ORIGINAL PAPER
ON A SOLVABLE SYSTEM OF NON-LINEAR DIFFERENCE
EQUATIONS WITH VARIABLE COEFFICIENTS
MERVE KARA1, YASIN YAZLIK2
_________________________________________________
Manuscript received: 07.11.2020; Accepted paper: 12.01.2021; Published online: 30.03.2021.
Abstract. In this paper, we show that the system of difference equations
2 3
2 3
2 3
0 1 2 3 1 2 3 1 2 3 , , , , n n n n n n n n n n n n n n n n n n n n n n n n x z y x z y x y z n z a b x z x y x y A B z y where the sequences
0, n n a
0, n n b
0, n n
0, n n
0, n n A
0, n nB and the initial values xj,yj,zj, j
1, 2,3
are non-zero real numbers, can be solved in the closed form.Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.
Keywords: periodicity; system of difference equations; forbidden set.
1. INTRODUCTION
Solving non-linear difference equations and their systems is a very hot topics that continue to attract the attention of a wide range of researchers, we can consult the following papers [1-23]. One of important non-linear solvable difference equation is the following difference equation
1 2
1 0 1 2 , . 1 n n n n n n x x x n x x x (1.1)El-Metwally et al. obtained the solutions of the equation (1.1) and studied the behavior of the solutions long time ago in [24].
In an earlier paper, Ibrahim et al. in [25] studied the solutions of the rational difference equation
1 2
1 0 1 2 , , n n n n n n n n x x x n x a b x x (1.2) where
0 n n a and
0 n nb are real two-periodic sequences and initial values x2,x1,x0are nonzero real numbers.
1 Karamanoglu Mehmetbey University, Kamil Ozdag Science Faculty, Department of Mathematics, 70100
Karaman, Turkey. E mail: mervekara@kmu.edu.tr.
2
Nevsehir Haci Bektas Veli University, Faculty of Science and Arts, Department of Mathematics, 50300 Nevsehir, Turkey. E-mail: yyazlik@nevsehir.edu.tr.
A few years ago, in [26], Ahmed et al. investigated the periodic character and the form of the solutions of some rational difference equations systems of order-three
1 2
1 2
1 1 0 1 2 1 2 , , , 1 1 n n n n n n n n n n n n x y y x x y n y x y x y x (1.3)by induction with x2,x1,x y0, 2,y1 and y are nonzero real numbers. When the assumption 0
of xn yn and x2 y2,x1 y1, x0 y0 in system (1.3), system (1.3) is reduced special case of the equation (1.2).
Recently, in [27] we showed that the following difference equations system
2 3
2 3
0 1 2 3 1 2 3 , , , n n n n n n n n n n n n n n n n x y y x x y n y a b x y x y x (1.4)where the sequences
0, n n a
0, n n b
0, n n
0 n n and the initial values xj,yj,
1, 2,3
j are non-zero real numbers can be solved in closed-form. In addition, we obtained the forbidden set of the initial values xj,yj, j
1, 2,3
for system (1.4) and give a study of the long-term behavior of its solutions when for every n 0, all the sequences
an ,
bn ,
n ,
n are constant.Quite recently in [28], was found exact formulas for the solutions of the system
1
1
1 1 0 1 1 , , , n k n k n k n k n n n n n n k n k n n n n k n k x y y x x y n y a b x y x c d y x (1.5) where
0, n n a
0, n n b
0 n n c and
0 n nd are non-zero real sequences. System (1.4) can obtain by taking k2 in system (1.5).
Finally, we showed that the following higher-order system of nonlinear difference equations
,
, 0, n k n k l n k n k l n n n l n n n k n k l n l n n n k n k l x y y x x y n y a b x y x y x (1.6) where k l, ,
0 , n n a
0 , n n b
0 , n n
0 n n and the initial values xj,yj,
1, ,
j kl are real numbers can be solved in [29]. Also, by using the solutions of system (1.6), we investigate the asymptotic behavior of well-defined solutions of the above difference equations system for the case k2, lk.
A natural question is to study three-dimensional form of equation (1.2) and system (1.4) solvable in closed-form. Here we study such a system. That is, we deal with the following system of difference equations
2 3
2 3
2 3
0 1 2 3 1 2 3 1 2 3 , , , , n n n n n n n n n n n n n n n n n n n n n n n n x z y x z y x y z n z a b x z x y x y A B z y (1.7)where the sequences
0, n n a
0, n n b
0, n n
0, n n
0, n n A
0 n nB and the initial values xj,yj,zj, j
1, 2,3
are non-zero real numbers.Definition 1.1. (Periodicity) Let
x y zn, n, n
n3 be solutions to difference equations system (1.7). The solutions
x y zn, n, n
n3 is said to be eventually periodic pif xn p xn, yn p yn,n p n
z z for all nn0. If n0 3 is said that the solutions are periodic with period p. Lemma 1.2. [30] Let
0 n n a and
0 n nb be two sequences of real numbers and the sequences y2m i,i
0,1 be solutions of the equations
2m i 2m i 2m 1 i 2m i, 0.
y a y b m (1.8)
Then, for each fixed i
0,1 and m 1, equation (1.8) has the general solutions2 2 2 2 2 0 0 1 . m m m m i i j i l i j i l j j l y y a b a
Further, if
0 n n a and
0 n nb are constant and i
0,1 , then
1 1 1 2 1 2 2 , if 1, 1 , if 1. m m a i a m i i a y b a y y b m a 2. CLOSED-FORM SOLUTIONS OF SYSTEM (1.7)
Let
3
, ,
n n n n
x y z be solutions of system (1.7). If at least one of the initial values
, , , 1, 2,3,
i i i
x y z i is equal to zero, then the solutions of system (1.7) is not defined. For example, if x3 0,then y0 0and so z is not defined. Similarly, if 1 y30 (or z3 0), then z0 0 (or x0 0) and so x (or 1 y ) is not defined. For 1 i1, 2, the other cases are similar.
On the other hand, if
0 0
n
x for some n0 0,then according to the first equation in (1.7) we have that 0 2 0 n x or 0 3 0. n z If 3 n0 2 1or 3 n0 3 1,then we have a
0 1, 2,3 , j such that 0 0 j x or 0 0. jz If n0 3 then by using the equations in (1.7) we
have that 0 4 0 n x or 0 5 0 n z if 0 2 0, n x or 0 5 0 n z or 0 6 0 n y if 0 3 0. n z If 0 3 n 4 1
or 3 n0 5 1in the first case, or 3 n0 5 1or 3 n0 6 1 in the second case, then we have a j1
1, 2,3
such that1 0
j
x or yj1 0or zj1 0.
Repeating this procedure we find a p
1, 2,3
such that xp 0or yp 0or zp 0.As we have proved above, such solutions are not defined.2 3 2 3 2 3 0 1 2 3 1 2 3 1 2 3 1 1 1 , , , . n n n n n n n n n n n n n n n n n n n n n n n n a b x z y x A B z y n x z x z y x y x z y z y (2.1)
Applying the substitution
1 1 1 1 1 1 , , , 2, n n n n n n n n n u v w n x z y x z y (2.2)
then system (2.1) reduce to the following linear difference equations of order two
2 , 2 , 2 , 0.
n n n n n n n n n n n n
u a u b v v w A w B n (2.3)
In view of Lemma 1.2, for i
0,1 , the general solutions of equations in (2.3) are2 2 2 2 2 0 0 1 2 2 2 2 2 0 0 1 2 2 2 2 2 0 0 0 1 , , , . m m m m i i j i l i j i l j j l m m m m i i j i l i j i l j j l m m m m i i j i l i j i l j j l u u a b a v v w w A B A m
(2.4)From equations in (2.2) we have that
2 1 2 3 2 5 2 2 3 2 2 2 2 4 2 1 2 3 2 5 2 2 3 2 2 2 2 4 2 1 2 3 2 5 2 2 3 2 2 2 2 4 , , , , m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i w u v x x u v w u v w y y v w u v w u z z m w u v (2.5)
6 1 6 3 6 5 6 6 1 0 6 6 2 6 4 6 1 6 3 6 5 6 6 1 0 6 6 2 6 4 6 1 6 3 6 5 6 6 1 0 6 6 2 6 4 , , , , , , m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j m j w u v x x m u v w u v w y y m v w u v w u z z m w u v (2.6)
where j
3, 4,5, 6, 7,8 .
