... sufficient to evaluate the sign of the first ν of them, where ⎧ ⎪k ⎪ ⎪ ⎨ ⎪k +1 ⎪ ⎩ ν=⎪ for even k, (2 .18) for odd k 6 Difference equations and the stability problem By posing ρk−1 (z) = k −1 j =0 a(k−1) ... where υ j and w j are the roots of ρk (z) and σk (z), respectively, and ν is defined according to (2 .18) Considering (2.12), the previous relation becomes qk e iθ = = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... 1)2 ρk−1 (z) − ρk−2 (z), 4 − k −2 k ≥ 1, (3.17) with initial conditions ρ−1 (z) = 0, ρ0 (z) = (3 .18) Proof From Theorem 3.4 and relation (2.15) the proof immediately follows We are now ready to...
... 3, we study the existence of one-sign solutions of the nonlinear problem x f t, x , t ∈ 0, T , 1 .18 x x T , x x T The proofs of the main results are based on the properties of G and the Dancer’s ... α s u T −1 u s v 2.17 Applying this and Lemma 2.2 iii , it follows that αs u s v s v −u T v 2 .18 Denote M: max G t, s , 0≤t,s≤T m: G t, s 0≤t,s≤T 2.19 Boundary Value Problems Finally, we state ... Obviously, for f t, s a t u a t s a t u, a t μ0 γ Γt j x x a t u u T 3.17 μ· μ0 a t x T , of j , 3 .18 μ0 Γ 1, 2, the principal eigenvalue μ0 1/8 x u T , s, we have that γ t For j u j x · x, x T t...
... integrating 3.2 with λ Fn t, J un t 2π un − ρun 0 ≤ρ 2π ρ n ≥ g0 nρ ≥ g0 un t dt − > ρ2 r ρH 3 .18 from to 2π, we obtain that ρ2 un − Fn t, J un t nρ 2π −ω − ρ2 dt n 3.19 − ω dt < Fn t, J un t...
... above from k0 to T to obtain u k0 − u(T + 1) ≤ ϕ−1 h(M) p T j =k0 j ϕ−1 p q(k) (2 .18) k=k0 Now (2.15) and (2 .18) imply M− ≤ b0 ϕ−1 h(M) p n0 (2.19) This contradicts (2.3) Thus ≤ u(k) ≤ M n0 ... F Zanolin, Upper and lower solutions for a generalized Emden-Fowler equation, J Math Anal Appl 181 (1994), no 3, 684–700 D Jiang, D O’Regan, and R P Agarwal, A generalized upper and lower solution ... nonlinear boundary value problems for the oneu dimensional p-Laplacian, Appl Math Lett 14 (2001), no 2, 189 –194 214 [10] [11] Discrete initial and boundary value problems R Man´ sevich and F Zanolin,...
... Finally, from 18 Boundary Value Problems 2.12 and 2.35 follows that the inequalities x t ≤ x t ˙ ≤ √ 2. 618 x 2.2361 x 6.213 √ x √ 1. 6180 x 4.5945 x ˙ e−0.00001559t/2 0.7071 x ˙ √ 2. 618 √ √ 4.5945 ... λmin G2 takes the form ⎛ 1.3000 ⎜ ⎜ ⎜ 1.1000 ⎜ ⎜ ⎜ ⎜−0.3106 S ⎜ ⎜ ⎜−0 .186 4 ⎜ ⎜ ⎜−0.0300 ⎜ ⎝ −0.0500 1.1000 −0.3106 −0 .186 4 −0.0300 −0.0500 ⎞ ⎟ ⎟ 1.1000 −0.3728 −0.1243 −0.0200 −0.0600⎟ ⎟ ⎟ −0.3728 ... depending on the positive value of β : either β> λmin S λmax H 2 .18 β≤ λmin S λmax H 2.19 is valid or holds 8 Boundary Value Problems Let 2 .18 be valid From 2.3 follows that − x t ≤− V0 x t , t λmax...
... β a , , β n−2 a , Case t1 , 3.17 < Mn−2 c In this case, max x n−2 t : x n−2 c ≥ Mn−2 > , 3 .18 t∈ a,c and x n−1 c ≥ For λ 0, by 3.6 we have the following contradiction: < Mn−2 ≤ x n−2 c 3.19 ... and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol 185 , no 2, pp 302– 320, 1994 A Cabada, M R Grossinho, and F Minhos, “Extremal solutions for third-order ... “Solvability of a third-order two-point boundary value problem,” Applied Mathematics Letters, vol 18, no 9, pp 1034–1040, 2005 M R Grossinho and F M Minhos, “Existence result for some third order...
... 16, no 6, pp 857–862, 2003 18 Z H Li, “The asymptotic estimates of solutions of difference equations,” Journal of Mathematical Analysis and Applications, vol 94, no 1, pp 181 –192, 1983 19 W F Trench, ... \ {0} and K i : qi , i ∈ Z Then, 3.7 has the following form: n Δx n Ax n qn−j x j n∈Z g n, 4 .18 j −∞ It is clear that r : limn→∞ |qn |1/n Gμ |q|, and the function G defined in 3.13 is given ... every N ≥ n lim sup n→∞ H n, j − V j N n max lim sup n→∞ N H n, j − j H∞ j − V, −lim inf n→∞ j n 5 .18 N H n, j H∞ j j n V , j ∞ and hence n H n, j − V lim sup lim sup N→∞ n→∞ j N n max lim sup n→∞...
... q1,τ (η) p0,τ (s) a(s)x(s)ds p0,τ (1) + ≥ q1,τ (η) p0,τ (1) − γ p0,τ (η) (Lτ x)(η) = η p0,τ (η) (2 .18) p0,τ (s)a(s)x(s)ds, η q1,τ (η) p0,τ (s) a(s)x(s)ds p0,τ (1) + p0,τ (η) p0,τ (1) − γ p0,τ (η) ... η p0,τ (η) q1,τ (s) a(s)x(s)ds q1,τ (0) η q1,τ (s)a(s)x(s)ds (2.19) X Xian and D O’Regan By (2 .18) and Lemma 2.6, we have for any t ∈ [0,η], Lτ x (t) = t qη,τ (t) p0,τ (s) a(s)x(s)ds p0,τ (η) ... and (3.17), we have Tλ x (t) ≤ Tλ x0 (t) + τ0 τ LλM z0 (t) ≤ u∗ (t) − LλM z0 (t), 2 t ∈ [0,1], (3 .18) for any x ∈ QλM with x − x0 < δ This implies that x ∈ Ωλ , and so Ωλ is an open set Now we will...
... (2.17) Now we change the index of summation j by j + Then ΔeBk = B I + m −1 Bj · j =1 k − jm j (2 .18) and due to (2.15) we conclude that formula (2.16) is valid The case k = (m + 1) In this case ... two discrete variables be given Then Δk k k F(k, j) = F(k + 1,k + 1) + j =1 j =1 Δk F(k, j) (3 .18) J Dibl´k and D Y Khusainov 11 ı Now we are ready to find a particular solution x p (k), k ∈ Z∞m ... k Δ B(k−m− j) em k −m ω( j) = B B(k−2m− j) em j =1 ω( j) + f (k) (3.23) j =1 With the aid of (3 .18) we obtain eB((k+1)−m−(k+1)) ω(k + 1) + m k B(k−m− j) Δ em k −m ω( j) = B B(k−2m− j) em j =1...
... authors (see, e.g., [2, 18] ), and present some related properties for our purposes For almost periodic and asymptotically almost periodic functions, we recommend [19, 18] Let X and Y be two Banach ... by Theorem 3.1, then u(n) = p(n) + q(n), n ≥ 0, (3 .18) where { p(n)}n∈Z is almost periodic and q(n) → as n → ∞ It follows from (3.17) and (3 .18) that → p(n) − p(n + mω) − as n −→ ∞, (3.19) which ... omitted here because it is not difficult for readers giving proofs by the similar arguments in [19, 18] for continuous (uniformly) almost periodic function φ : R × Ω → X (see also [2] for the case...
... write Φ instead of Φ(δ) and Φ instead of Φ (δ) Lemma 2.3 yields −Φ = + O(1)Φ−β δ f (Φ) (2 .18) Using (2 .18) and the equation Φ = −(2F(Φ))1/2 we find lim δ →0 Φ(δ) δ Φ(δ) 1−β −β f (Φ) = lim δ →0 ... = o(1)δ Φ(δ) −β f (Φ), (2.20) where o(1) denotes a quantity which tends to zero as δ → Using (2 .18) again we find lim δ →0 −β Φ(δ) −β δ Φ(δ) Φ f (Φ) = −1 (2.21) Therefore, δ Φ(δ) 1−β 1−β = Φ(δ) ... (2.26) Second-order estimates Denoting by M1 a nonnegative constant independent of α and using (2 .18) , (2.20), (2.22), (2.25), (2.26), by (2.17) we get Δw < f (Φ) + Hδ + M1 δΦ−β + αM1 δΦ−2β (2.27)...
... 1085–1105 R M¨ rz, On linear differential-algebraic equations and linearizations, Appl Numer Math 18 a (1995), no 1–3, 267–292 Pham Ky Anh: Department of Mathematics, Mechanics, and Informatics,...
... solutions of (1.1) such that lim un = lim wn = 0, n n lim u[1] = cu , n n [1] lim wn = dw , n (3 .18) where cu ,dw ∈ R \ {0}, then there exists λ ∈ R \ {0} such that u = λw Proof Let z = {zn } be...
... singular We will treat iterative methods only incompletely in this book, in §2.7 and in Chapters 18 and 19 These methods are important, but mostly beyond our scope We will, however, discuss in...
... A·x−C·y=b (2.3.17) C·x+A·y=d which can be written as a 2N × 2N set of real equations, · x y = b d (2.3 .18) and then solved with ludcmp and lubksb in their present forms This scheme is a factor of inefficient ... work of an N × N one If you can tolerate these factor-of-two inefficiencies, then equation (2.3 .18) is an easy way to proceed CITED REFERENCES AND FURTHER READING: Golub, G.H., and Van Loan, C.F ... Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations...