... with respect to the continuous initial value problem, it is not necessary to approximate the latter with a discrete problem of the same type In the case of first order differential equation, for ... limitations on the choice of the methods in the class of LMMs used as initial value methods (IVMs) As matter of fact, there are not methods with order greater than two having the critical point asymptotically ... terminology used in Numerical Analysis is often different from the one used in the Difference Equations setting In order to avoid confusion, it is worth to note that the terms such as “stable methods” refer...
... 2 Boundary Value Problems always has a unique solution which, moreover, can be written as T x t 1.4 G t, s h s ds, where G t, s is the Green’s function related to 1.1 , 1.2 In recent ... 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...
... two-point boundary value problems with nonlinearities across several eigenvalues J Appl Math Lett 18, 587–595 (2005) [4] Rabinowitz, PH: Some global results for nonlinear eigenvalue problems J ... Ma, R, Thompson, B: Nodal solutions for nonlinear eigenvalue problems Nonlinear Anal TMA 59, 707– 718 (2004) [2] Walter, W: Ordinary Differential Equations Springer, New York (1998) [3] Ma, R, Thompson, ... u(0) = u(1) = 0} with the norm u E = max |u| + max |u | t∈[0,1] t∈[0,1] Define L : D(L) → Y by setting Lu := −u (t), t ∈ [0, 1], u ∈ D(L), where D(L) = {u ∈ C [0, 1] | u(0) = u(1) = 0} Then L−1...
... Furthermore, unk satisfies the integral equation 2π unk t − ω ds G t, s F s, J unk s nk 3.12 Letting k → ∞, we obtain that 2π ut G t, s F s, J u s − ω ds, 3.13 where the uniform continuity of ... 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 ... existence theorem for a singular third-order boundary value problem on 0, ∞ ,” Applied Mathematics Letters, vol 15, no 4, pp 445–451, 2002 12 L Kong, S Wang, and J Wang, “Positive solution of a singular...
... 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 ... singular 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, 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...
... 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...
... Liu, “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 ... Grace, and D O’Regan, “Semipositone higher-order differential equations,” Applied Mathematics Letters, vol 17, no 2, pp 201–207, 2004 A Cabada, “The method of lower and upper solutions for second, ... 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...
... Mathematics Letters, vol 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, ... ∞ −1 i K i μ−i , 3.17 i iii if G r < 1, then the constant A satisfies either ∞ A≥r−1− K i r −i 3 .18 i or ∞ A ≤ −r − − −1 i K i r −i 3.19 i Remark 3.2 Let K : Z →R be a sequence such that K n0 ... \ {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...
... 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...
... omitted) we suppose that all used expressions are well defined Lemma 3.4 Let a function F(k,n) of 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) ... it by the method of steps, we conclude that the solution of the problem (2.1), (2.2) can be written in the form ⎧ ⎪1 if k ∈ Z0 , ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪ ⎪1 + b · if k ∈ Z4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k k−3 ... (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...
... 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 ... : Z+ → X a The proof of the above results is omitted here because it is not difficult for readers giving proofs by the similar arguments in [19, 18] for continuous (uniformly) almost periodic function ... 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...
... de l’Hˆ pital’s rule Using (2.11), (2.10) can be rewritten o as 2F(s) 1/2 f (s) −1 ∞ = 2F(t) s −1/2 dt + O(1)s−β ∞ s 2F(t) −1/2 dt (2.13) Putting s = Φ(δ) and using the equation −Φ (δ) = (2F(Φ(δ)))1/2 ... 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 ... 2F(s) −1/2 dt −1/2 = (3.9) Using (3.9), (3.8) can be rewritten as 2F(s) 1/2 f (s) −1 = ∞ s 2F(t) −1/2 dt + O(1)e−s ∞ s 2F(t) −1/2 dt (3.10) Putting s = Ψ(δ) and recalling that −Ψ (δ) = (2F(Ψ(δ)))1/2...
... respectively As in the DAEs case, we only need to initialize the P0 -component of x0 Further, putting n = in (2.8) and noting that V−1 = V0 , u0 = P−1 x0 = P0 x0 , we find that a consistent initial ... 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,...
... vn+1 or ∆vn + wn vn+1 ≥ Since wn ≥ and ∞ n=N wn < ∞, we have < ∞ −1 j =N (1 + w j ) (2.10) < ∞ Putting ∞ hn = , + wj j =n (2.11) we have hn > and ∆hn = hn wn Multiplying (2.10) by hn , we obtain ... bk Φ xk+1 − Φ zk+1 ≤H k=n ∞ ≤ HM bk Ak+1 k=n p−2 ∞ Φ∗ a j j =k+1 Φ∗ z[1] − Φ∗ x[1] j j (3.15) Putting un = sup [1] [1] Φ∗ xk − Φ∗ zk :k≥n , (3.16) 200 Recessive solutions for half-linear equations ... 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...
... Equation (2.0.1) can be written in matrix form as A·x=b (2.0.2) Here the raised dot denotes matrix multiplication, A is the matrix of coefficients, and b is the right-hand side written as a column vector, ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk...
... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk...
... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk...
... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk...
... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk ... any servercomputer, is strictly prohibited To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk...