Research Article On the Difference Equation x n a n x n k / b n c n x n 1 x n k

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1 Abstract and Applied Analysis Volume 2012, Article ID , 20 pages doi: /2012/ Research Article On the Difference Equation x n a n x n k / b n c n x n 1 x n k Stevo Stević, 1 Josef Diblík, 2, 3 Bratislav Iričanin, 4 and Zdeněk Šmarda 3 1 Mathematical Institute of The Serbian Academy of Sciences and Arts, Knez, Mihailova 36/III, Beograd, Serbia 2 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic 3 Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic 4 Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, Beograd, Serbia Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs Received 27 May 2012; Accepted 2 July 2012 Academic Editor: Jean Pierre Gossez Copyright q 2012 Stevo Stević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The behavior of well-defined solutions of the difference equation x n a n x n k / b n c n x n 1 x n k,n N 0,wherek N is fixed, the sequences a n, b n and c n are real, b n,c n / 0, 0, n N 0, and the initial values x k,...,x 1 are real numbers, is described. 1. Introduction Recently there has been a huge interest in studying nonlinear difference equations and systems see, e.g., 1 33 and the references therein. Here we study the difference equation x n x n k b n c n x n 1 x n k, n N 0, 1.1 where k N is fixed, the sequences b n and c n are real, b n,c n / 0, 0, n N 0, and the initial values x k,...,x 1 are real numbers. Equation 1.1 is a particular case of the equation x n â n x n k b n ĉ n x n 1 x n k, n N 0, 1.2

2 2 Abstract and Applied Analysis with real sequences â n, b n and ĉ n. For â n 0, n N 0, the equation is trivial, and, for â n / 0, n N 0, it is reduced to equation 1.1 with b n b n /â n and c n ĉ n /â n. Equation x n x n k b cx n 1 x n k, n N 0, 1.3 where b, c R, which was treated in 32, is a particular case of equation 1.1. As in 32, here, we employ our idea of using a change of variables in equation 1.1 which extends the one in our paper 21 and is later also used, for example, in 4. For similar methods see also 22, 25. Equation 1.3 in the case k 2 was also studied in 1, 2, in adifferent way. The case when the sequences b n and c n are two-periodic was studied in 31 some related results are also announced in talk 3. For related symmetric systems of difference equations, see 27, 29. For some other recent results on difference equations and systems which can be solved, see, for example, 6, 7, 20 22, 30, 31, 33. Some classical results can be found, for example, in 11. Equation 1.1 is a particular case of the equation y n f y n 1,...,y n k,n ) y n k, n N 0, 1.4 where f : R k 1 R is a continuous function. Numerous particular cases of 1.4 have been investigated, for example, in 9, 21, 23. In this paper we adopt the customary notation k i k 1 g i 1and k i k 1 g i Case c n 0, n N 0 Here we consider the case c n 0, n N 0. In this case equation 1.1 becomes x n x n k b n, n N 0, 2.1 b n / 0, n N 0, from which it follows that for each i {1,...,k} x km i x i m b, m N kj i Using formula 2.2 the following theorem can be easily proved.

3 Abstract and Applied Analysis 3 Theorem 2.1. Consider equation 1.1 with c n 0, b n / 0, n N 0. Then the following statements are true: a if inf m b km i p i > 1, 2.3 for some i {1,...,k}, thenx km i 0 as m ; b if, for each i {1,...,k}, the its p i in 2.3 are greater than 1,thenx n 0 as n ; c if b km i 1, for every m N and for some i {1,...,k}, thenx km i x i, m N 0 ; d if b km i 1, for every m N and for some i {1,...,k}, thenx km i 1 m x i, m N 0 ; e if sup b km i q i 0, 1, 2.4 m and x i / 0, forsomei {1,...,k}, then x km i,asm ; f if, for each i {1,...,k}, the its q i in 2.4 belong to the interval 0, 1 and x i / 0,then x n as n. 3. Case b n 0, n N 0 In this section we consider the case b n becomes 0, n N 0. Note that in this case equation 1.1 x n x n k c n x n 1 x n k 1 x n k, n N 0, 3.1 where c n / 0, n N 0.Ifx n is a well-defined solution of equation 3.1 i.e., a solution with initial values x i / 0, i 1,...,k, which implies x n / 0, n N 0, then x n 1 c n x n 1 x n 2 x n k 1 c n 1x n 2 x n k c n x n 2 x n k 1 c n 1 c n x n k, n N. 3.2 Hence for each i {0, 1,...,k 1} x km i x i c kj i 1 c kj i, m N Using formula 3.3 we easily prove the next theorem.

