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Strong convergence theorem for common fixed point for finite family{J i}N i=1is proved in a real Banch space.. As an application, a new convergence theorem for finite family of Lipschitz

R E S E A R C H Open Access

Convergence theorem for finite family of

lipschitzian demi-contractive semigroups

Bashir Ali*and Godwin Chidi Ugwunnadi

* Correspondence:

bashiralik@yahoo.com

Department of Mathematical

Sciences, Bayero University, Kano,

Nigeria

Abstract

Let E be a real Banach space and K be a nonempty, closed, and convex subset of E Let{J i}N

i=1be a finite family of Lipschitzian demi-contractive semigroups of K, with sequences of bounded measurable functions Li : [0,∞) ® (0, ∞) and bounded functionsli: [0,∞) ® (0, ∞), respectively, whereJ i:={T i (t) : t≥ 0}, i = 1,2, , N Strong convergence theorem for common fixed point for finite family{J i}N

i=1is proved in a real Banch space As an application, a new convergence theorem for finite family of Lipschitzian demi-contractive maps is also proved

Mathematics subject classification (2000) 47H09, 47J25 Keywords: Demi-contractive maps, Demi-contractive semigroup, Demicompact maps, Fixed point

1 Introduction

Let E be a real Banach space and E* be the dual space of E The normalized duality mapping J : E→ 2E∗is defined by, xÎ E,

Jx = {x∗∈ E∗:x, x∗ = ||x|| ||x∗||, ||x∗|| = ||x||},

where〈., 〉 denotes the normalized duality pairing For any x Î E, an element of

Jxis denoted by j(x)

Let K be a nonempty, closed and convex subset of E Let T : K® K be a map, a point xÎ K is called a fixed point of T if Tx = x, and the set of all fixed points of T is denoted by F(T) The mapping T is called L-Lipschitzian or simply Lipschitz if∃L >0, such that ||Tx -Ty||≤ L||x - y|| ∀x, y Î K and if L = 1, then the map T is called nonexpansive

A one parameter familyJ = {T(t) : t ≥ 0}of self mapping of K is called a nonexpan-sive semigroupif the following conditions are satisfied,

(i) T(0)x = x∀ x Î K;

(ii) T(t + s) = T(t)∘ T(s) ∀ t, s ≥ 0;

(iii) for each xÎ K, the mapping t ® T(t)x is continuos;

(iv) for x, yÎ K and t ≥ 0, ||T(t)x -T(t)y|| ≤ ||x - y||

If the familyJ = {T(t) : t ≥ 0}satisfies conditions (i) - (iii), then it is called (a) pseudocontractive semigroup if for any x, y Î K, there exists j(x - y) Î J(x - y) such that

T(t)x − T(t)y, j(x − y) ≤ ||x − y||2;

© 2011 Ali and Ugwunnadi; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

(b) strictly pseudocontractive semigroup if there exists a bounded functionl : [0, ∞) ® (0,∞) and j(x - y) Î J(x - y) such that

T(t)x − T(t)y, j(x − y) ≤ ||x − y||2− λ(t)||(I − T(t))x − (I − T(t))y||2

for all x, yÎ K;

(c) demi-contractive semigroup if F(T(t))≠ ∅ ∀t ≥ 0, there exists a bounded function l : [0,∞) ® (0, ∞), and j(x - y) Î J(x - y) such that

T(t)x − q, j(x − q) ≤ ||x − q||2− λ(t)||x − T(t)x||2

for any x Î K and q Î F(T(t));

(d) Lipschitzian semigroup if there is a bounded measurable function

L: [0,∞) ® (0, ∞) such that for x, y Î K and t ≥ 0,

||T(t)x − T(t)y|| ≤ L(t)||x − y||.