From (2.6), we get that6 1 6 3 6 5 6 6 0 6 6 2 6 4 6 1 6 3 6 5 6 6 0 6 6 2 6 4 6 1 6 3 6 5 6 6 0 6 6 2 6 4 , , , m j l j l j l m l l j j l j l j l m j l j l j l m l l j j l j l j l m j l j l j l m l l j j l j l j l w u v x x u v w u v w y y v w u v w u z z w u v
(2.7)where m 1 and l
3, 4,5, 6, 7,8 .
From (2.7)6 2 1 6 2 3 6 2 5 6 2 2 6 0 6 2 6 2 2 6 2 4 6 2 1 6 2 3 6 2 5 6 2 2 6 0 6 2 6 2 2 6 2 4 6 2 1 6 2 6 2 2 6 6 2 , , m j i k j i k j i k m i k i k j j i k j i k j i k m j i k j i k j i k m i k i k j j i k j i k j i k j i k j i m i k i k j i k w u v x x u v w u v w y y v w u v w z z w
3 6 2 5 0 6 2 2 6 2 4 , m k j i k j j i k j i k u u v
(2.8)3 3 3 3 1 3 1 3 1 2 2 2 2 2 2 2 2 0 0 0 1 0 1 6 2 1 2 5 3 3 3 3 1 3 1 3 1 0 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l w A B A u a b a x x u a b a v
3 2 3 2 3 2 2 2 2 2 0 0 1 3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 3 3 2 2 3 2 2 0 0 1 1 1 1 2 5 3 3 3 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l v w A B A A z y B A x y z x x y z a x z b a
3 1 3 1 3 1 2 2 3 2 2 0 0 1 3 3 3 1 3 1 3 1 0 2 1 1 2 2 1 2 1 0 0 0 0 1 3 2 3 2 3 2 2 2 3 2 2 0 0 1 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l a x z b a y x y x A z y B A
3 2 3 2 2 0 0 , j i j i j i l s
(2.9) 3 3 3 3 1 3 1 3 1 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 6 2 2 2 4 3 1 3 1 3 1 3 3 3 0 2 2 2 2 2 2 2 2 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l w A B A u a b a x x u a b a v
3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 1 3 1 3 1 2 2 2 2 0 0 1 3 3 3 2 1 1 2 2 1 2 1 0 3 3 3 0 1 2 4 3 1 1 1 2 2 3 2 2 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l v w A B A A z y B A x y z x x y z a x z b a
3 1 2 1 1 23 1 2 13 1 2 1 0 0 1 3 1 3 1 1 3 3 3 0 2 2 3 2 2 0 0 0 0 1 3 2 3 2 3 2 2 1 1 2 2 1 2 1 0 0 1 3 2 2 3 2 2 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l a x z b a y x y x A z y B A
3 1 3 1 1 0 0 , j i j i j i l s
(2.10)3 3 3 3 1 3 1 3 1 2 2 2 2 2 2 2 2 0 0 0 1 0 1 6 2 1 2 5 3 3 3 3 1 3 1 3 1 0 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l u a b a v y y v w A B A
3 2 3 2 3 2 2 2 2 2 0 0 1 3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 3 3 2 2 3 2 2 0 0 1 1 1 1 2 5 3 3 3 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l w A B A u a b a a x z b a x y z y x y z y x
3 1 3 1 3 1 2 2 3 2 2 0 0 1 3 3 3 1 3 1 3 1 0 2 1 1 2 2 1 2 1 0 0 0 0 1 3 2 3 2 3 2 2 2 3 2 2 0 0 1 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l y x A z y B A A z y B A a x z b a
3 2 3 2 2 0 0 , j i j i j i l s
(2.11) 3 3 3 3 1 3 1 3 1 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 6 2 2 2 4 3 1 3 1 3 1 3 3 3 0 2 2 2 2 2 2 2 2 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l u a b a v y y v w A B A
3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 1 3 1 3 1 2 2 2 2 0 0 1 3 3 3 2 1 1 2 2 1 2 1 0 3 3 3 0 1 2 4 3 1 1 1 2 2 3 2 2 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l w A B A u a b a a x z b a x y z y x y z y x
3 1 2 1 1 23 1 2 13 1 2 1 0 0 1 3 1 3 1 1 3 3 3 0 2 2 3 2 2 0 0 0 0 1 3 2 3 2 3 2 2 1 1 2 2 1 2 1 0 0 1 3 2 2 3 2 2 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l y x A z y B A A z y B A a x z b a
3 1 3 1 1 0 0 , j i j i j i l s
(2.12)3 3 3 3 1 3 1 3 1 2 2 2 2 2 2 2 2 0 0 0 1 0 1 6 2 1 2 5 3 3 3 3 1 3 1 3 1 0 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l v w A B A z z w A B A u a b a
3 2 3 2 3 2 2 2 2 2 0 0 1 3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 3 3 2 2 3 2 2 0 0 1 1 1 1 2 5 3 3 3 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l u a b a v y x x y z z x y z A z y B A
3 1 3 1 3 1 2 2 3 2 2 0 0 1 3 3 3 1 3 1 3 1 0 2 1 1 2 2 1 2 1 0 0 0 0 1 3 2 3 2 3 2 2 2 3 2 2 0 0 1 3 2 1 1 2 2 1 2 1 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l A z y B A a x z b a a x z b a y x
3 2 3 2 2 0 0 , j i j i j i l s
(2.13) 3 3 3 3 1 3 1 3 1 1 2 1 2 1 2 1 1 2 1 2 1 2 1 0 0 0 1 0 1 6 2 2 2 4 3 1 3 1 3 1 3 3 3 0 2 2 2 2 2 2 2 2 0 0 0 1 0 1 j i j i j i j i j i j i s l s s l s m l l s s l s s l m i i j i j i j i j i j i j i j s l s s l s l l s s l s s l v w A B A z z w A B A u a b a
3 2 3 2 3 2 1 2 1 2 1 2 1 0 0 1 3 1 3 1 3 1 2 2 2 2 0 0 1 3 3 3 2 1 1 2 2 1 2 1 0 3 3 3 0 1 2 4 3 1 1 1 2 2 3 2 2 1 j i j i j i s l s l s s l j i j i j i s l s l s s l j i j i j i s l s l s s l i j i s l s s l u a b a v y x x y z z x y z A z y B A
3 1 2 1 1 23 1 2 13 1 2 1 0 0 1 3 1 3 1 1 3 3 3 0 2 2 3 2 2 0 0 0 0 1 3 2 3 2 3 2 2 1 1 2 2 1 2 1 0 0 1 3 2 2 3 2 2 1 j i j i j i s l s m l s s l j i j i j i j i j i j s l s l l s s s l j i j i j i s l s l s s l s l s s l A z y B A a x z b a a x z b a y x
3 1 3 1 1 0 0 , j i j i j i l s
(2.14) for every m 1, i
1, 2,3 .
The forbidden set of the initial values for system (1.7) can be given in the following theorem.
