Assume that every nonempty closed bounded convex subset ofK has the fixed point property for nonexpansive map-pings.. A strong convergence theorem is proved for a common fixed point of
Trang 1POINT OF A FINITE FAMILY OF NONSELF
NONEXPANSIVE MAPPINGS
C E CHIDUME, HABTU ZEGEYE, AND NASEER SHAHZAD
Received 10 September 2003 and in revised form 6 July 2004
LetK be a nonempty closed convex subset of a reflexive real Banach space E which has a
uniformly Gˆateaux differentiable norm Assume that K is a sunny nonexpansive retract
ofE with Q as the sunny nonexpansive retraction Let Ti:K → E, i =1, ,r, be a
fam-ily of nonexpansive mappings which are weakly inward Assume that every nonempty closed bounded convex subset ofK has the fixed point property for nonexpansive
map-pings A strong convergence theorem is proved for a common fixed point of a family of nonexpansive mappings provided thatTi,i =1, 2, ,r, satisfy some mild conditions.
1 Introduction
LetK be a nonempty closed convex subset of a real Banach space E A mapping T : K → E
is called nonexpansive if Tx − T y ≤ x − y for allx, y ∈ K Let T : K → K be a
non-expansive self-mapping For a sequence{ αn }of real numbers in (0, 1) and an arbitrary
u ∈ K, let the sequence { x n }inK be iteratively defined by x0∈ K,
xn+1:= αn+1 u +
1− αn+1
Halpern [5] was the first to study the convergence of the algorithm (1.1) in the framework
of Hilbert spaces Lions [6] improved the result of Halpern, still in Hilbert spaces, by proving strong convergence of{ xn }to a fixed point ofT if the real sequence { αn }satisfies the following conditions:
(i) limn →∞ α n =0;
(ii)∞
n =1αn = ∞;
(iii) limn →∞((αn − αn −1)/α2
n)=0.
It was observed that both Halpern’s and Lions’ conditions on the real sequence{ α n } ex-cluded the natural choice,αn:=(n + 1) −1 This was overcome by Wittmann [12] who proved, still in Hilbert spaces, the strong convergence of{ xn }if{ αn }satisfies the follow-ing conditions:
(i) limn →∞ αn =0;
(ii)∞
n =1αn = ∞;
(iii)∗∞
n =0 α n+1 − α n< ∞ .
Copyright©2005 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2005:2 (2005) 233–241
DOI: 10.1155/FPTA.2005.233
Trang 2Reich [9] extended this result of Wittmann to the class of Banach spaces which are uni-formly smooth and have weakly sequentially continuous duality maps Moreover, the se-quence{ αn }is required to satisfy conditions (i) and (ii) and to be decreasing (and hence also satisfying (iii)∗) Subsequently, Shioji and Takahashi [10] extended Wittmann’s re-sult to Banach spaces with uniformly Gˆateaux differentiable norms and in which each nonempty closed convex subset ofK has the fixed point property for nonexpansive
map-pings and{ αn }satisfies conditions (i), (ii), and (iii)∗
Xu [13] showed that the results of Halpern holds in uniformly smooth Banach spaces if
{ αn }satisfies the following conditions:
(i) limn →∞ αn =0;
(ii)∞
n =1α n = ∞;
(iii)∗∗limn →∞((αn − αn −1)/αn)=0.