4 4 Abstract and Applied Analysis Theorem 3.1. Consider equation 1.1 with b n 0, c n / 0, n N 0. Then the following statements are true: a if inf m c km i p i > 1, c km i for some i {0, 1,...,k 1}, thenx km i 0 as m ; b if, for each i {0, 1,...,k 1}, the its p i in 3.4 are greater than 1, thenx n 0 as n ; c if c km i 1 c km i, for every m N and for some i {0, 1,...,k 1}, thenx km i x i, m N 0 ; d if c km i 1 c km i, for every m N and for some i {0, 1,...,k 1}, thenx km i 1 m x i, m N 0 ; e if sup c km i q i 0, 1, m c km i and x i / 0 for some i {0, 1,...,k 1}, then x km i,asm ; f if, for each i {0, 1,...,k 1}, x i / 0 and the its q i in 3.5 belong to the interval 0, 1, then x n as n. 4. Case b n / 0 and c n / 0 The case when b n / 0andc n / 0 for every n N 0 is considered in this section. If x i0 0 for some i 0 {1,...,k}, then from 1.1 we have that x km i0 0, for m N From 4.1 and 1.1 we have that for each i {1,...,k}\{i 0 } x km i x k m 1 i b km i x i m b, m N kj i From 4.1 we see that, for i i 0, 4.2 also holds. Hence Theorem 2.1 can be applied in this case. Note that if x n 0 for some n N 0, then 1.1 implies that there is an i 0 {1,...,k} such that x i0 0, and by the previous consideration we have that 4.2 also holds. If x i / 0, for each i {1,...,k}, then for every well-defined solution we have x n / 0 for n k note that there are solutions which are not well defined, that is, those for which x n 1 x n k b n /c n, for some n N 0.

5 Abstract and Applied Analysis 5 Multiplying equation 1.1 by x n 1 x n k 1 and using the transformation we obtain equation y n 1 x n x n 1 x n k 1, n 1, 4.3 y n b n y n 1 c n, n N Note that from 4.3, for every well-defined solution x n n k of equation 1.1 such that x i / 0, for each i {1,...,k}, it follows that y n / 0, n 1. Since b n / 0, n N 0, we have that n ) n c j y n b i y 1 j, n N i 0 i 0 b i From 4.3 and 4.5 we have that 1 x n y n 1 x n k y n x n 1 x n k 1 y n y 1 n 1 b n y 1 n c j / j i 0 b i ) c j / j i 0 b i ))x n k, 4.6 for every n N 0. Hence, from 4.6, weobtainthat 1/α kl i 1 c j / j x mk i x i l 1 b kl i 1/α kl i i 0 b i ) c j / j i 0 b i )), 4.7 for every m N 0 and each i 1, 2,...,k, where α k x l. l Case b n 1, n N 0 Here we consider the case b n i {1,...,k} 1, n N 0. In this case, from 4.7 we have that for each x mk i x i l 1 1 α kl i 1 c j 1 α kl i c j, m N Note that this formula includes also the case when x i0 0 for some i 0 {1,...,k}.

6 6 Abstract and Applied Analysis Now we formulate and prove a result in this case by using formula 5.1. Theorem 5.1. Consider equation 1.1 with b n 1, n N 0, sign c n sign c 0, n N, α / 0, and n α c j / 1, n N Then the following statements hold: a if for some i {1,...,k} αc kl i l 1 1 α kl i c, j l αc kl i 1 α kl i c j 0, then x mk i 0 as m ; b if 5.3 and 5.4 hold for every i {1,...,k}, thenx n 0 as n ; c if for some i {1,...,k} the sum αc kl i l 1 1 α kl i c j 5.5 converges, then the sequence x mk i is also convergent; d if the sum in 5.5 is finite for every i {1,...,k}, then the sequences x km i are convergent. Proof. Let x n n k be a solution of equation 1.1. Using condition sign c n sign c 0, n N, it is easy to see that if 5.4 holds for some i {1,...,k}, there is an m 0 N such that for j m 0 1 the terms in the product in 5.1 are positive and that the following asymptotic formula ln 1 x x O x 2) 5.6

7 Abstract and Applied Analysis 7 can be used with x being the fraction in the it 5.4. From 5.1 and 5.6 we have that x km i x i l 1 1 α kl i 1 c j 1 α kl i c j x i c m 0 exp x i c m 0 exp x i c m 0 exp m l m 0 1 m l m 0 1 ln 1 α kl i 1 c j 1 α kl i c j αc kl i ln 1 1 α kl i c j m l m 0 1 αc kl i 1 o 1 1 α, kl i c j 5.7 where m 0 c m 0 l 1 1 α kl i 1 c j 1 α kl i c j. 5.8 Using formula 5.7, the assumptions regarding the sum j m 0 1 αc kl i/ 1 α kl i c j and the comparison test for the series whose terms are of eventually the same sign, the results in the theorem easily follow. 6. Case b n 1, n N 0 Here we consider the case b n 1, n N 0. In this case from 4.7 we have x mk i 1 m x i l 1 1 α kl i 1 1 c j 1 α, 6.1 kl i 1 c j for every m N 0 and each i 1, 2,...,k, where α is defined by 4.8.

8 8 Abstract and Applied Analysis Theorem 6.1. Consider equation 1.1 with α / 0, b n 1, n N 0, and n α 1 c j / 1, n N Then the following statements hold: a if for some i {1,...,k} α 1 kl i 1 c kl i l 1 1 α, kl i 1 c j l 1 α 1 kl i 1 c kl i l 1 α 0, kl i 1 c j c 2 kl i 1 α kl i 1 c j ) 2 <, then x mk i 0 as m ; b if for every i {1,...,k}, 6.3, 6.4, and 6.5 hold, then x n 0 as n ; c if for some i {1,...,k} α 1 kl i 1 c kl i l 1 1 α, kl i 1 c j 6.6 conditions 6.4 and 6.5 hold, and x i / 0, then x mk i as m ; d if for every i {1,...,k}, conditions 6.4, 6.5, and 6.6 hold, and x i / 0, i {1,...,k}, then x n as n ; e if for some i {1,...,k} the sum α 1 kl i 1 c kl i l 1 1 α kl i 1 c j 6.7 converges and condition 6.5 holds, then the sequences x 2mk i and x 2m 1 k i are also convergent; f if for every i {1,...,k} the sum in 6.7 converges and condition 6.5 holds, then the sequences x 2km j, j 1,...,2k are convergent.

9 Abstract and Applied Analysis 9 Proof. Let x n n k be a solution of equation 1.1. By 6.4 we see that irrespectively on i {1,...,k}, there is an m 1 N such that for j m 1 1 the terms in the product in 6.1 belong to the interval 1/2, 3/2 and that asymptotic formulae ln 1 x x x2 2 O x 3) 6.8 can be used with x being the fraction in 6.4. From this and 6.1 we have that 1 α kl i 1 1 c j x km i x i l 1 1 α kl i 1 c j x i c 1 m 1 exp m ln 1 α kl i 1 1 c j l m α kl i 1 c j x i c 1 m 1 exp x i c 1 m 1 exp m l m 1 1 l m 1 1 α 1 ln 1 kl i 1 c kl i 1 α kl i 1 c j m α 1 kl i 1 c kl i 1 α kl i 1 c j α 2 c 2 kl i 1 o α ) 2 kl i 1 c j, 6.9 where m 1 c 1 m 1 l 1 1 α kl i 1 1 c j. 1 α kl i 1 c j 6.10 Using formula 6.9, the assumptions of the theorem and some well-known convergence tests for series, the results in a f easily follow. 7. Case b n b n k,c n c n k,n N 0 In this section we consider equation 1.1 for the case b n b n k, c n c n k, n N 0,thatis, when the sequences b n and c n are k-periodic. First we show the existence of k-periodic solutions of equation 4.4. If y0, y 1,...,y k 1 ) 7.1 is such a solution, then we have that y 1 b 1 y 0 c 1, y 2 b 2 y 1 c 2,..., y 0 b k y k 1 c k. 7.2

10 10 Abstract and Applied Analysis By successive eination, or by Kronecker theorem note that system 7.2 is linear, weget y i k 1 c j 1 σ j i 1 k b j s 0 b σ s i, i 1,k, 7.3 if k b j / 1, where σ is the permutation defined by σ i i 1, i 2,k, σ 1 k, 7.4 and σ i σ σ i 1,σ 0 Id, where Id denotes the identity. It is easy to see that 4.4 along with k periodicity of sequences b n and c n implies y km i k k 1 b j y k m 1 i j 1 c σ j i b σ s i, s for every m N 0 and i {1, 2,...,k}, such that k m 1 i 1. Since 7.5 is a linear first-order difference equation, we have that when k b j / 1, its general solution is y km i k m k 1 b j b j y i 1 k b j ) m k 1 j 1 c σ j i b σ s i. s By letting m in 7.6 we obtain the following corollary. Corollary 7.1. Consider equation 4.4 with b n b n k, c n c n k, n N 0. Assume that k b j < Then for every solution y n of the equation we have that m y km i y i, 7.8 for every i {1, 2,...,k}, that is, y n converges to the k-periodic solution in formula 7.3. Let k 1 L i : c σ j i s 0 j 1 b σ s i, i 1,k, q : k b j. 7.9

11 Abstract and Applied Analysis 11 From now on we will use the following convention: if i, j N 0, then we regard that L j L i,ifi j mod k. Also if a sequence m j j N0 is defined by the relation m j f L j, where f is a real function, then we will assume that m j m i,ifi j mod k. Using 7.6 and notation 7.9 in the relation x n y n 1 /y n x n k see 4.6, forthe case q / 1, we have that yi 1 L i 1 / 1 q )) q j L i 1 / 1 q ) x km i x i k yi L i / 1 q )) q j L i / 1 q ) x i k L i 1 L i 1 ) 1 q yi 1 /L i 1 1 ) q j 1 1 q ) y i /L i 1 ), q j 7.10 for every m N 0 and each i {2,...,k}, and yk L k / 1 q )) q j 1 L k / 1 q ) x km 1 x 1 k y1 L 1 / 1 q )) q j L 1 / 1 q ) L k x 1 k 1 ) 1 q yk /L k 1 ) q j 1 L q ) y 1 /L 1 1 ). q j 7.11 Now we present some results, which are applications of formulae 7.10 and Case q 1 If q 1, then by 7.5 we get y km i y k m 1 i L i L i L i y k m 2 i ) yk m 2 i, m N, 7.12 for k m 2 i 1; that is, y km i is two-periodic for each i {1,...,k}. Hence y n is a 2k-periodic solution of equation 4.4, in this case. Hence from the relation x n y n 1 /y n x n k see 4.6, for each i {1, 2,...,k}, we have x km i y km i 1 x km i k y km i 1 y k m 1 i 1 x km i 2k, 7.13 y km i y km i y k m 1 i for k m 1 i. From 7.13 and by 2k periodicity of y n,weget ) l yj 1 y j k 1 x 2kl j x j, l N 0, 7.14 y j y j k for each j { k 1,..., 1, 0, 1,...,k}.

12 12 Abstract and Applied Analysis From 7.14, the behavior of solutions of equation 1.1, in this case, easily follows. For example, if p j : y j 1y j k 1 y j y j k 1, 7.15 for each j { k 1,..., 1, 0, 1,...,k}, then the solution x n n k of 1.1 is 2k-periodic Case q 1 If q 1andα / 0, then from 7.5 we obtain y km i y k m 1 i L i, m N 0,i 1,k, 7.16 when k m 1 i 1, from which along with 4.6, it follows that x km i x i x km 1 x 1 y i 1 jl i 1 y i jl i, m N, i 2,k, y k j 1 ) L k y 1 jl 1, m N Corollary 7.2. Consider equation 1.1. Letq 1, α / 0, and p i : L i 1 /L i, i {1,...,k}. Then the following statements hold true. a If p i < 1, forsomei {1,...,k}, thenx km i 0 as m. b If p i > 1, orl i 0 and L i 1 / 0, forsomei {1,...,k}, then x km i as m, if x i / 0. c If p i 1, forsomei {2,...,k}, and y i 1 y i /L i > 0, then x km i as m, if x i / 0. d If p i 1, forsomei {2,...,k}, and y i 1 y i /L i < 0, thenx km i 0 as m. e If p i 1, forsomei {2,...,k}, and y i 1 y i, then the sequence x km i m N0 is convergent. f If p 1 1, and y k L 1 y 1 /L 1 > 0, then x km 1 as m,ifx 1 / 0. g If p 1 1, and y k L 1 y 1 /L 1 < 0, thenx km 1 0 as m. h If p 1 1, and y k L 1 y 1, then the sequence x km 1 m N0 is convergent. Proof. The statements in a and b follow from the facts that y i 1 jl i 1 j y i jl pi, i {2,...,k} 7.18 i