Theorem 2.1. Assume that an 0,bn 0,n 0, n 0, An 0, Bn 0, n 0. Then the forbidden set of the initial values for system (1.7) is given by the set
0 1 9 3 2 1 3 2 1 3 2 1 2 3 2 3 2 3 0 3 9 3 2 1 3 2 1 3 2 1 1 1 1 1 { , , , , , , , , : , , } { , , , , , , , , : 0, 0, 0}, i i i i i i m i m m m j j j j F x x x y y y z z z x z y x z y c d e x x x y y y z z z x y z where 1 1 1 2 2 2 0 2 0 2 0 2 0 2 0 2 0 2 1 1 1 : 0, : 0, : 0. j j j m m m j i j i j i m m m j j i l l i j j i l l i j j i l l i b B c d e a a A A
Proof: At the beginning of Section 2, we have acquired that the set
3 9 3 2 1 3 2 1 3 2 1 1 { , , , , , , , , : j 0, j 0, j 0} j x x x y y y z z z x y z belongs to the forbidden set of the initial values for system (1.7). Now, we assume that 0,
n
x yn 0 and zn 0.Note that the system (1.7) is undefined, when the conditions
2 3 0, n n n n a b x z nnyn2xn30 or AnB zn n2yn3 0, that is, 2 3 , n n n n a x z b 2 3 n n n n y x or 2 3 , n n n n A z y B
for some n 0, are satisfied(Here we consider that
0,
n
b n 0 and An 0 for some n 0). From this and equations in (2.2), we get
2 2 2 2 1 2 1 2 1 2 2 2 , , , m i m i m i m i m i m i m i m i m i b B u v w a A (2.15)
for some m 0 and i
0,1 . Hence, we can determine the forbidden set of the initial values for system (1.7) by using the equations in (2.2). Now, we consider the functions
2 2 2 2 2 2 2 2 2 0 : , : , : , , 0,1 , m i m i m i m i m i m i m i m i m i f t a t b g t t h t A t B m i (2.16)which correspond to the equations of (2.3). From (2.3) and (2.16), we can write
2m i 2m i 2m1 i i i 2 ,
2m i 2m i 2m 1 i i i 2 , v g g g v (2.18)
2m i 2m i 2m1 i i i 2 , w h h h w (2.19)where m 0 and i
0,1 . By using (2.15) and implicit forms (2.17)-(2.19) and considering
1 2 2 2 0 m i , m i m i b f a 1
2 2 2 0 m i, m i m i g 1
2 2 2 0 m i, m i m i B h A for m 0 and i
0,1 , we have
1 1 1 1 1 1 2 2 0 , 2 2 0 , 2 2 0 , i i m i i i m i i i m i u f f v g g w h h (2.20) where 21
2 2 , m i m i m i t b f t a 1
2 2 2 , m i m i m i t g t 1
2 2 2 , m i m i m i t B h t A for m 0,i
0,1 . From (2.20) we obtain 1 2 2 0 2 0 2 1 , j m j i i j j i l l i b u a a
2 1 2 0 2 0 2 1 , j m j i i j j i l l i v
1 2 2 0 2 0 2 1 , j m j i i j j i l l i B w A A
for some m 0 and i
0,1 . This means that if one of the conditions in (2.20) holds, then mth iteration or
m 1
th iteration in system (1.7) can not be calculated.3. CASE OF CONSTANT COEFFICIENTS
In this section, we examine the forms of solutions of system (1.7) for the case when ,
n
a a bn b, n , n , An A, and Bn B, for every n 0.Then, the system (1.7) becomes
2 3
2 3
2 3
0 1 2 3 1 2 3 1 2 3 , , , . n n n n n n n n n n n n n n n n n n x z y x z y x y z n z a bx z x y x y A Bz y (3.1)We start the following theorem describing the form of well-defined solutions of system (3.1).
Theorem 3.1. Let
3
, ,
n n n n
x y z be well-defined solutions of system (3.1). Then, for 1
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 1 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k A A z y B z y B x y z x x x y z a a x z b x z b a a x z b x z b y x y x y x y x
3 2 1 1 , 1 j i k k k k k A A z y B z y B (3.2)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 1 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k a a x z b x z b x y z y y x y z y x y x y x y x A A z y B z y B A A z y B z y
3 2 1 1 , 1 j i k k k k k B a a x z b x z b (3.3)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 1 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k y x y x x y z z z x y z A A z y B z y B A A z y B z y B a a x z b x z b a a x z b x z
3 2 1 1 , 1 j i k k k k k b y x y x (3.4) when a1,1,A1,
2 3 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 3 1 1 1 1 1 1 3 k m k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k k k z y B j i x y z x x x y z a a x z b x z b a a x z b x z b y x y x y x y x z y B