As has been remarked in [13], conditions (iii) and (iii)∗are not comparable Also condi-tions (iii)∗and (iii)∗∗are not comparable However, condition (iii) does not permit the natural choiceαn:=(n + 1) −1for all integersn ≥0 Hence, conditions (iii)∗and (iii)∗∗ are preferred
In [2], Chidume et al extended the results of Xu to Banach spaces which are more general than uniformly smooth spaces
Next considerr nonexpansive mappings T1,T2, ,Tr For a sequence{ αn } ⊆(0, 1) and an arbitraryu0∈ K, let the sequence { x n }inK be iteratively defined by x0∈ K,
xn+1:= αn+1u +
1− αn+1
whereTn = Tn(modr)
In 1996, Bauschke [1] defined and studied the iterative process (1.2) in Hilbert spaces with conditions in (i), (ii), and (iii)∗on the parameter{ α n }
Recently, Takahashi et al [11] extended Bauschke’s result to uniformly convex Banach spaces More precisely, they proved the following result
Theorem 1.1 [11] Let K be a nonempty closed convex subset of a uniformly convex Banach space E which has a uniformly Gˆateaux differentiable norm Let T i:K → K, i =1, ,r, be a family of nonexpansive mappings with F : = r
i =1F(Ti) = ∅ and r
i =1F(Ti) =
F(TrTr −1··· T1)= F(T1Tr ··· T2)= ··· = F(Tr −1Tr −2··· T1Tr ) For given u,x0∈ K, let
{ x n } be generated by the algorithm
xn+1:= αn+1u +
1− αn+1
where Tn:= Tn(modr) and { αn } is a real sequence which satisfies the following conditions:
(i) limn →∞ α n = 0; (ii)∞
n =1α n = ∞ , and (iii) ∗∞
n =1| α n+r − α n | < ∞ Then { x n } converges strongly to a common fixed point of { T1,T2, ,Tr } Further, if Px0=limn →∞ xn for each
x0∈ K, then P is a sunny nonexpansive retraction of K onto F.
More recently, O’Hara et al [8] proved the following complementary result to Bauschke’s theorem [1] with condition (iii)∗replaced with (iii)∗∗ limn →∞((α n+r − α n)
/α n+r)=0 (or equivalently, limn →∞(α n /α n+r)=1)
Trang 3Theorem 1.2 [8] Let K be a nonempty closed convex subset of a Hilbert space H and let
T i:K → K, i =1, ,r, be a family of nonexpansive mappings with F : =r
i =1F(T i) = ∅ and
r
i =1F(Ti)= F(TrTr −1··· T1)= F(T1Tr ··· T2)= ··· = F(Tr −1Tr −2··· T1Tr ) For given u,x0∈ K, let { xn } be generated by the algorithm
x n+1:= α n+1 u +
1− α n+1
where Tn:= Tn(modr) and { αn } is a real sequence which satisfies the following conditions: (i)
limn →∞ α n = 0; (ii) ∞
n =1α n = ∞ , and (iii) ∗∗ limn →∞(α n /α n+r)= 1 Then { x n } converges strongly to Pu, where P is the projection of K onto F.
In the above work, the mappingsT1,T2, ,T r remain self-mappings of a nonempty closed convex subsetK either of a Hilbert space or a uniformly convex space If, however,
the domain ofT1,T2, ,Tr,D(Ti)= K, i =1, 2, ,r, is a proper subset of E and Timaps
K into E, then the iteration process (1.4) may fail to be well defined (see also (1.3))
It is our purpose in this paper to define an algorithm for nonself-mappings and to obtain a strong convergence theorem to a fixed point of a family of nonself nonexpansive mappings in Banach spaces more general than the spaces considered by Takahashi et al [11] with { αn }satisfying conditions (i), (ii), and (iii)∗ We also show that our result holds if{ αn }satisfies conditions (i), (ii), and (iii)∗∗ Our results extend and improve the corresponding results of O’Hara et al [8], Takahashi et al [11], and hence Bauschke [1]
to more general Banach spaces and to the class of nonself -maps.
2 Preliminaries
LetE be a real Banach space with dual E ∗ We denote byJ the normalized duality mapping
fromE to 2 E ∗
defined by
Jx : =f ∗ ∈ E ∗:
x, f ∗
= x 2= f ∗ 2
where ·,· denotes the generalized duality pairing It is well known that ifE ∗is strictly convex, thenJ is single valued In the sequel, we will denote the single-valued normalized
duality map byj.
The norm is said to be uniformly Gˆateaux di fferentiable if for each y ∈ S1(0) := { x ∈
E : x =1}, limt →0(( x + ty − x )/t) exists uniformly for x ∈ S1(0) It is well known thatLp spaces, 1< p < ∞, have uniformly Gˆateaux differentiable norm (see, e.g., [4]) Furthermore, ifE has a uniformly Gˆateaux differentiable norm, then the duality map is
norm-to-w ∗uniformly continuous on bounded subsets ofE.
A Banach spaceE is said to be strictly convex if (x + y)/2 < 1 for x, y ∈ E with x =
y =1 andx = y In a strictly convex Banach space E, we have that if x = y =
λx + (1 − λ)y , forx, y ∈ E and λ ∈(0, 1), thenx = y.
