C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 67, N umb er 1, Pages 116–128 (2018) D O I: 10.1501/C om mua1_ 0000000835 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
STANCU TYPE (p; q)-SZÁSZ-MIRAKYAN-BASKAKOV
OPERATORS
TUNCER ACAR, MOHAMMAD MURSALEEN, AND S. A. MOHIUDDINE
Abstract. In the present paper, we introduce Stancu type generalization of (p; q)-Szasz-Mirakyan-Baskakov operators and investigate their approximation properties such as weighted approximation, rate of convergence and pointwise convergence.
1. Introduction
In the last two decades, there has been intensive research on the approximation of functions by positive linear operators introduced by making use of q-calculus. Many q-generalizations of approximation operators and their approximation behaviors were intensively studied. Nowadays, approximation by linear positive operators in post-quantum calculus, namely the (p; q)-calculus is very active area. Mursaleen et al. introduced the (p; q)-analogues of some well-known operators such as Bernstein operators [15], Bernstein-Stancu operators [16], Bleimann-Butzer-Hahn operators [17], Bernstein-Schurer operators [18]. They investigated the approximation prop-erties of above mentioned operators using the techniques of (p; q)-calculus. Also we can refer the readers to some recent papers: (p; q)-Baskakov-Kantorovich operators [3], bivariate (p; q)-Bernstein Kantorovich operators [4], bivariate (p; q)-Baskakov-Kantorovich operators [12].
Let us recall certain notations and de…nitions of (p; q)-calculus. Let 0 < q < p 1: For each nonnegative integer k; n, n k 0, the (p; q)-numbers [k]p;q, (p; q)-factorial [k]p;q! and (p; q)-binomial are de…ned by
[k]p;q := p
k qk
p q ;
[k]p;q! := [k]p;q[k 1]p;q:::1 ; k 1;
1; k = 0
Received by the editors: December 26, 2016, Accepted: April 03, 2017. 2010 Mathematics Subject Classi…cation. 41A25, 41A35, 41A36.
Key words and phrases. (p; q)-integers, (p; q)-Szász-Mirakyan operators, weighted approximation.
c 2 0 1 8 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
and
n k p;q :=
[n]p;q! [n k]p;q! [k]p;q!:
In the case of p = 1, the above notations reduce to q-analogues and one can easily see that [n]p;q= pn 1[n]
q=p:
Further, the (p; q)-power basis is de…ned by
(x a)np;q= (x + a)(px + qa)(p2x + q2a) (pn 1x + qn 1a); and
(x a)np;q= (x a)(px qa)(p2x q2a) (pn 1x qn 1a): Also the (p; q)-derivative of a function f , denoted by Dp;qf; is de…ned by
(Dp;qf ) (x) :=
f (px) f (qx)
(p q) x ; x 6= 0; (Dp;qf ) (0) := f
0(0)
provided that f is di¤erentiable at 0: The formula for the (p; q)-derivative of a product is
Dp;q(u (x) v (x)) := Dp;q(u (x)) v (qx) + Dp;q(v (x)) u (px) :
For more details on (p; q)-calculus, we refer the readers to [9, 10, 20] and the ref-erences therein. There are two (p; q)-analogues of the exponential function, see [10], ep;q(x) = 1 X n=0 pn(n21)xn [n]p;q! ; and Ep;q(x) = 1 X n=0 qn(n21)xn [n]p;q! ; (1.1)
which satisfy the equality ep;q(x) Ep;q( x) = 1: For p = 1, ep;q(x) and Ep;q(x)
reduce to q-exponential functions. Here we note that the interval of convergence of ep;q(x) is jxj < 1= (p q), jpj < 1 and jqj < 1 and the series (1.1) is convergent for
all x 2 R, jpj < 1 and jqj < 1:
In the recent paper [1], Acar introduced (p; q)-Szász-Mirakyan operators as Sn;p;q(f ; x) := 1 E [n]p;qx 1 X k=0 f [k]p;q qk 2[n] p;q ! qk(k21) [n]kp;qxk [k]p;q! : (1.2) The operators (1.2) are linear and positive and for p = 1; the operators (1.2) turn out to be q-Szász-Mirakyan operators de…ned in [14]. King type modi…cation of (p; q)-Szász-Mirakyan operators was introduced in [2].