j i k 2
, (3.5)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 1 3 4 3 2 1 1 1 1 1 3 1 1 3 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k k k k k j i k k k a a x z b x z b x y z y y x y z y x y x y x y x z y B j i k z y B j i a a x z b x
1 , kz k b (3.6)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 3 1 1 1 3 1 3 4 3 4 3 2 1 1 1 3 1 3 1 1 1 k j i m k k k k m i k i k j k k k k j i k k k k k j i k k k k j i k k k y x y x x y z z z x y z z y B j i k z y B j i a a x z b x z b a a x z b x z b y x y
1 , kx k (3.7) when a1,1,A1,
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 1 3 4 3 2 1 1 1 1 1 3 1 1 3 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k k k k k j i k k k A A z y B z y B x y z x x x y z a a x z b x z b a a x z b x z b y x j i k y x j i A A z y B z
1 , ky k B (3.8)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 3 1 1 1 3 1 3 4 3 4 3 2 1 1 1 3 1 3 1 1 1 k j i m k k k k m i k i k j k k k k j i k k k k k j i k k k k j i k k k a a x z b x z b x y z y y x y z y x j i k y x j i A A z y B z y B A A z y B z y B a a x z b x
1 , kz k b (3.9)
2 3 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 3 1 1 1 1 1 1 3 k m k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k k k y x j i x y z z z x y z A A z y B z y B A A z y B z y B a a x z b x z b a a x z b x z b y x
j i k 2
, (3.10)when a1,1,A1,
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 3 1 1 1 3 1 3 4 3 4 3 2 1 1 1 3 1 3 1 1 1 k j i m k k k k m i k i k j k k k k j i k k k k k j i k k k k j i k k k A A z y B z y B x y z x x x y z x z b j i k x z b j i y x y x y x y x A A z y B z
1 , ky k B (3.11)
2 3 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 3 1 1 1 3 1 3 4 3 4 1 1 3 1 1 1 1 1 1 3 k m k k m i k i k j i k j k k k k j i k k k k j i k k k k k j i k k k k k k x z b j i x y z y y x y z y x y x y x y x A A z y B z y B A A z y B z y B x z b
j i k 2
, (3.12)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 1 3 4 3 2 1 1 1 1 1 3 1 1 3 1 1 k j i m k k k k m i k i k j i k j k k k k j i k k k k k k k k j i k k k y x y x x y z z z x y z A A z y B z y B A A z y B z y B x z b j i k x z b j i y x y
1 , kx k (3.13) when a1, 1,A1,
2 3 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 1 3 4 1 1 3 1 1 1 1 3 1 1 3 1 , 1 3 2 k m k k m i k i k j i k j k k k k j i k k k k k k k k k k z y B j i x y z x x x y z a a x z b x z b a a x z b x z b y x j i k y x j i z y B j i k
(3.14)
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 1 3 4 3 2 1 1 1 1 3 1 3 1 3 1 1 3 1 , 1 k j i m k k k k m i k i k j k k k k k k k k j i k k k k k a a x z b x z b x y z y y x y z y x j i k y x j i z y B j i k z y B j i a a x z b x z b
(3.15)
2 3 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 3 1 1 1 3 1 3 4 3 4 1 1 3 1 1 3 1 3 1 1 , 1 3 2 k m k k m i k i k j k k k k j i k k k k k j i k k k k k k y x j i x y z z z x y z z y B j i k z y B j i a a x z b x z b a a x z b x z b y x j i k
(3.16) when a1, 1,A1,
2 3 3 1 3 4 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 1 3 4 3 2 1 1 1 1 3 1 3 1 3 1 1 3 1 , 1 k j i m k k k k m i k i k j k k k k k k k k j i k k k k k A A z y B z y B x y z x x x y z x z b j i k x z b j i y x j i k y x j i A A z y B z y B
(3.17)
2 3 3 4 3 3 3 6 2 2 6 0 1 1 1 1 3 4 3 1 1 1 3 1 3 4 3 4 1 1 3 1 1 3 1 3 1 1 , 1 3 2 k m k k m i k i k j k k k k j i k k k k k j i k k k k k k x z b j i x y z y y x y z y x j i k y x j i A A z y B z y B A A z y B z y B x z b j i k
(3.18)
2 3 3 4 3 3 3 6 2 2 6 3 0 1 1 1 1 1 3 3 4 3 4 1 3 4 1 1 3 1 1 1 1 3 1 1 3 1 , 1 3 2 k m k k m i k i k j i k j k k k k j i k k k k k k k k k k y x j i x y z z z x y z A A z y B z y B A A z y B z y B x z b j i k x z b j i y x j i k
(3.19)when a1,1,A1,