0 þ hÞ ÿ Aðx0 Þ ÿ Ah ; t ¼ m2 ð0; tÞ ¼ 0; m1 ðh; tÞ

for all t > 0. Proposition 3.1. If f is sif(wif)-Fréchet differentiable at x0 2 U then it is strong (weak) intuitionistic fuzzy continuous at x0 . Proof. We use the following inequalities. For given t > 0,

l2 ðf ðxÞ ÿ f ðx0 Þ; tÞ ¼ l2 ðf ðxÞ ÿ f ðx0 Þ ÿ Th þ Th; tÞ P l2 ðf ðxÞ ÿ f ðx0 Þ ÿ Th; tð1 ÿ l1 ðh; tÞÞÞ l2 ðTh; tl1 ðh; tÞÞ ¼ l2 and

f ðxÞ ÿ f ðx0 Þ ÿ Th Th ; t l2 ;t ; ð1 ÿ l1 ðh; tÞÞ l1 ðh; tÞ

m2 ðf ðxÞ ÿ f ðx0 Þ; tÞ ¼ m2 ðf ðxÞ ÿ f ðx0 Þ ÿ Th þ Th; tÞ 6 m2 ðf ðxÞ ÿ f ðx0 Þ ÿ Th; tð1 ÿ m1 ðh; tÞÞÞ}m2 ðTh; tm1 ðh; tÞÞ ¼ m2

f ðxÞ ÿ f ðx0 Þ ÿ Th Th ; t }m2 ;t : ðm1 ðh; tÞÞ 1 ÿ m1 ðh; tÞ

Since f is sif(wif)-Fréchet differentiable at x0 2 U, it follows that

l2 ðf ðxÞ ÿ f ðx0 Þ; tÞ P 1 l2

Th ;t ; l1 ðh; tÞ

and

m2 ðf ðxÞ ÿ f ðx0 Þ; tÞ 6 0}m2

Th ;t ; 1 ÿ m1 ðh; tÞ

where T ¼ dsifðwifÞ f ½x0 . Therefore f is strong(weak) intuitionistic fuzzy continuous. Theorem 3.2. Let ðX; l1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two IFNSs, U # X be an intuitionistic fuzzy open subset and f : U ! Y be a mapping. If f is sif-Fréchet differentiable at some x0 2 U then it is wif-Fréchet differentiable at x0 with the same derivative but not conversely. The proof of the above theorem follows directly from the Proposition 2.1. For the converse part, let us consider the following example. Example ÿP 3.1. pLet 1=p X ¼ Y ¼ ‘p ð1 6 p < 1Þ, the Banach space of all absolutely p-summable sequences with the norm . Define the functions kxkp ¼ n jxn j

l1 ðx; tÞ ¼

(

m1 ðx; tÞ ¼

8
0;

0

if t < 0;

l2 ðx; tÞ ¼

(

1 if t > 0 and t 2 > kxkpp ; 0

if t < 0 and t 2 6 kxkpp ;

m2 ðx; tÞ ¼

(

0

if t > 0 and t 2 > kxkpp ;

and

:

p

2kxkp

t 2 þ2kxkpp

0

if t > 0; if t < 0;

1 if t < 0 and t 2 6 kxkpp :

Then ðl1 ; m1 Þ and ðl2 ; m2 Þ are intuitionistic fuzzy norms on ‘p . Consider the shift operator SðxÞ ¼ Sðfx1 ; x2 ; gÞ ¼ f0; x1 ; x2 ; g on ‘p . It is easy to see that linear operator S is weakly bounded hence is weakly continuous from ð‘p ; l1 ; m1 ; ; }Þ into ð‘p ; l2 ; m2 ; ; }Þ but not strongly continuous, since S is linear we get from Theorem 3.1 that S ¼ dwif S½x for all x 2 ‘p , while dsif S½x does not exist. Definition 3.2. A subset B in an intuitionistic fuzzy normed space ðX; l; m; ; }Þ is called sifðwifÞ-compact if each sequence of elements of B has a sifðwifÞ-convergent subsequence. Definition 3.3. Let ðX; mu1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two IFNSs and f : X ! Y be a mapping. Then f is called sifðwifÞ-compact if for every intuitionistic fuzzy bounded subset B of X the subset f ðBÞ is relatively sifðwifÞ-compact, that is, the closure of f ðBÞ is sifðwifÞ-compact.


1015

Remark 3.1. By the same way as in the proof of [15, Theorem 5] we can show that f is sif(wif)-compact if and only if it maps every intuitionistic fuzzy bounded sequence onto a sequence which has a sif(wif)-convergent subsequence. Therefore, a sifcompact operator is wif-compact but not conversely. For example, the identity operator on ðco ; l; m; ; }Þ, in Example 2.2, is ÿ 1 not sif-compact while it is wif-compact. Because 1k k¼1 cannot have sif-convergent subsequence.

Theorem 3.3. Let ðX; l1 ; m1 ; ; }Þ and ðY; l2 ; m2 ; ; }Þ be two wif-complete IFNSs and f : X ! Y be nonlinear wif-compact operator. Suppose that, for some x0 2 X, dwif f ½x0 ¼ A exists. Then the linear operator A is also wif-compact. Proof. Let ðxk Þ ðX; l1 ; m1 ; ; }Þ be an arbitrary intuitionistic fuzzy bounded sequence. This means, there exists some t 0 > 0 and r 2 ð0; 1Þ such that l1 ðxk ; t20 Þ P 1 ÿ r and m1 ðxk ; t20 Þ 6 r, for every positive integer k. Consider the sequence ðx0 þ xk Þ1 k¼1 and let us show that it is intuitionistic fuzzy bounded. If we take 1 ÿ r1 ¼ l1 ðx0 ; t20 Þ 1 ÿ r and r 1 ¼ l1 ðx0 ; t20 Þ}r, then

l1 ðx0 þ xk ; t0 Þ P l1 x0 ;

t0 t0 t0 l 1 xk ; > l1 x0 ; ð1 ÿ rÞ ¼ 1 ÿ r 1 ; 2 2 2

and

m1 ðx0 þ xk ; t0 Þ 6 m1 x0 ;

t0 t0 t0 }m1 xk ; < m1 x0 ; }r ¼ r 1 ; 2 2 2

for every positive integer k. Rest of the proof can be done on the same lines as in [22]. Remark 3.2. Theorem 3.3 can also be proved for sif-compact operators. 4. Conclusion The present work is an improvement and generalization of the work of Yilmaz [23] as we have presented it in a more general setting, i.e. in an intuitionistic fuzzy normed space which is more general than the fuzzy normed space. So that one may expect it to be more useful functional tool in the field of fuzzy topology in modelling the vagueness and inexactness of various problems occuring in many areas of science, engineering and economics. Acknowledgement Research of the first author is supported by the Department of Science and Technology, New Delhi, under Grant No. SRnS4nMS:505n07. Research of the second author is supported by the Department of Atomic Energy, Government of India under the NBHM-Post Doctoral Fellowship Programme No. 28/2008/810. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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