linear difference equations with discrete transforms method abdul j jerri

... = ∞  j= 0 ε j y ( j) t,k + ε k ∞  j= 0 ε j w ( j) 0,k + ···+ ε k−m+1 ∞  j= 0 ε j w ( j) m −1,k ,0≤ k ≤ N, (3.8) where  ∞ j= 0 ε j y ( j) t,k is the outer series and  ∞ j= 0 ε j w ( j) s,k , s = 0, ... (3.8 )wesee that y k (ε) = +∞  j= 0 ε j y ( j) k ,0≤ k ≤ N, (3.19) where y ( j) k = y ( j) t,k + w ( j k) 0,k + w ( j k+1) 1,k + ···+ w ( j k+m−1) m −1,k ,0≤ k ≤ N, j ≥ 0. (3.20) Proposition 3.4. ... j) s,k+m + a m−1 w ( j) s,k+m −1 + ···+ a 0 w ( j) s,k =−a m+1 w ( j 1) s,k+m+1 − a m+2 w ( j 2) s,k+m+2 −··· − a m+n w ( j n) s,k+m+n , k = 0,1, , (3.16) w ( j) s,s =−y ( j) t,s , w ( j) s,k = 0ifk...

Ngày tải lên: 22/06/2014, 22:20

... 1+ ∞  j= 1 λ − j  1 −  1 − 1 λ  j  G j + 1 λ ∞  j= 1 λ − j jK j ≥ 1 − ∞  j= 1 λ − j  1+  1+ 1 λ  j    G j   − 1 λ ∞  j= 1 λ − j j   K j   > 1 − ∞  j= 1 γ − j  1+  1+ 1 γ  j    G j   − 1 γ ∞  j= 1 γ − ... λ>1 F  (λ) = 1 − ∞  j= 1 λ − j   G j   +  1 − 1 λ  ∞  j= 1 λ − j j   G j   − 1 λ ∞  j= 1 λ − j j   K j   > 1 − ∞  j= 1   G j   − ∞  j= 1 j   K j   . (4.75) Hence, by ... e γ  λ 0  = ∞  j= 1 λ − j 0  1 −  1 − 1 λ 0  j  G j + 1 λ 0 ∞  j= 1 λ − j 0 jK j , μ  λ 0  = ∞  j= 1 λ − j 0  1+     1 − 1 λ 0     j    G j   + 1 λ 0 ∞  j= 1 λ − j 0 j   K j   . (2.12) Property...

Ngày tải lên: 22/06/2014, 22:20

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Ngày tải lên: 22/03/2014, 11:20

... M 2 x 0  K d , (9) for all x 0 ∈ K d . VNU Journal of Science, Mathematics - Physics 26 (2010) 175-184 Stability Radii for Difference Equations with Time-varying Coefficients Le Hong Lan* Department ... Vietnam Received 10 August 2010 Abstract. This paper deals with a formula of stability radii for an linear difference equation (LDEs for short) with the coefficients varying in time under structured ... operator. The result is considered as an discrete version of a previous result for time-varying ordinary differential equations [1]. Keywords: Robust stability, Linear difference equation, Input-output...

Ngày tải lên: 22/03/2014, 11:20

... with distributed delays ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x i (m +1)=a i x i (m)+ n  j= 1 b ij f j (x j (m)) + n  j= 1 c ij ∞  k=1 μ ij (k)g j (x j (m − k)), m ∈ Z + ∞ , m = m k , x i (m +1)=H im (x 1 (m), , x m (m)), ... doi:10.1016 /j. camwa.2007.10.020 10. Song, QK, Cao, JD: Dynamical behaviors of discrete- time fuzzy cellular neural networks with variable delays and impulses. J Franklin Inst. 345,39–59 (2008). doi:10.1016 /j. jfranklin.2007.06.001 11. ... et al.: Difference inequality for stability of impulsive difference equations with distributed delays. Journal of Inequalities and Applications 2011 2011:8. Submit your manuscript to a journal...

Ngày tải lên: 21/06/2014, 03:20

... approximation of j q b j y 1 j /y 1 j  2 is j q b  j  y 1  j  y 1  j  2   b 0 j ρ 2 1  1  α 2 − α 1 j  σ  j   , 3.39 where σ j is the reminder and σ 0  sup j {j 2 |σ j |} is a ... equation” h  n   ∞  j n K  n, j  R  j  − j p φ  j  h  j  1  − j q ψ  j  h  j  . 2.1 Lemma 2.1. Let Kn, j , φ j , ψ j , R j be real or complex functions of integer variables n, j; p and q ... h s  n  |  ∞  j n   K  n, j     j p   φ  j      h s  j  1  h s−1  j  1     j q   ψ  j      h s  j  −h s−1  j      ∞  j n 2C R C 2 K P  n   ςN  θ  p  j  j −θ  j p−s C φ λ s−1 P  j...

Ngày tải lên: 21/06/2014, 11:20

... get w k  ∞  j k p j  ∞  j k w 2 j 1  w j ; 3.5 8 Advances in Difference Equations k ≥ a. Thanks to 3.22 and 3.24,weget K 2 k ≤ 1 h k ∞  j k h j j    j  Gw  j    ≤ 1 h k ∞  j k h j j ·  1 ... where H 1 k  |1/h k   ∞ j k h j /j w 2 j −v 2 j |, H 2 k  |1/h k   ∞ j k h j /j Gw j − Gv j |,andH 3 k  |1/h k   ∞ j k Δh j Δw j − v j |.Inview of 3.22, we have H 1 k        1 h k ∞  j k h j j  w j − ... that lim k  ∞ j k   Δ 2 h j   h k  0. 3.19 Indeed, first note that  ∞ j k |Δh j |≤1 −2 1  2sup j k |ψ j |  ∞ j k h j /j < ∞,andso  ∞ j k |Δ 2 h j |≤  ∞ j k |Δh j |  |Δh j 1 | <...

Ngày tải lên: 21/06/2014, 11:20

... k 12  N−1  j 0 w j ψ j  Δϕ N−1 ψ j − ϕ j Δψ N−1  −  k 21 − k 22  N−2  j 0 w j ψ j  ϕ N−1 ψ j − ϕ j ψ N−1  − k 12 N−1  j 0 w j ϕ j  Δϕ N−1 ψ j − ϕ j Δψ N−1   N−1  j 0 w j δ j , 3.31 where δ j :   k 11 − ... that Δϕ  n−1  λ   n−1  j 0 w j ϕ j  λ   Δϕ n−1  λ  ψ j  λ  − ϕ j  λ  Δψ n−1  λ   , Δψ  n−1  λ   n−1  j 0 w j ψ j  λ   Δϕ n−1  λ  ψ j  λ  − ϕ j  λ  Δψ n−1  λ   . 3.30 6 ... 3.27–3.28, we have ϕ  n  λ   n−1  j 0 w j ϕ j  λ   ϕ n  λ  ψ j  λ  − ϕ j  λ  ψ n  λ   , ψ  n  λ   n−1  j 0 w j ψ j  λ   ϕ n  λ  ψ j  λ  − ϕ j  λ  ψ n  λ   . 3.29 It...

Ngày tải lên: 21/06/2014, 20:20

... condition” φ i  0   a i φ i  −τ 1   n  j 1 a ij f j  φ j  −τ 1    n  j 1 b ij g j  φ j  −τ 2     0 −∞ p ij  −s  h j  φ j  s   ds  I i . 2.2 By the method of steps, one can easily ... N j   n  j 1  ∞ 0 q rj  s  1  ε   z j e −λ  t ∗ −s−t 0   N j  ds  I r  n  j 1  p rj e λτ 1  w rj e λτ 2   ∞ 0 q rj  s  e λs ds  z j  1  ε  e −λ  t ∗ −t 0   n  j 1  p rj  ... in Difference Equations By using 3.1, 3.7–3.10,andq ij t ≥ 0, we have u r  t ∗  ≤ n  j 1 p rj  1  ε   z j e −λ  t ∗ −τ 1 −t 0   N j   n  j 1 w rj  1  ε   z j e −λ  t ∗ −τ 2 −t 0  ...