Further, Aral et al. proposed (p; q)-Beta function of second kind for m; n 2 N as Bp;q(m; n) =
Z 1
0
xm 1
and considering the (p; q)-Beta function, Gupta [7] introduced (p; q)-Szász-Mirakyan-Baskakov operators as Dnp;q(f ; x) = [n 1]p;q 1 X k=0 sp;qn;k(x) q[k(k+1) 2]=2p(k+1)(k+2)=2 1 Z 0 bp;qn;k(t) f pkt dp;qt; (1.3) where sp;qn;k(x) = 1 Ep;q [n]p;qx qk(k21) [n]kp;qxk [k]p;q! ; b p;q n;k(t) n + k 1 k p;q tk (1 pt)n+kp;q : Similar consideration of the operators given by (1.3) in classical calculus, we refer the readers to [5].
In the present paper, we extend the operators (1.3) for 0 ; and every n 2 N, q 2 (0; 1) ; p 2 (q; 1] ; Stancu type modi…cation of the operators (1.3) can be de…ned by Dn; ;p;q (f ; x) = [n 1]p;q 1 X k=0 sp;qn;k(x) q[k(k+1) 2]=2p(k+1)(k+2)=2 1 Z 0 bp;qn;k(t) f p k[n] p;qt + [n]p;q+ ! dp;qt; (1.4) for x 2 [0; 1) and for every real valued continuous function f on [0; 1). In case
= = 0; the operators (1.4) reduce to (1.3).
We shall …rst give some lemmas which will be necessary to prove our main results. We obtain local approximation behaviors of the operators (1.3) in terms of second order modulus of smoothness and classical modulus of continuity. We also present uniform convergence theorems via weighted Korovkin theorem for the functions belong to weighted spaces. In the last section, we prove the pointwise estimates for the functions satisfying the Lipschitz conditions.
2. Auxiliary Results Lemma 1. ([7]) For x 2 [0; 1) ; 0 < q < p 1; we have
(i) Dp;qn (1; x) = 1 (ii) Dp;q n (t; x) = qp2[n 2]1 p;q + [n]p;qx pq2[n 2] p;q (iii) Dp;q n t2; x = [2]p;q p5q3[n 2] p;q[n 3]p;q+ [q([2]p;q+p)+p2][n]p;q p4q5[n 2] p;q[n 3]p;q x+ [n]2 p;q pq6[n 2] p;q[n 3]p;qx 2:
Lemma 2. For x 2 [0; 1) ; 0 < q < p 1; we have
Dn; ;p;q (t; x) = [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q ; (2.2) Dn; ;p;q t2; x = [n] 4 p;q pq6[n 2] p;q[n 3]p;q [n]p;q+ 2x 2 + 0 B @ q [2]p;q+ p + p2 [n]3 p;q+ 2 [n] 2 p;qp3q3[n 3]p;q p4q5[n 2] p;q[n 3]p;q [n]p;q+ 2 1 C A x + [2]p;q[n] 2 p;q p5q3[n 2] p;q[n 3]p;q [n]p;q+ 2 + 2 [n]p;q [n]p;q+ 2qp2[n 2] p;q + 2 [n]p;q+ 2 :
Proof. Using Lemma 1, we have Dp;qn; ; (1; x) = 1: Also we get
Dn; ;p;q (t; x) = [n 1]p;q 1 X k=0 sp;qn;k(x) q[k(k+1) 2]=2p(k+1)(k+2)=2 1 Z 0 bp;qn;k(t) p k[n] p;qt + [n]p;q+ ! dp;qt = [n]p;q [n]p;q+ D p;q n (t; x) + [n]p;q+ D p;q n (1; x) = [n]p;q [n]p;q+ 1 qp2[n 2] p;q + [n]p;qx pq2[n 2] p;q ! + [n]p;q+ = [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q and Dn; ;p;q t2; x = [n 1]p;q 1 X k=0 sp;qn;k(x) q[k(k+1) 2]=2p(k+1)(k+2)=2 1 Z 0 bp;qn;k(t) p k [n]p;qt + [n]p;q+ !2 dp;qt (2.3) = [n]p;q [n]p;q+ !2 Dp;qn t 2 ; x + 2 [n]p;q [n]p;q+ 2 Dp;qn (t; x) + 2 [n]p;q+ 2 Dp;qn (1; x) = [n]p;q [n]p;q+ !20 @q 3 [2]p;q+ pq q [2]p;q+ p + p2 [n]p;qx + p4[n]2p;qx2 p5q6[n 2] p;q[n 3]p;q 1 A
+ 2 [n]p;q [n]p;q+ 2 q + p [n]p;qx q2p2[n 2] p;q ! + 2 [n]p;q+ 2 = [n] 4 p;q pq6[n 2] p;q[n 3]p;q [n]p;q+ 2x 2 + 0 B @ q [2]p;q+ p + p2 [n]3 p;q+ 2 [n] 2 p;qp 3q3[n 3] p;q p4q5[n 2] p;q[n 3]p;q [n]p;q+ 2 1 C A x + [2]p;q[n] 2 p;q p5q3[n 2] p;q[n 3]p;q [n]p;q+ 2 + 2 [n]p;q [n]p;q+ 2qp2[n 2] p;q + 2 [n]p;q+ 2:
Corollary 1. Using Lemma 2 we set
Dn; ;p;q (t x)2; x = x2 0 B @ [n] 4 p;q pq6[n 2] p;q[n 3]p;q [n]p;q+ 2 2 [n]2p;qx [n]p;q+ pq2[n 2] p;q + 1 1 C A +x 0 B @ h q [2]p;q+ p + p2i[n]3 p;q p4q5[n 2] p;q[n 3]p;q [n]p;q+ 2 + 2 [n] 2 p;q [n]p;q+ 2 pq2[n 2] p;q 2 [n]p;q+ 2 qp2[n 2] p;q [n]p;q+ qp2[n 2] p;q 1 C A + [2]p;q[n] 2 p;q p5q3[n 2] p;q[n 3]p;q [n]p;q+ 2 + 2 [n]p;q [n]p;q+ 2 qp2[n 2] p;q + 2 [n]p;q+ 2 = : n(p; q; x) and n(p; q; x) := 0 @ [n] 2 p;q [n]p;q+ pq2[n 2] p;q 1 1 A x + [n]p;q+ qp 2 [n 2]p;q [n]p;q+ qp2[n 2] p;q :
By CB[0; 1) ; we denote the space of real-valued uniformly continuous and
bounded functions f de…ned on the interval [0; 1) : The norm k k on the space CB[0; 1) is given by
kfk = sup
0 x<1jf (x)j :
Further let us consider the following K-functional: K2(f; ) = inf
g2W2fkf gk + kg 00kg ;
where > 0 and W2 = fg 2 C
B[0; 1) : g0; g002 CB[0; 1)g : By [6, p. 177,
Theo-rem 2.4] there exists an absolute constant C > 0 such that K2(f; ) C!2 f; p ; (2.4) where !2(f; ) = sup 0<h p sup x2[0;1)jf (x + 2h) 2f (x + h) + f (x)j
is the second order modulus of smoothness of f 2 CB[0; 1) : The usual modulus
of continuity of f 2 CB[0; 1) is de…ned by
! (f; ) = sup
0<h
sup
x2[0;1)jf (x + h) f (x)j :
Let Bm[0; 1) be the set of all functions satisfying the condition jf (x)j Mf(1 + xm) ;
x 2 [0; 1); m > 0 with some constant depending on f: Cm[0; 1) = Bm[0; 1) \
C[0; 1) endowed with the norm
kfkm= sup x 0 jf (x)j 1 + xm and Cm[0; 1) = f 2 Cm[0; 1) : lim x!1 jf (x)j 1 + xm < 1 :
Lemma 3. Let f 2 CB[0; 1) : Then for all g 2 CB2 [0; 1) ; we have
^
Dp;qn; ; (g; x) g (x) kg00k n(p; q; x) + 2n(p; q; x) ; (2.5)
where
^
Dn; ;p;q (g; x) = Dp;qn; ; (g; x) + g (x) g Dn; ;p;q (t; x) : (2.6) Proof. By the de…nition of ^Dp;qn; ; and Lemma 2, it is obvious that
^
Dp;qn; ; (t x; x) = 0: (2.7)