LetK be a nonempty subset of a Banach space E For x ∈ K, the inward set of x, IK(x),
is defined byIK(x) : = { x + λ(u − x) : u ∈ K, λ ≥1} A mappingT : K → E is called weakly inward if Tx ∈cl[I K(x)] for all x ∈ K, where cl[I K(x)] denotes the closure of the inward
set Every self-map is trivially weakly inward
Trang 4LetK ⊆ E be closed convex and Q a mapping of E onto K Then Q is said to be sunny
ifQ(Qx + t(x − Qx)) = Qx for all x ∈ E and t ≥0 A mappingQ of E into E is said to be a retraction if Q2= Q If a mapping Q is a retraction, then Qz = z for every z ∈ R(Q), range
of Q A subset K of E is said to be a sunny nonexpansive retract of E if there exists a sunny
nonexpansive retraction ofE onto K and it is said to be a nonexpansive retract of E if there
exists a nonexpansive retraction ofE onto K If E = H, the metric projection PK is a sunny nonexpansive retraction from H to any closed convex subset of H.
In the sequel, we will make use of the following lemma
Lemma 2.1 Let { a n } be a sequence of nonnegative real numbers satisfying the relation
an+1 ≤1− αn
where (i) 0 < αn < 1; (ii)∞
n =1αn = ∞ Suppose, either (a)σn = o(αn ), or ( b)∞
n =1σn < ∞ ,
or (c)limsup n →∞ σ n ≤ 0 Then a n → 0 as n → ∞ (see, e.g., [13]).
We will also need the following results
Lemma 2.2 (see, e.g., [7]) Let E be a real Banach space Then the following inequality holds For each x, y ∈ E,
x + y 2≤ x 2+ 2
y, j(x + y)
∀ j(x + y) ∈ J(x + y). (2.3) Theorem 2.3 [7, Theorem 1, Proposition 2(v)] Let K be a nonempty closed convex subset
of a reflexive Banach space E which has a uniformly Gˆateaux differentiable norm Let T :
K → E be a nonexpansive mapping with F(T) = ∅ Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings Then there exists a continuous path t → zt , 0 < t < 1, satisfying zt = tu + (1 − t)Tzt , for arbitrary but fixed u ∈ K, which converges strongly to a fixed point of T Further, if Pu =limt →0zt for each u ∈ K, then P is a sunny nonexpansive retraction of K onto F(T).
3 Main results
We now prove the following theorem
Theorem 3.1 Let K be a nonempty closed convex subset of a reflexive real Banach space E which has a uniformly Gˆateaux differentiable norm Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction Assume that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive mappings Let
Ti:K → E, i =1, ,r, be a family of nonexpansive mappings which are weakly inward with
F : =r
i =1F(T i) = ∅ andr
i =1F(QT i)= F(QT r QT r −1··· QT1)= F(QT1QT r ··· QT2)=
··· = F(QTr −1QTr −2··· QT1QTr ) For given u,x0∈ K, let { xn } be generated by the algo-rithm
x n+1:= α n+1 u +
1− α n+1
where Tn:= Tn(modr) and { αn } is a real sequence which satisfies the following conditions:
(i) limn →∞ α n = 0; (ii)∞
n =1α n = ∞ ; and either (iii) ∗ ∞
n =1| α n+r − α n | < ∞ , or (iii) ∗∗
limn →∞((α n+r − α n)/α n+r)= 0 Then { x n } converges strongly to a common fixed point
Trang 5of { T1,T2, ,T r } Further, if Pu =limn →∞ x n for each u ∈ K, then P is a sunny nonex-pansive retraction of K onto F.
Proof For x ∗ ∈ F, one easily shows by induction that x n − x ∗ ≤max{ x0− x ∗ , u −
x ∗ }, for all integersn ≥0, and hence{ xn }and{ QTn+1 xn }are bounded But this implies that xn+1 − QTn+1xn = αn+1 u − QTn+1 xn →0 asn → ∞ Now we show that
xn+r − xn −→0 asn −→ ∞ (3.2) From (3.1), we get that
xn+r − xn = αn+r − αn
u − QTnxn −1
+
1− αn+r
QTn+r xn+r −1− QTnxn −1
= αn+r − αn
u − QTnxn −1
+
1− αn+r
QTnxn+r −1− QTnxn −1
≤1− αn+r xn+r −1− xn −1 +αn+r − αnM, (3.3) for someM > 0 We consider two cases.
Case 1 Condition (iii) ∗is satisfied Then,
xn+r − xn ≤ 1 − αn+r xn+r −1− xn −1 +σn, (3.4) whereσn:= M | αn+r − αn |so that∞
n =1σn < ∞
Case 2 Condition (iii) ∗∗is satisfied Then,
x
n+r − x n ≤ 1 − α n+r x n+r −1− x n −1 +σ
whereσn:= αn+r βnandβn:=(| αn+r − αn | M/αn+r) so thatσn = o(αn+r)
In either case, byLemma 2.1, we conclude that limn →∞ xn+r − xn =0 Next we prove that
lim
n →∞ xn − QTn+r ··· QTn+1xn =0. (3.6)
In view of (3.2), it suffices to show that limn →∞ x n+r − QT n+r ··· QT n+1 x n =0 Since
xn+r − QTn+r xn+r −1 = αn+r u − QTn+r xn+r −1and limn →∞ αn =0, we have thatxn+r −
QTn+r xn+r −1→0 From
xn+r − QTn+rQTn+r −1xn+r −2 ≤ xn+r − QTn+r xn+r −1
+ QTn+r xn+r −1− QTn+rQTn+r −1xn+r −2
≤ xn+r − QTn+r xn+r −1 + xn+r −1− QTn+r −1xn+r −2
= x n+r − QT n+r x n+r −1 +α n+r −1 u − QT n+r −1x n+r −2 ,
(3.7)
we also have xn+r − QTn+r QTn+r −1xn+r −2→0 Similarly, we obtain the conclusion Let
z n
t ∈ K be a continuous path satisfying
z t n = tu + (1 − t)QTn+r QTn+r −1··· QTn+1z t n (3.8)
Trang 6guaranteed by Theorem 2.3 Also by Theorem 2.3, z n
t → Pu as t →0+, where P is
the sunny nonexpansive retraction of K onto r
i =1F(QT i) (notice r
i =1F(QT i)=
F(QTn+r QTn+r −1 QTn+1)) and hence asTi,i =1, ,r, is weakly inward by [2, Remark 2.1],Pu ∈ F =r
i =1F(Ti) Leta =lim supn →∞ u − Pu, j(xn − Pu) Now we show that
a ≤0 We can find a subsequence{ x n i }of{ x n }such thata =limi →∞ u − Pu, j(x n i − Pu)
We assume thatni ≡ k(modr) for some k ∈ {1, 2, ,r } UsingLemma 2.2, we have that
z k
t − x n i 2
= t
u − x n i
+
1− t
QT n i+r QT n+r −1··· QT n i+1z k
t − x n i 2
≤(1− t)2 QT
n i+r QT n i+r −1··· QT n i+1z k
t − x n i 2
+ 2t
u − x n i,j
z k
t − x n i
≤(1− t)2 QT n
i+r QT n i+r −1··· QT n i+1z k
t − QT n i+r QT n i+r −1··· QT n i+1x n i + QTn
i+r QTn i+r −1··· QTn i+1xn i − xn i 2
+ 2t
z t k − xn i 2
+
u − z k t,j(z t k − xn i
≤1 +t2 z t − x n
i 2
+ QT
n i+r QT n i+r −1··· QT n i+1x n i − x n i
×2 z k
t − x n i + QT
n i+r QT n i+r −1··· QT n i+1x n i − x n i + 2t
u − z k
t,j
z k
t − x n i
,
(3.9) and hence,
u − z k
t,j
xn i − z k
t
≤ t
2 z k
t − xn i 2
+ QTn
i+rQTn i+r −1··· QTn i+1xn i − xn i
2t
×2 z k
t − xn i + QTn
i+rQTn i+r −1··· QTn i+1xn i − xn i . (3.10) Since{ x n i }is bounded, we have that{ QT n+r QT n i+r −1··· QT n i+1x n i }is bounded and by (3.6), xn i − QTn i+rQTn i+r −1··· QTn i+1xn i →0 as i → ∞, then it follows from the last inequality that