It is known that every strictly pseudocontractive semigroup is Lipschitzian, and every strictly pseudocontractive semigroup with fixed point is demi-contractive semi-group

Let E be a real Banach space and let K be a nonempty closed convex subset of E A mapping T : K® K is demicompact if for every bounded sequence {xn} in K such that

{xn - Txn} converges, and there exists a subsequence, say{x n j}of {xn} that converges

strongly to some x* in K T is said to be demi-contractive if F(T)≠ ∅, and there exists l

>0 such that〈Tx- q, j(x - q)〉 ≤ ||x - q||2

-l||x - Tx||2∀ x Î K, q Î F(T) and j(x - q) Î J (x - q)

Let T1, T2, , TN be a family of self-mappings of K such that F :=∩N

i=1 F(T i) Then, the family is said to satisfy conditionCif there exists a nondecreasing function f

: [0, ∞) ® [0, ∞) with f (0) = 0 and f (r) >0 ∀ r Î (0, ∞) such that f (d(x, F)) ≤ ||x

-Tsx|| for some s in {1, 2, , N} and for all xÎ K, where d(x, F) = inf {||x - q|| : q Î F}

Existence theorems for family of nonexpansive mappings are proved in [1-5] and actually many others Recently, Suzuki [6] proved the equivalence between the fixed

point property for nonexpansive mappings and that of the nonexpansive semi-groups

Both implicit and explicit, Mann, Ishikawa, and Halpern-type schemes were studied for approximation of common fixed points of family of nonexpansive semigroups and

their generalizations in various spaces; see, for example [6-13], to list but a few

In 1998, Shoiji and Takahashi [7] introduced and studied a Halpern-type scheme for common fixed point of a family of asymptotically nonexpansive semigroup in the

fra-mework of a real Hilbert space Suzuki [8] proved that the implicit scheme defined by

x, x1Î K,

converges strongly to a common fixed point of the family of nonexpansive semigroup

in a real Hilbert space Xu [9] extended the result of Suzuki to a more general real

uni-formly convex Banach space having a weakly sequentially continuous duality mapping

In 2005, Aleyner and Reich [10] proved the strong convergence of an explicit Halpern-type scheme defined by x, x1Î K,

to a common fixed point of the family {T(t) : t≥ 0} of nonexpansive semigroup in a reflexive Banach space with uniformly Gatéuax differentiable norm Recently, Zhang

et al [11] introduced and studied a composite iterative scheme defined by x, x1Î K,

x n+1=α n y n+ (1− α n )x; y n=β n T(t n )x n+ (1− β n )x n Those authors proved strong convergence of the sequence {xn} to a common fixed point of the family {T(t) : t≥ 0} of nonexpansive semigroup

Very recently, Chang et al [12] proved a strong convergence theorem which extended and improved the results in [10,9] and some others They proved the

follow-ing theorem

Theorem 1.1 Chang et al [12]Let K be a nonempty, closed, and convex subset of a real Banach space E: LetJ := {T(t) : t ≥ 0}be a Lipschitzian demi-contractive

semi-group of K with bounded measurable function L : [0,∞) ® (0, ∞) and bounded

func-tionl : [0, ∞) ® (0, ∞) such that

L := sup

t≥0{L(t)} < ∞, λ := inf

t≥0{λ(t)} > 0 and F := ∩

Let {tn} be an increasing sequence in [0, ∞) and {an} be a sequence in (0,1) satisfying the following conditions,

(i)∞

n=1(1− α n) =∞; (ii)∞

subset C of E such that∪t ≥0T(t)(K)⊂ C and for any bounded set D ⊂ K

lim

n→∞x ∈D,s∈Rsup +||T(s + t n )x − T(t n )x|| = 0

Let{xn} be generated by x1Î K,

Then, the sequence {xn} converges strongly to some element in F

The purpose in this article is to prove a strong convergence theorem for common fixed point for finite families{J i}N

i=1of demi-contractive semigroups in a real Banach space As application, we also prove convergence theorem for finite family of

demi-contractive mappings Our theorems generalize and improve several recent results For

instance, Theorem 1.1, which generalized, extended and improved several recent

results, is a special case of our Theorem

2 Preliminaries

We shall make use of the following lemmas

Lemma 2.1 Let E be a real normed linear space Then, the following inequality holds:

||x + y||2≤ ||x||2+ 2y, j(x + y), ∀ x, y ∈ E and j(x + y) ∈ J(x + y).

Lemma 2.2 (Xu [14]) Let {an} and {bn} be sequences of nonnegative real numbers satisfying the inequality

a n+1 ≤ (1 + b n )a n, n≥ 1

If∞

con-verges strongly to zero, thennlim→∞a n= 0.

Lemma 2.3 (Suzuki [15]) Let {xn} and {yn} be bounded sequences in a Banach space E and let{bn} be a sequence in [0, 1] with 0 <lim infbn≤ lim supbn<1 Suppose xn+1=

bnyn+(1 -bn)xnfor all integers n≥ 1 and lim sup(||yn+1- yn|| - ||xn+1- xn||)≤ 0 Then,

lim ||yn- xn|| = 0

3 Main Results

Let E be a real Banach space, and K be a nonempty, closed convex subset of E For

some fixed i Î N, letJ i:={T i (t) : t≥ 0}be a Lipschitzian demi-contractive

semi-group with bounded measurable function Li: [0, ∞) ® (0, ∞) and bounded function

li: [0,∞) ® (0, ∞) such that

L i:= sup

t≥0{L i (t) } < ∞, λ i := inf

t≥0{λ i (t) } > 0 and F i:= ∩

t≥0F(T i (t))

Then, for x, yÎ K, q Î Fi

and t≥ 0,

T i (t)x − q, j(x − q) ≤ ||x − q||2− λ i ||x − T i (t)x||2

and

||T i (t)x − T i (t)y || ≤ L i ||x − y||.

Consider a family{J i}N

i=1of Lipschitzian demi-contractive semigroups of K and let

L := max

1≤i≤N {λ i}Clearly L <∞ and l >0 and for x, y Î

K,q∈F, t≥ 0 and any i Î {1, 2, , N},

T i (t)x − q, j(x − q) ≤ ||x − q||2− λ||x − T i (t)x||2

and

||T i (t)x − T i (t)y || ≤ L||x − y||.

For a fixedδ Î (0, 1) and t ≥ 0 define a family Si(t) : K® K i = 1, 2, , N by

S i (t)x := (1 − δ2

)x + δ2

Then, for x, yÎ K andq∈F,

S i (t)x − q, j(x − q) = (1 − δ2)x − q, j(x − q) + δ2T i (t)x − q, j(x − q)

≤ (1 − δ2)||x − q||2+δ2[||x − q||2− λ||x − T i (t)x||2]

=||x − q||2− λδ2||x − T i (t)x||2 Let ¯λ = λδ2> 0,then

S i (t)x − q, j(x − q) ≤ ||x − q||2− ¯λ||x − T i (t)x||2 (3:2) Also,

||S i (t)x − S i (t)y|| = ||(1 − δ2)(x − y) + δ2(T i (t)x − T i (t)y)||

≤ (1 − δ2)||x − y|| + δ2L ||x − y||

= [1− δ2+δ2L] ||x − y||

≤ (1 + δ2L) ||x − y||.

Let ¯L = 1 + δ2L.

Hence, for each iÎ {1, 2, N}, Siis Lipschitz with Lipschitz constant ¯L > 0 Lemma 3.1 Let E be a real Banach space and K be a nonempty closed convex subset

of E Let{J i}N

i=1be a finite family of Lipschitzian demi-contractive semigroups of K with sequences of bounded measurable functions Li: [0,∞) ® (0, ∞) and bounded functions

li: [0, ∞) ® (0, ∞) i = 1, 2, , N such that for each i = 1, 2, , N,

L i:= sup

t≥0{L i (t) } < ∞, λ i := inf

t≥0{λ i (t) } > 0 and F i:= ∩

t≥0F(T i (t))

1≤i≤N{ ∩

t≥0F(T i (t)) , {tn}be an increasing sequence in [0,∞) and {an} be a sequence in (0,1) satisfying the following conditions:

(i)∞

n=1(1− α n) =∞, (ii)∞

Assume∀ i Î {1,2, , N} for any bounded set D ⊂ K the relation lim

n→∞x ∈D,s∈Rsup +||T i (s + t n )x − T i (t n )x|| = 0 (3:4) holds Let{xn} be a sequence generated by x1Î K,

where Tn(tn) = Tn modN(tn)

Then, (a)nlim→∞||x n − q||exists for allq∈F (b)lim infn→∞ ||x n − T i (t n )x n|| = 0for all iÎ {1,2,3, , N}

Proof For any fixedq∈F using (3.5), we have

x n+1 − q = (x n − q) + (1 − α n+1 )(S n+1 (t n+1 )x n − x n)

Thus,

||x n+1 − q||2= ||(x n − q) + (1 − α n+1 )(S n+1 (t n+1 )x n − x n)||2

≤ ||x n − q||2+ 2(1− α n+1)S n+1 (t n+1 )x n − x n , j(x n+1 − q)

=||x n − q||2

+ 2(1− α n+1)



S n+1 (t n+1 )x n − S n+1 (t n+1 )x n+1 , j(x n+1 − q)

+S n+1 (t n+1 )x n+1 − q, j(x n+1 − q) − x n+1 − q, j(x n+1 − q)

+xn+1 − x n , j(x n+1 − q)

≤ ||x n − q||2+ 2(1− α n+1 )( ¯L + 1) ||x n − x n+1 ||x n+1 − q||

−2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2

≤ ||x n − q||2+ 2(1− α n+1)2(1 + ¯L)2||S n+1 (t n+1 )x n − x n || ||x n − q||

−2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2

≤ ||x n − q||2+ 2(1− α n+1)2(1 + ¯L)3||x n − q||2

−2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2

= (1 +σ n+1)||x n − q||2− 2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2

≤ (1 + σ n+1)||x n − q||2,

(3:6)

whereσ n+1 = 2(1 + ¯L)3(1− α n+1)2 Since∞

Hence, {xn} is bounded, which implies that {Tn(tn)xn} and {Sn(tn)xn} are also bounded

From (3.6)

||x n+1 − q||2≤ ||x n − q||2+ 2(1− α n+1)2(1 + ¯L)3||x n − q||2

−2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2

≤ ||x n − q||2− 2(1 − α n+1)¯λ||x n+1 − T n+1 (t n+1 )x n+1||2+ 2(1− α n+1)2M,

where,M := (1 + ¯L)3sup

n∈N(||x n − q||2) Hence, for some mÎ N, 2¯λ

m



n=1

(1− α n+1)||xn+1 − T n+1 (t n+1 )x n+1||2≤

m



n=1

(||xn − q||2− ||x n+1 − q||2)

+ 2M

m



n=1

(1− α n+1)2

≤ ||x1− q||2

+ 2M

m



n=1

Since m Î N is arbitrary, we have

2¯λ

∞



n=1

which implies lim inf

Next, we show that, lim

n→∞||x n+1 − x n|| = 0

Let {bn} and {yn} be two sequences define by bn := δ(1 - δ)an+1 + δ2

and

y n:=x n+1 −x n+β n x n

β n Then, using the definition of {bn} and {Sn} we obtain that

y n:=δα n+1 x n+δ2 (1−α n+1 )T n+1 (t n+1 )x n

β n Then,

[x n+1 − x n] +δ

α

n+2



x n

+δ2(1− α n+2)

[T n+2 (t n+2 )x n+1 − T n+2 (t n+2 )x n] +δ2



1− α n+2

β n



T n+2 (t n+2 )x n

+δ2(1− α n+1)

β n

[T n+2 (t n+2 )x n − T n+1 (t n+1 )x n]

||y n+1 − y n || − ||x n+1 − x n|| ≤

δα

n+2

+ δ2L(1 − α n+2)

||x n+1 − x n||

+δ α n+2

+δ2 1− α n+2

(t n+2 )x n||

+δ2(1− α n+1)

Hence, lim sup

n→∞ (||y n+1 − y n || − ||x n+1 − x n||) ≤ 0, and by lemma 2.3,

lim

n→∞||y n − x n|| = 0

Thus,

||x n+1 − x n || = β n ||y n − x n || → 0 as n → ∞.

This implies that,

||x n+i − x n || → 0 as n → ∞, ∀ i ∈ {1, 2, 3, , N}.

But, for iÎ {1,2,3, , N},

||x n − S n+i (t n+i )x n || ≤ δ2

||x n − x n+i || + ||x n+i − T n+i (t n+i )x n+i||

+||T n+i (t n+i )x n+i − T n+i (t n+i )x n||

≤ δ2[(1 + L)||x n+i − x n || + ||x n+i − T n+i (t n+i )x n+i||]

Therefore, lim inf

n→∞ ||x n − S n+i (t n+i )x n|| = 0

Hence,

lim inf

n→∞ ||T n+i (t n+i )x n − x n|| = lim inf

n→∞ [

1

δ2||S n+i (t n+i )x n − x n||] = 0

From the relation,

||T n+i (t n )x n − x n || ≤ ||T n+i (t n )x n − T n+i ((t n+i − t n ) + t n )x n||

+||T n+i (t n+i )x n − x n||

z ∈{x n },s∈R+||T n+i (t n )z − T n+i (s + t n )z|| + ||T n+i (t n+i )x n − x n||, and condition (3.4) we get

lim inf

It follows from (3.8) thatlim infn→∞ ||T l (t n )x n − x n || = 0 ∀ l ∈ {1, 2, 3, , N} This

com-pletes the proof □

Theorem 3.2 Let E, K,F, {an}, {tn},{J i}N

i=1and{xn} be as in lemma 3.1 Assume that, for at least one i Î {1, 2, , N}, there exists a compact subset C of E such that ∪t ≥0Ti(t)

(K)⊂ C Then, the sequence {xn} converges to some elementF

Proof By Lemma 3.1, we havelim infn→∞ ||T l (t n )x n − x n || = 0 ∀ l ∈ {1, 2, 3, , N}

If ∪t≥0Ts(t)(K) ⊂ C for some compact subet C of E and some s Î {1, 2, , N}, then there exists a subsequence{x n k}, of {xn} and q*Î K, such that

x n k → q∗and||T s (t n k )x n k − x n k || → 0 as n → ∞. (3:9) Observe that for t >0,

||T s (t)x n k − x n k || ≤ ||T s (t)x n k − T s (t)T s (t n k )x n k||

+||T s (t)T s (t n k )x n k − T s (t n k )x n k || + ||T s (t n k )x n k − x n k||

≤ ||T s (t + t n k )x n k − T s (t n k )x n k || + (1 + L)||T s (t n k )x n k − x n k||

From the above we haveklim→∞||T s (t)x n k − x n k|| = 0 Using (3.9) and the fact that Ts is Lipschitzian, we get q*Î ∩t ≥0F(Ts(t))

Now, for any l Î {1,2, ,N }, sincelim inf

k→∞ ||T l (t n k )x n k − x n k|| = 0, there exists a subse-quence{x n kj}of{x n k}such that

lim

j→∞||T l (t n kj )x n kj − x n kj|| = lim inf

k→∞ ||T l (t n k )x n k − x n k|| = 0 Then, similarly for t≥ 0, we obtain

||T l (t)x n kj − x n kj || ≤ ||T l (t)x n kj − T l (t)T l (t n kj )x n kj||

+||T l (t)T l (t n kj )x n kj − T l (t n kj )x n kj || + ||T l (t n kj )x n kj − x n kj||

≤ ||T l (t + t n kj )x n kj − T l (t n kj )x n kj || + (1 + L)||T l (t n kj )x n kj − x n kj||

This implies that jlim→∞||T l (t)x n kj − x n kj|| = 0and hence q*Î ∩t≥0F(Tl(t)) Since lÎ {1,

2, N} is arbitrarily chosen, we have q∗∈F As the limitnlim→∞||x n − q∗||exists, the

conclusion of the theorem follows immediately and this completes the proof □

Remark3.3 Observe that considering a single one-parameter family of demi-contrac-tive semigroup in Theorem 3.2, we obtain the conclusion of Theorem 1.1

Let T1, T2, , TNbe a finite family of Lipschitzian demi-contractive self-mapping of

Kwith positive constantsl1,l2, ,lNand Lipschitz constants L1,L2, , LN,

respectively Let F := ∩

1≤i≤N F(T i) .

For a fixedδ Î (0, 1), define Sn: K® K by

Then, it follows that for x, y Î K and i Î F,

S n x − q, j(x − q) ≤ ||x − q||2− ¯λ||x − T n x||2and

||S n x − S n y || ≤ ¯L||x − y||,

where ¯λ = λδ2> 0, ¯L = 1 + δ2L,λ := min

1≤i≤N{λ i}andL := max

1≤i≤N{L i}. The following Theorem is a consequence of Lemma 3.1

Theorem 3.4 Let E, K and {an} be as in Lemma 3.1 Let T1, T2, , TN : K® K be Lipschitzian demi-contractive mappings with Tsdemicompact for at least one sÎ {1, 2,

, N} Let {xn] be a sequence generated by x1 Î K

x n+1=α n+1 x n+ (1− α n+1 )S n+1 x n, (3:11) where Tn = Tn modN Then, {xn} converges strongly to a common fixed point of the family{T i}N

i=1 Proof Following the line of proof of lemma 3.1 we immediately obtain lim

n→∞||x n − q||qk exists for any qÎ F andlim inf

n→∞ ||T i x n − x n|| = 0, ∀i Î {1,2, N} Let

{x n k}be a subsequence of {xn} such that

lim

k→∞||T i x n k − x n k|| = lim inf

n→∞ ||T i x n − x n|| = 0

Therefore lim

k→∞||T s x n k − x n k|| = 0and, by demicompactness of Ts, there exists a sub-sequence{x n kj}of{x n k}and q*Î K, such thatx n kj → q∗as j® ∞

Since,

0 = lim

j→∞||T i x n kj − x n kj || = ||T ilim

j→∞x n kj − lim

j→∞x n kj||

=||T i q∗− q∗||,

we obtain q* Î F But, lim

n→∞||x n − q∗||exists, thus xn® q* Î F and this completes the proof □

The following corollaries follow from Theorem 3.4 Corollary 3.5 Let E, K and {an} be as in Theorem 3.4 Let T1, T2, , TN: K® K be Lipschitzian demi-contractive mappings Suppose there exists a compact subset C in E

such that ∪N

i=1 T i (K) ⊂ C Let {xn} be defined by (3.11) Then, {xn} converges strongly to a common fixed point of the family{T i}N

i=1 Corollary 3.6 Let E; K and {an} be as in Theorem 3.4 Let T1, T2, , TN: K® K be Lipschitzian demi-contractive mappings satisfying condition C Let {xn} be defined by

(3.11) Then, {xn} converges strongly to a common fixed point of the family{T i}N

i=1 Proof Following the line of proof of lemma 3.1, we obtainlim infn→∞ ||x n − T i x n|| = 0for all i Î {1, 2, 3, , N} and ||xn+1 - q||2 ≤ (1 + sn+1) ||xn - q||2, where

σ n+1 = 2(1 + ¯L)3(1− α n+1)2 Since∞

exists and consequently nlim→∞d(x n , F)exists Let {x n k}be a subsequence of {xn} such

that klim→∞||x n k − T i x n k|| = lim inf

n→∞ ||x n − T i x n|| = 0 Then, by using condition C, there exists s Î {1, 2, , N} such that0 = lim

k→∞||x n k − T s x n k|| ≥ lim

the property of f, we get that klim→∞d(x n k , F) = 0, and since the limit lim

n→∞d(x n , F)exists

we have that nlim→∞d(x n , F) = 0 We next show that {xn} is Cauchy Letε > 0 be given,

then there exists p* Î F and n* Î N such that ∀n ≥ n*,||x n − p∗|| < ε

2 Hence, for n≥ n* and k Î N, we have

||x n+k − x n || ≤ ||x n+k − p∗|| + ||x n − p∗||

< ε.

Thus, {xn} is Cauchy and so xn ® q* Î K We now show that q* is in F Since lim

n→∞d(x n , F) = 0, there exists m0Î N large enough and p* Î F such that for all n ≥ m0,

and||x n − p∗|| < ε

6(1+L) Hence,

||q∗− T l q∗|| ≤ ||x n − q∗|| + ||x n − p∗|| + ||p∗− T l q∗||

6(1 + L)+

ε 6(1 + L) + L ||p∗− q∗||

6(1 + L)+

ε 6(1 + L) +

3L ε 6(1 + L)

< ε.

Thus, q*Î F(Tl) and since l Î {1, 2, , N} is arbitrary, we have q* Î F This com-pletes the proof □

Acknowledgements

This study was conducted when the first author was visiting the AbdusSalam International Center for Theoretical

Physics Trieste Italy as an Associate, and the hospitality and financial support provided by the centre is gratefully

acknowledged.

Authors ’ contributions

BA conceived the study, GCU carried out the computations for Theorem 3.4 BA Modified Theorem 3.4 to obtain

Theorem 3.2 Both authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 6 March 2011 Accepted: 23 July 2011 Published: 23 July 2011

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