Since g 2 CB2[0; 1) ; using the Taylor’s expansion for x 2 [0; 1) we have
g (t) = g (x) + g (x) (t x) +
t
Z
x
Applying the operators ^Dp;qn; ; to both sides of above equality and considering the fact (2.7) we obtain ^ Dn; ;p;q (g; x) g (x) = ^Dn; ;p;q 0 @ t Z x (t u) g00(u) du; x 1 A = Dp;q n; ; 0 @ t Z x (t u) g00(u) du; x 1 A (2.8) Dn; ;p;qZ (t;x) x 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q u 1 A g00(u) du: Also we get t Z x (t u) g00(u) du t Z x jt uj jg00(u)j du kg00k t Z x jt uj du kg00k (t x)2 (2.9) and Dp;qn; ;Z (t;x) x 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q u 1 A g00(u) du 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q x 1 A 2 kg00k := 2n(p; q; x) kg00k : (2.10) Using the inequalities (2.9) and (2.10) in (2.8) we immediately have
^
Dp;qn; ; (g; x) g (x) kg00k n(p; q; x) + 2n(p; q; x) :
Lemma 4. For f 2 CB[0; 1), one has
Dn; ;p;q f kfk :
Proof. In view of (1.4) and Lemma 2, the proof easily follows. 3. Local Approximation
Theorem 1. Let f 2 CB[0; 1) : Then for every x 2 [0; 1) ; there exists a constant
L > 0 such that
Dp;qn; ; (f ; x) f (x) L!2 f ;
q
Proof. By (2.6), for every g 2 C2
B[0; 1) one can obtain
Dn; ;p;q (f ; x) f (x) D^p;qn; ; (f ; x) f (x) + f (x) f 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q 1 A ^ Dn; ;p;q (f g; x) (f g) (x) + f (x) f 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q 1 A + ^Dp;qn; ; (g; x) g (x) :
Taking into account Lemma 4 and Lemma 3 we get Dn; ;p;q (f ; x) f (x) 4 kf gk + f (x) f 0 @ [n] 2 p;qx [n]p;q+ pq2[n 2] p;q + [n]p;q+ qp 2[n 2] p;q [n]p;q+ qp2[n 2] p;q 1 A + kg00k n(p; q; x) + 2n(p; q; x)
and taking in…mum on the right-hand side over all g 2 C2
B[0; 1) and using (2.4), we deduce Dn; ;p;q (f ; x) f (x) 4K2 f ; n(p; q; x) + 2n(p; q; x) + ! (f ; n(p; q; x)) 4!2 f ; q n(p; q; x) + 2n(p; q; x) + ! (f ; n(p; q; x)) = L!2 f ; q n(p; q; x) + 2n(p; q; x) + ! (f ; n(p; q; x)) ; where L = 4M > 0:
Theorem 2. Let f 2 C2[0; 1) ; pn; qn 2 (0; 1) such that 0 < qn < pn 1 and
!a+1(f; ) be its modulus of continuity on the …nite interval [0; a + 1] [0; 1) ;
where a > 0: Then, for every n > 2;
Dp;qn; ; (f ; x) f (x) 4Mf 1 + a2 n(pn; qn; x) + 2!a+1 f;
p
n(pn; qn; x) :
Proof. By [11], !a+1( ; ) has the property
Applying Cauchy-Schwarz inequality and choosing =p n(pn; qn; x), we have Dn; ;p;q (f ; x) f (x) 4Mf 1 + a2 Dn; ;p;q (t x) 2 ; x +!a+1(f; ) 1 + 1 Dn; ;p;q (t x)2; x 1=2 = 4Mf 1 + a2 n(pn; qn; x) + 2!a+1 f; p n(pn; qn; x) ;
which completes the proof.
4. Weighted Approximation
First, let us recall the de…nitions of weighted spaces and corresponding modulus of continuity. Let C [0; 1) be the set of all continuous functions f de…ned on [0; 1) and B2[0; 1) the set of all functions f de…ned on [0; 1) satisfying the
condition jf (x)j M 1 + x2 with some positive constant M which may depend
only on f: C2[0; 1) denotes the subspace of all continuous functions in B2[0; 1) :
By C2[0; 1), we denote the subspace of all functions f 2 C2[0; 1) for which
limx!11+xf (x)2 is …nite: B2[0; 1) is a linear normed space with the norm kfk2 =
supx 0jf (x)j1+x2:
Theorem 3. Let q = qn 2 (0; 1), p = pn 2 (q; 1] such that qn ! 1; pn ! 1 as
n ! 1: Then for each function f 2 C2[0; 1) we get
lim
n!1 D pn;qn n; ; f f
2= 0:
Proof. According to weighted Korovkin theorem proved in [8], it is su¢ cient to verify the following three conditions
lim
n!1 D pn;qn n; ; ei ei
2= 0; i = 0; 1; 2: (4.1)
By (2.1), (4.1) holds for i = 0: By (2.2) and (2.3) we have Dpn;qn n; ; e1 e1 2 = supx 0 n(pn; qn; x) 1 + x2 0 @ [n] 2 pn;qn [n]pn;qn+ pnq2n[n 2]pn;qn 1 1 A sup x 0 x 1 + x2 +[n]pn;qn+ qnp 2 n[n 2]pn;qn [n]p;q+ qp2[n 2] p;q 0 @ [n] 2 pn;qn [n]pn;qn+ pnq2n[n 2]pn;qn 1 1 A
+ [n]pn;qn+ qnp 2
n[n 2]pn;qn [n]pn;qn+ qnp2n[n 2]pn;qn and by similar consideration we have
Dpn;qn n; ; e2 e2 2 [n]4p n;qn pnq6n[n 2]pn;qn[n 3]pn;qn [n]pn;qn+ 2 1 + h qn [2]pn;qn+ pn + p 2 n i [n]3pn;qn p4 nqn5[n 2]pn;qn[n 3]pn;qn [n]pn;qn+ 2 + 2 [n] 2 p;q [n]pn;qn+ 2pnq2n[n 2]pn;qn + [2]pn;qn[n] 2 pn;qn p5 nqn3[n 2]pn;qn[n 3]pn;qn [n]pn;qn+ 2 + 2 [n]p;q [n]pn;qn+ 2qnp2n[n 2]pn;qn + 2 [n]p n;qn+ 2:
Last two inequality mean that (4.1) holds for i = 1; 2: Hence, the proof is completed. Theorem 4. Let p = pnand q = qn satis…es 0 < qn< pn 1 and for n su¢ ciently
large pn! 1, qn! 1 and qnn ! 1 and pnn ! 1: For each f 2 Cx2[0; 1) ; we have
lim
n!1x2[0; 1)sup
Dpn;qn
n; ; (f; x) f (x)
(1 + x2)1+ = 0:
Proof. For any …xed x0> 0;
sup x2[0; 1) Dpn;qn n; ; (f; x) f (x) (1 + x2)1+ sup x x0 Dpn;qn n; ; (f; x) f (x) (1 + x2)1+ + supx x 0 Dpn;qn n; ; (f; x) f (x) (1 + x2)1+ Dpn;qn n; ; (f ) f C[0; x0]+ kfkx 2 sup x x0 Dpn;qn n; ; 1 + t2; x (1 + x2)1+
+ sup
x x0
jf (x)j (1 + x2)1+ :
Since jf (x)j M 1 + x2 ; we have sup x x0 jf (x)j (1+x2)1+ kf k2 (1+x2 0) 1+ Let " > 0 be arbitrary. We can choose x0 to be so large that
kfk2 (1 + x2 0) 1+ < " 3 (4.2)
On the other hand, in view of Lemma 2 we get kfk2nlim !1 Dpn;qn n; ; 1 + t2; x (1 + x2)1+ = 1 + x2 (1 + x2)1+ kfk2 kfk2 (1 + x2) kfk2 (1 + x2 0) < " 3: (4.3) Also, the …rst term of the above inequality tends to zero by well known Korovkin’s theorem, that is,