lim sup
t →0 +
lim sup
i →∞
u − z k
t,j
x n i − z k t
Moreover, j is norm-to-w ∗uniformly continuous on bounded subsets of E Thus, we
obtain from (3.11) that
lim sup
i →∞
u − Pu, j
x n i − Pu
and hence lim supn →∞ u − Pu, j(xn − Pu) ≤0 Furthermore, from (3.1), we havexn+1 −
Pu = αn+1(u − Pu) + (1 − αn+1)(QTn+1xn − Pu) Thus usingLemma 2.2, we obtain that x
n+1 − Pu 2
≤1− α n+1 2 QT
n+1 x n − Pu 2
+ 2α n+1
u − Pu, j
x n+1 − Pu
≤1− αn+1 xn − Pu 2
whereσn+1:= αn+1βn+1and lim supn →∞ σn+1 ≤0, forβn+1:= u − Pu, j(xn+1 − Pu) Thus,
byLemma 2.1,{ x n }converges strongly to a common fixed pointPu of { T1,T2, ,T r }
Trang 7If inTheorem 3.1,T i,i =1, ,r, are self-mappings then the projection operator Q is
replaced withI, the identity map on E Moreover, each T i fori ∈ {1, 2, ,r }is weakly inward Thus, we have the following corollary
Corollary 3.2 Let K be a nonempty closed convex subset of a reflexive real Banach space
E which has a uniformly Gˆateaux differentiable norm Assume that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive mappings Let
Ti:K → K, i =1, ,r, be a family of nonexpansive mappings withr
i =1F(Ti) = ∅ and
r
i =1F(Ti)= F(TrTr −1··· T1)= F(T1Tr ··· T2)= ··· = F(Tr −1Tr −2··· T1Tr ) For given u,x0∈ K, let { x n } be generated by the algorithm
x n+1:= α n+1 u +
1− α n+1
T n+1 x n, n ≥0, (3.14)
where Tn:= Tn(modr) and { αn } is a real sequence which satisfies the following conditions:
(i) limn →∞ α n = 0; (ii)∞
n =1α n = ∞ ; and either (iii) ∗ ∞
n =1| α n+r − α n | < ∞ , or (iii) ∗∗
limn →∞((αn+r − αn)/αn+r)= 0 Then { xn } converges strongly to a common fixed point of { T1,
T2, ,Tr } Further, if Pu =limn →∞ xn for each u ∈ K, then P is a sunny nonexpansive re-traction of K onto F.
In the sequel, we will use the following lemma
Lemma 3.3 Let K be a nonempty closed convex subset of a strictly convex real Banach space E Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonex-pansive retraction Let Ti:K → E, i =1, ,r, be a family of nonexpansive mappings which are weakly inward with r
i =1F(T i) = ∅ Let S i:K → E, i =1, ,r, be a family of map-pings defined by S i:=(1− λ i)I + λ i T i , 0 < λ i < 1 for each i =1, 2, ,r Thenr
i =1F(T i)=
r
i =1F(Si)=r
i =1F(QSi ) andr
i =1F(Si)= F(QSrQSr −1··· QS1)= F(QS1QSr ··· QS2)=
··· = F(QS r −1QS r −2··· QS1QS r ).
Proof We note that, since T ifor eachi ∈ {1, 2, ,r }is weakly inward, then by [3, Remark 3.3],S i, is weakly inward Moreover, by [2, Remark 2.1],F(QS i)= F(S i) Furthermore, one easily shows thatF(Si)= F(Ti) for eachi =1, 2, ,r Now we show thatr
i =1F(Si)=
F(QSrQSr −1··· QS1) = F(QS1QSr ··· QS2) = ··· = F(QSr −1QSr −2··· QS1QSr) For simplicity, we prove forr =2 It is clear thatF(S1)
F(S2)⊆ F(QS2QS1) Now, we show that F(QS2QS1)⊆ F(S1)
F(S2) Let z ∈ F(QS2QS1) and w ∈ F(S1)
F(S2)= F(T1)
F(T2) Then,
z − w = QS2QS1z − w
≤ 1− λ2
Q 1− λ1
z + λ1T1z
+λ2T2
Q 1− λ1
z + λ1T1z
− w
≤1− λ2 1− λ1
z + λ1T1z − w +λ2 1− λ1
z + λ1T1z − w
= 1− λ1
(z − w) + λ1
T1z − w
≤1− λ1
z − w +λ1 T1z − w ≤ z − w
(3.15)
Thus from the preceding inequalities and strict convexity ofE, we obtain that z − w =
T1z − w and T2(Q[(1 − λ1)z + λ1T1z]) − w = z − w Therefore, we obtain that z = T1z =
Trang 8Theorem 3.4 Let K be a nonempty closed convex subset of a strictly convex reflexive real Banach space E which has a uniformly Gˆateaux differentiable norm Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction Assume that every nonempty closed bounded convex subset of K has the fixed point property for nonex-pansive mappings Let T i:K → E, i =1, ,r, be a family of nonexpansive mappings which are weakly inward withr
i =1F(Ti) = ∅ Let Si:K → E, i =1, ,r, be a family of mappings defined by Si:=(1− λi)I + λiTi , 0 < λi < 1 for each i =1, 2, ,r For given u,x0∈ K, let
{ x n } be generated by the algorithm
xn+1:= αn+1 u +
1− αn+1
where Sn:= Sn(modr) and { αn } is a real sequence which satisfies the following conditions:
(i) limn →∞ α n = 0; (ii)∞
n =1α n = ∞ ; and either (iii) ∗ ∞
n =1| α n+r − α n | < ∞ , or (iii) ∗∗
limn →∞((αn+r − αn)/αn+1)= 0 Then, { xn } converges strongly to a common fixed point of
{ T1,T2, ,Tr } Further, if Pu =limn →∞ xn for each u ∈ K, then P is a sunny nonexpansive retraction of K onto F.
Proof By Lemma 3.3, r
i =1F(T i) = r
i =1F(S i) = r
i =1F(QS i) and r
i =1F(QS i) =
F(QSrQSr −1··· QS1)= F(QS1QSr ··· QS2)= ··· = F(QSr −1QSr −2··· QS1QSr) Thus, as
in the proof ofTheorem 3.1,xn → x ∗ ∈r
i =1F(Ti) The proof is complete
If inTheorem 3.4,Ti,i =1, ,r, are self-mappings, the following corollary follows Corollary 3.5 Let K be a nonempty closed convex subset of a strictly convex reflex-ive real Banach space E which has a uniformly Gˆateaux differentiable norm Assume that every nonempty closed bounded convex subset of K has the fixed point property for non-expansive mappings Let Ti:K → K, i =1, ,r, be a family of nonexpansive mappings withr
i =1F(Ti) = ∅ Let Si:K → K, i =1, ,r, be a family of mappings defined by Si:=
(1− λ i)I + λ i T i , 0 < λ i < 1 for each i =1, 2, ,r For given u,x0∈ K, let { x n } be generated
by the algorithm
xn+1:= αn+1u +
1− αn+1
where S n:= S n(modr) and { α n } is a real sequence which satisfies the following conditions:
(i) limn →∞ αn = 0; (ii) ∞
n =1αn = ∞ ; and either (iii) ∗ ∞
n =1| αn+1 − αn | < ∞ , or (iii) ∗∗
limn →∞((αn+1 − αn)/αn+r)= 0 Then { xn } converges strongly to a common fixed point of
{ T1,T2, ,T r } Further, if Pu =limn →∞ x n for each u ∈ K, then P is a sunny nonexpansive retraction of K onto F.
Remark 3.6 Corollaries3.2and3.5are improvements of Theorems1.1and1.2to more general Banach spaces (having a uniformly Gˆateaux differentiable norm) than uniformly convex spaces Moreover, IfE is a Hilbert space,Corollary 3.2 reduces to the result of Bauschke [1]
Acknowledgments.
This work was done while the authors Habtu Zegeye and Naseer Shahzad were visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, the first as
Trang 9a Postdoctoral Fellow and the second as a Junior Associate They would like to thank the Centre for hospitality and financial support The authors also thank the referee for valuable remarks
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C E Chidume: Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
E-mail address:chidume@ictp.trieste.it
Habtu Zegeye: Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
E-mail address:habtuzh@yahoo.com
Naseer Shahzad: Department of Mathematics, Faculty of Sciences, King Abdul Aziz University, P.O Box 80203, Jeddah 21589, Saudi Arabia
E-mail address:nshahzad@kau.edu.sa
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