Ngày tải lên: 21/06/2014, 20:20

... Project MSM0022162409 of the Czech Ministry of Education. References 1 P. ˇ Reh ´ ak, “Hartman-Wintner type lemma, oscillation, and conjugacy criteria for half -linear difference equations, ” Journal ... half -linear difference equations, ” Czechoslovak Mathematical Journal, vol. 51, no. 2, pp. 303–321, 2001. 3 P. ˇ Reh ´ ak, “Generalized discrete Riccati equation and oscillation of half -linear ... Discrete Oscillation Theory, Hindawi, New York, NY, USA, 2005. 7 O. Do ˇ sl ´ yandP. ˇ Reh ´ ak, “Recessive solution of half -linear second order difference equations, ” Journal of Difference Equations...

Ngày tải lên: 21/06/2014, 20:20

... case, A j  p  q if j  0, ,r, A j  q if j  r  1, ,m, B j  1ifj  0, ,r, B j  0ifj  r  1, ,m. 4.4 In the second case, A j  p  q if j  0, ,m, A j  p if j  m  1, ,r, B j  1ifj  0, ... fractional difference equation x n1  μ   k j 0 a j x n j λ   k j 0 b j x n j ,n∈ Z  {0, 1, }, 1.1 with the initial condition x j  φ j ,j Z 0  {−k,−k  1, ,0}, 1.2 B. Paternoster ... the equilibrium point of 2.1. Together with nonlinear equation 2.2 we will consider and its linear part z n1  k  j 0 γ j z n j  σz n ξ n1 ,γ j  a j − b j x λ  B x . 2.3 Two following...

Ngày tải lên: 22/06/2014, 11:20

... equations with several delays. For comparison, we will cite the following two results. Theorem A 9–14. Suppose that a j  n  ≥ 0, 0 ≤ j ≤ m, m  j 0 a j  n  > 0, ∞  n0 m  j 0 a j  n   ... 0, ∞  n0 m  j 0 a j  n   ∞, sup n≥m n  kn−m m  j 0 a j  k  < 3 2  1 2  m  1  . 3.14 Then the equation with several delays x  n  1  − x  n   − m  j 0 a j  n  x  n − j  ,n 0, 1, 2, 3.15 is ... I a l  n  x  h l  n    k∈I a k  n  n−1  j h k n  x  j  1  − x  j   x  n  −  k∈I a k  n  x  n  −  l / ∈ I a l  n  x  h l  n   −  k∈I a k  n  n−1  j h k n m  l1 a l  j  x  h l  j   x  n   1...

Ngày tải lên: 22/06/2014, 11:20

... numbers a j, i , not all equal to zero for j = 1, ,k and i = 0, ,l j such that k  j= 1 l j  i=0 a j, i P (i) n  λ j  = 0 (3.10) holds with a j, l j = 0. Translating (3.10)by1wehave k  j= 1 l j  i=1 a j, i  λ j P (i) n  λ j  + ... P (l 1 ) n (λ 1 ): l 1 −1  i=0 c 1,i P (i) n  λ 1  + k  j= 2 a j, l j  λ 1 − λ j  P (l j ) n  λ j  + k  j= 2 l j −1  i=0 c j, i P (i) n  λ j  = 0 (3.12) with some constants c j, i , and this means that either k ... a j, l j = 0. Translating (3.10)by1wehave k  j= 1 l j  i=1 a j, i  λ j P (i) n  λ j  + P (i−1) n  λ j  + k  j= 1 a j, 0 λ j P n  λ j  = 0, (3.11) and if we subtract (3.10)timesλ 1 from this...

Ngày tải lên: 22/06/2014, 22:20

... ON kTH-ORDER LINEAR FUNCTIONAL EQUATIONS WITH CONSTANT COEFFICIENTS JITKA LAITOCHOV ´ A Received 30 January 2006; Accepted 18 May 2006 Abel functional equations are associated to a linear homogeneous ... < λ j λ 1 < 1, j = 2,3, ,k, (3.13) imply lim x→∞ c j (x)  λ j λ 1  X ∗ α(x) = 0. (3.14) Formally, at least, lim x→∞ f (x) = lim x→∞ c 1 (x) λ X ∗ α(x) 1 . (3.15) 8 A remark on linear ... element Φ in S according to multiplication ◦ in group Ᏻ. 2. Linear functional equations of the kth order with constant coefficients Let a j ∈ R, j = 0,1,2, ,k. Then the equation a k f ◦ Φ k (x)+a k−1 f...

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