Dpn;qn n; ; (f ) f
C[0; x0] <"
3: (4.4)
Therefore, combining (4.2)-(4.4) we get the desired result. 5. Pointwise Estimates
Theorem 5. Let 0 < 1 and E be any subset of the interval [0; 1) : Then, if f 2 CB[0; 1) is locally Lip ( ) ; i.e., the condition
jf (y) f (x)j L jy xj ; y 2 E and x 2 [0; 1) (5.1) holds, then, for each x 2 [0; 1) ; we have
Dn; ;p;q (f ; x) f (x) Ln 2
n (x) + 2 (d (x; E))
o ;
where L is a constant depending on and f ; and d (x; E) is the distance between x and E de…ned by
d (x; E) = inf fjt xj : t 2 Eg :
Proof. Let E denote the closure of E in [0; 1) : Then, there exists a point x02 E
such that jx x0j = d (x; E). Using the triangle inequality
jf (t) f (x)j jf (t) f (x0)j + jf (x) f (x0)j
we immediately have by (5.1) that
Dn; ;p;q (f ; x) f (x) Dp;qn; ; (jf (t) f (x0)j ; x) + Dp;qn; ; (jf (x) f (x0)j ; x) LnDn; ;p;q (jt x0j ; x) + jx x0j o LnDn; ;p;q (jt xj + jx x0j ; x) + jx x0j o = LnDn; ;p;q (jt xj ; x) + 2 jx x0j o :
Using Hölder inequality with p = 2= ; q = 2= (2 ) ; we obtain Sn;p;q(f ; x) f (x) L h Dp;qn; ; (jt xj p; x) i1 p + 2 (d (x; E)) = L h Dp;qn; ; jt xj2; x i2 + 2 (d (x; E)) L n 2 n (x) + 2 (d (x; E)) o
Next we obtain the local direct estimate of the operators Dp;qn; ; , using the Lipc-shitz type maximal function of order introduced by Lenze [13] as
~
!a(f; x) = sup t6=x;t2[0;1)
jf (t) f (x)j
jt xj ; x 2 [0; 1) and 2 (0; 1] : (5.2) Theorem 6. Let f 2 CB[0; 1) and 0 < 1: Then, for all x 2 [0; 1) we have
Dp;qn; ; (f ; x) f (x) !~a(f; x) n2 (x) :
Proof. From the Eq. (5.2), we have
Dn; ;p;q (f ; x) f (x) !~a(f; x) Dn; ;p;q (jt xj ; x) :
Applying the Hölder inequality with p = 2= ; q = 2= (2 ) ; we get Dp;qn; ; (f ; x) f (x) !~a(f; x) h Dn; ;p;q jt xj2; x i2 ~ !a(f; x) n2 (x) :
Acknowledgement 1. The …rst author is partially supported by Research Project of Kirikkale University, BAP, 2017/014 (Turkey).
References
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Current address : Tuncer Acar: Kirikkale University, Faculty of Science and Arts, Department of Mathematics, Yahsihan, 71450, Kirikkale, Turkey
E-mail address : tunceracar@ymail.com
Current address : Mohammad Mursaleen: Department of Mathematics, Aligarh Muslim Uni-versity, Aligarh–202002, India
E-mail address : mursaleenm@gmail.com
Current address : S. A. Mohiuddine: OperatorTheory and Applications Research Group, De-partment of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