13 Abstract and Applied Analysis 13 if L i / 0, y i 1 jl i 1 j y i jl, i {2,...,k} 7.19 i if L i 0andL i 1 / 0, y k j 1 ) L k j y 1 jl 1 p1, 7.20 if L 1 / 0, and y k j 1 ) L k j y 1 jl 1, 7.21 if L 1 0andL k / 0. Now assume that p i 1andlet x n n k be a solution of equation 1.1.Itiseasytosee that there is an m 2 N such that for j m 2 1 the terms in the products in 7.17 are positive and that the following asymptotic formulae 1 x 1 1 x O x 2), ln 1 x x O x 2) 7.22 can be applied with x y i 1 y i / jl i, when i {2,...,k} or with x y k L 1 y 1 / jl 1. Using these formulae, for the case i {2,...,k}, we have that x km i x i y i 1 jl i 1 y i jl i x i c m 2 exp x i c m 2 exp x i c m 2 exp m j m 2 1 m j m 2 1 m j m 2 1 ln y i 1 jl i 1 y i jl i ln 1 y )) i 1 y i 1 O jl i j 2 yi 1 y i jl i )) 1 O, j where m 2 y i 1 jl i 1 c m y i jl i

14 14 Abstract and Applied Analysis Letting m in 7.23, using the facts that m 1 j as m 7.25 j m 2 1 and that the series j m 2 1 O 1/j2 converges, we get statements c e. If p 1 1, that is L 1 L k / 0, then by using 7.22 we get x km 1 x 1 y k j 1 ) L k y 1 jl 1 x 1 d m 2 exp x 1 d m 2 exp x 1 d m 2 exp m j m 2 1 m j m 2 1 m j m 2 1 ln y k j 1 ) L k y 1 jl 1 ln 1 y )) k L 1 y 1 1 O jl 1 j 2 yk L 1 y 1 jl 1 )) 1 O, j where m 2 y k j 1 ) L k d m y 1 jl 1 Letting m in 7.26,using 7.25 and the fact that the series j m 2 1 O 1/j2 converges, we get statements f h, as desired Case q / ± 1 If q / ± 1, then from 7.6 we get y km i q m s i t i, m N 0, 7.28 where from 4.6 it follows that s i y i x km i x i L i q 1, t i L i, i 1,k, q q j s i 1 t i 1 q j s i t i, m N 0, 7.30

15 Abstract and Applied Analysis 15 for i {2,...,k}, and x km 1 x 1y k q j 1 s k t k, m N qs 1 t 1 q j s 1 t 1 j 2 Note that t i t j,ifi j mod k. Corollary 7.3. If 0 < q < 1, α / 0, and q j s i t i / 0, for every j N 0 and i {1,...,k}, then the following statements hold true. a If t i 1 < t i,forsomei {1,...,k}, we have that x km i 0 as m. b If t i 1 > t i, and s i / 0 if t i 0 for some i {1,...,k}, we have that x km i as m,ifx i / 0. c If t i 1 t i / 0, forsomei {1,...,k}, thenx km i is convergent. d If t i 1 t i 0, and s i 1 < s i for some i {2,...,k}, then x km i 0 as m. e If t 1 t k 0, and s k < qs 1,thenx km 1 0 as m. f If t i 1 t i 0, and s i 1 > s i for some i {2,...,k}, then x km i as m,if x i / 0. g If t 1 t k 0, and s k > qs 1,then x km 1 as m,ifx 1 / 0. h If t i 1 t i 0, and s i 1 s i / 0 for some i {2,...,k}, thenx km i is constant. i If t 1 t k 0, and s k qs 1 / 0, thenx km 1 is constant. j If t i 1 t i 0, and s i 1 s i / 0 for some i {2,...,k}, thenx km i 1 m x i. k If t 1 t k 0, and s k qs 1 / 0, thenx km 1 x 1 y k 1 m 1 / qs 1. l If t i 1 t i / 0, forsomei {1,...,k}, then the subsequences x 2km i and x 2km k i are convergent. Proof. Since we have that q j s i 1 t i 1 t i 1, i {2,...,k} 7.32 j q j s i t i t i when t i / 0, q j s i 1 t i 1 j q j, i {2,...,k} 7.33 s i t i when t i 1 >t i 0ands i / 0, q j 1 s k t k j q j s 1 t 1 t k t 1, 7.34

16 16 Abstract and Applied Analysis when t 1 / 0, and q j 1 s k t k j q j s 1 t 1, 7.35 when t k >t 1 0ands 1 / 0, the statements in a and b easily follow from c If t i 1 t i / 0, then x km i x i 1 q j si 1 s i t i ) o q j)), 7.36 for i {2,...,k}, andift 1 t k / 0, then x km 1 x 1y k qs 1 t 1 j 2 1 q j 1 sk qs 1 t 1 ) o q j)) 7.37 from which the statement in c easily follows. d k If t i 1 t i 0, then x km i x i s i 1 s i, 7.38 for i {2,...,k}, andift 1 t k 0, then x km 1 x 1y k qs 1 j 2 s k qs from which the statements in d k easily follow. l If t i 1 t i / 0, then we have that [ x km i x i 1 q j si 1 s i t i ) o q j))] 7.40 for i {2,...,k}, andift 1 t k / 0, then x km 1 x 1y k [ qs 1 t 1 j 2 1 q j 1 sk qs 1 t 1 ) o q j))] 7.41 from which the statement in l easily follows.

17 Abstract and Applied Analysis 17 Corollary 7.4. If q > 1 and α / 0, and q j s i t i / 0, for every j N 0 and i {1,...,k}, then the following statements hold true. a If s i 1 < s i,forsomei {2,...,k}, thenx km i 0 as m. b If s k < qs 1,thenx km 1 0 as m. c If s i 1 > s i,ors i 0, s i 1 / 0 and t i / 0, forsomei {2,...,k}, then x km i as m,ifx i / 0. d If s k > qs 1,orifs 1 0, s k / 0 and t 1 / 0, then x km 1 as m,ifx 1 / 0. e If s i 1 s i / 0, forsomei {2,...,k}, then the sequence x km i m N0 is convergent. f If s i 1 s i 0 and t i 1 < t i for some i {2,...,k}, thenx km i 0 as m. g If s 1 s k 0 and t k < t 1,thenx km 1 0 as m. h If s i 1 s i 0 and t i 1 > t i for some i {2,...,k}, then x km i as m,if x i / 0. i If s 1 s k 0 and t k > t 1,then x km 1 as m,ifx 1 / 0. j If s i 1 s i 0 and t i 1 t i for some i {2,...,k}, then the sequence x km i is constant. k If s 1 s k 0 and t 1 t k, then the sequence x km 1 is constant. l If s i 1 s i 0 and t i 1 t i for some i {2,...,k}, then the sequence x km i is twoperiodic. m If s 1 s k 0 and t 1 t k, then the sequence x km 1 is two periodic. n If s k qs 1 / 0, then the sequence x km 1 m N0 is convergent. o If s i 1 s i / 0, for some i {2,...,k}, then the sequences x 2km i m N0 and x 2km k i m N0, are convergent. p If s k qs 1 / 0, then the sequences x 2km 1 m N0 and x 2km k 1 m N0, are convergent. Proof. a d These statements follow correspondingly from the next relations which are derived using formulae 7.30 and 7.31 : q j s i 1 t i 1 s i 1 j q j s i t i s i 7.42 for i {2,...,k} if s i / 0, and q j s i 1 t i 1 j q j s i t i 7.43 for i {2,...,k} if s i 0, s i 1 / 0andt i / 0; q j 1 s k t k j q j s 1 t 1 s k qs 1, 7.44

18 18 Abstract and Applied Analysis if s 1 / 0, and q j 1 s k t k j q j s 1 t if s 1 0, s k / 0andt 1 / 0. e If s i 1 s i / 0, then from 7.30 we get x km i x i 1 q j ti 1 t i s i 1 ) o q j)), 7.46 for i {2,...,k}, from which e follows. f m If s i 1 s i 0 for some i {2,...,k}, then for i {2,...,k} we have q j s i 1 t i 1 q j s i t i t i 1 t i 7.47 while when s 1 s k 0, we have q j 1 s k t k q j s 1 t 1 t k t from which the statements f i easily follow. n If s k qs 1 / 0, then we have x km 1 x 1y k qs 1 t 1 j 2 1 q j tk t 1 s 1 ) o q j)), 7.49 from which along with the assumption q > 1 the statement follows. o and p If s i 1 s i / 0, then [ x km i x i 1 q j ti 1 t i s i ) o q j))] 7.50 for i {2,...,k}, and x km 1 x 1y k [ qs 1 t 1 j 2 1 q j tk t 1 s 1 ) o q j))] From 7.50 and 7.51 the statements in o and p correspondingly follow.

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