basic concepts and classifying differential equations

Ngày tải lên: 31/03/2014, 15:56

... Multistep Initial and Boundary Value Methods, Stability and Control: Theory, Methods and Applications, vol 6, Gordon and Breach, Amsterdam, 1998 [3] G Dahlquist, Convergence and stability in the ... (z) and σk (z) be the polynomials defined in (2.11) and ν ≥ given by (2.18) Then, (1) ρk (z) and σk (z) are both of type (ν − 1,1,ν − 1), for k = 2ν − 1; (2) ρk (z) is of type (ν − 1,2,ν − 1) and ... Difference equations and the stability problem Corollary 3.9 Suppose that the polynomials ρk (z) and σk (z), defined in (2.11), admit roots on the unit circle Then, all of them are real and simple...

Ngày tải lên: 23/06/2014, 00:20

... positive solutions of singular second-order equations and systems; see for instance Chu and Torres , Chu et al , Franco and Torres , Jiang et al , and Torres Moreover, the positiveness of Green’s ... Equations, vol 232, no 1, pp 277–284, 2007 A Cabada, P Habets, and S Lois, “Monotone method for the Neumann problem with lower and upper solutions in the reverse order,” Applied Mathematics and ... 2.13 and 2.15 , we have Gt t, s α s u t v t − ⎧ ⎨v t u s , − v ⎩v s u t , v s , v 0 ≤ s < t ≤ T, ≤ t < s ≤ T, 2.16 and, for s ∈ 0, T , Gt 0, s α s v Gt T, s α s u T −1 u s v 2.17 Applying this and...

Ngày tải lên: 21/06/2014, 11:20

... k k actly k −1 zeros in (0,1) and is positive near 0, and u− has exactly k −1 zeros in (0,1) and is negative near k In [3], Ma and Thompson studied the existence and multiplicity of nodal solutions ... (H2 ), and (H3 ), then problem (1.4) possesses two solutions u+ and u− , such that u+ has exactly k − zeros in (0,1) and is positive near 0, and u− has exactly k k k k k − zeros in (0,1) and is ... α < β and f (r,s) s ≥ and a subinterval [α, β] of for all r ∈ [α, β] and s = In the celebrated study [4], Rabinowitz established Rabinowitz’s global bifurcation theory [4, Theorems 1.27 and 1.40]...

Ngày tải lên: 21/06/2014, 17:20

... g u and h u on 0, ∞ such that F t, u ≤ g u ∀ t, u ∈ 0, 2π × 0, ∞ hu 1.2 and g u > is nonincreasing and h u /g u is nondecreasing in u; G3 there exist continuous, nonnegative functions g1 u and ... Chu, and M Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations, ” Journal of Differential Equations, vol 211, no 2, pp 282–302, 2005 A C Lazer and S ... method of lower and upper solutions or second, third, fourth and higher order boundary value problem,” Applied Mathematics and Computation, vol 40, pp 135–145, 1990 10 J Chu and Z Zhou, “Positive...

Ngày tải lên: 22/06/2014, 00:20

... Agarwal, D O’Regan, and P J Y Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999 P Habets and F Zanolin, Upper and lower solutions ... > 0, ≤ β < 1, and A > Now (2.47) has a solution u ∈ C(N + , R) with u(k) > for k ∈ N To see this we will apply Theorem 2.3 with (2.24) Let n0 > (2A)1/α and c0 = A Then for k ∈ N and < u ≤ 1/n0 ... Agarwal, D Jiang, and D O’Regan, A generalized upper and lower solution method for singular discrete initial value problems, Demonstratio Math 37 (2004), no 1, 115–122 R P Agarwal, H L¨ , and D O’Regan,...

Ngày tải lên: 23/06/2014, 00:20

... t−τ where x is a solution of 1.1 , β and p are real parameters, the n × x matrices H, G1 , and G2 are positive definite, and t > The form of functionals 1.9 and 1.10 is suggested by the functionals ... 0021630529, and by Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication 20 Boundary Value Problems References V Kolmanovskii and A Myshkis, Introduction to the Theory and Applications ... Translated in Differential Equations, vol 24, no 4, pp 424–431 K Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Pynamics, vol 74 of Mathematics and its Applications,...

Ngày tải lên: 21/06/2014, 11:20

... x2 , 3, x2 3.68 Choose α t −t, β t t and γ t t It is easy to check that α t −t, and β t t are lower and upper solutions of BVP 3.66 , 3.67 respectively, and all the assumptions in Theorem 3.2 ... 3.72 13x2 , x1 9x2 t2 and γ t t2 It is easy to check that α t −t2 , and β t t2 Choose α t −t2 , β t are lower and upper solutions of BVP 3.70 , 3.71 , respectively, and all the assumptions ... 18 Y Shi and M Pei, “Two-point and three-point boundary value problems for nth-order nonlinear differential equations, ” Applicable Analysis, vol 85, no 12, pp 1421–1432, 2006 19 L Wang and M Pei,...

Ngày tải lên: 21/06/2014, 20:20

... no K73274 References R P Agarwal, Difference Equations and Inequalities: Theory, Methods, and Application, vol 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New ... Mathematical Analysis and Applications, vol 330, no 1, pp 195–206, 2007 Y Song and C T H Baker, “Admissibility for discrete Volterra equations, ” Journal of Difference Equations and Applications, vol ... type theorems and stability of difference equations, ” Journal of Mathematical Analysis and Applications, vol 304, no 2, pp 511–530, 2005 22 Advances in Difference Equations 10 S Bodine and D A Lutz,...

Ngày tải lên: 22/06/2014, 11:20

... Theorem 1.1 and the main results in [9] Using the upper and lower solutions technique and the fixed point index method, Dalmasso [2] proved the following result Theorem 1.2 (see [2]) Let a and f satisfy ... and s(1 − s)a(s)ds < ∞ (2.1) (H2 ) f (u) = g(u) + h(u), where g : (0, ∞) → (0, ∞) is continuous and nonincreasing, h : R+ → R+ is continuous, and h(u) ≥ b0 uw , u ∈ R+ , (2.2) for some b0 > and ... (η) (2.9) X Xian and D O’Regan Let P = {x ∈ C[0,1]|x(t) ≥ for t ∈ [0,1]} and Qτ = {x ∈ P |x(t) ≥ θτ x t for t ∈ [0,1]} It is easy to see that P and Qτ are cones in C[0,1] For τ ≥ and each n ∈ N,...

Ngày tải lên: 22/06/2014, 22:20

... Difference Equations and Applications, Chapman & Hall/CRC, Florida, 2005, pp 51–59 ˇ [6] J Cerm´ k, On the related asymptotics of delay differential and difference equations, Dynamic Sysa tems and Applications ... and M Pituk, Asymptotic estimates and exponential stability for higher-order monotone difference equations, Advances in Difference Equations 2005 (2005), no 1, 41–55 [12] M Migda, A Musielak, and ... 6, 461–470 ˚z c a [4] A Boichuk and M Ruˇ iˇ kov´ , Solutions of nonlinear difference equations bounded on the whole line, Colloquium on Differential and Difference Equations, CDDE 2002 (Brno), Folia...

Ngày tải lên: 22/06/2014, 22:20

... Agarwal, Difference Equations and Inequalities, vol 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000 [3] C T H Baker and Y Song, “Periodic ... [7] S Elaydi and I Gy¨ ri, “Asymptotic theory for delay difference equations, ” Journal of Difference o Equations and Applications, vol 1, no 2, pp 99–116, 1995 [8] S Elaydi, S Murakami, and E Kamiyama, ... difference equations with infinite delay,” Journal of Difference Equations and Applications, vol 5, no 1, pp 1–23, 1999 [9] I Gy¨ ri and G Ladas, Oscillation Theory of Delay Differential Equations, ...

Ngày tải lên: 22/06/2014, 22:20

... 731–738 [5] C Bandle and E Giarrusso, Boundary blow up for semilinear elliptic equations with nonlinear gradient terms, Advances in Differential Equations (1996), no 1, 133–150 [6] C Bandle and M Marcus, ... A Buttu, and G Porru, Boundary estimates for blow-up solutions of elliptic equations with exponential growth, to appear in Proceedings Differential and Difference Equations [3] C Anedda and G Porru, ... (δ) − Cδ Φ (δ) Φ(δ) β −1 (2.47) Take α and δ0 such that (2.46) holds, and put αδ0 (Φ(δ0 ))−β = q Decrease δ0 and increase α so that αδ0 (Φ(δ0 ))−β = q and β−1 Hδ − q − Cδ Φ (δ) Φ(δ) β −1

Ngày tải lên: 22/06/2014, 22:20

... N H Du, and P K Anh, On linear implicit non-autonomous systems of difference equations, J Difference Equ Appl (2002), no 12, 1085–1105 R M¨ rz, On linear differential-algebraic equations and linearizations, ... compute un+1 , , and xn (n 0) by (2.6), (2.7), and (2.8), 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 ... action of G−1 on both sides of (3.6) and using the last observations, we get n ∆u + Qn,n−1 ∆v = G−1 u,un + v q n (3.7) P K Anh and H T N Yen 199 Now, applying Pn and Qn to both sides of (3.7), respectively,...

Ngày tải lên: 23/06/2014, 00:20

... recurrences, J Differential Equations 107 (1994), no 2, 238–258 C D Ahlbrandt and A C Peterson, Discrete Hamiltonian Systems Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts ... 2000 R P Agarwal and D O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 2001 C D Ahlbrandt, Dominant and recessive solutions ... half-linear equations References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] R P Agarwal, Difference Equations and Inequalities, 2nd ed., Monographs and Textbooks in Pure and...

Ngày tải lên: 23/06/2014, 00:20

... sections For clarity, and to avoid writing endless ellipses (· · ·) we will write out equations only for the case of four equations and four unknowns, and with three different righthand side vectors ... (§2.4), band diagonal (§2.4), Toeplitz (§2.8), Vandermonde (§2.8), sparse (§2.7) • Strassen’s “fast matrix inversion” (§2.11) 36 Chapter Solution of Linear Algebraic Equations Coleman, T.F., and ... method For solving sets of linear equations, Gauss-Jordan elimination produces both the solution of the equations for one or more right-hand side vectors b, and also the matrix inverse A−1 However,...

Ngày tải lên: 15/12/2013, 04:15

... where A and Y are square matrices, the bi ’s and xi ’s are column vectors, and is the identity matrix, simultaneously solves the linear sets A · x1 = b A · x2 = b2 A · x3 = b3 (2.1.3) and A·Y ... rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill), Program B-2, p 298 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations...

Ngày tải lên: 15/12/2013, 04:15

... set of n linear equations A · x = b a[1 n][1 n], c[1 n], and d[1 n] are input as the output of the routine qrdcmp and are not modified b[1 n] is input as the right-hand side vector, and is overwritten ... R = Q · (R + u ⊗ v) (2.10.8) One can go back and forth between equations (2.10.7) and (2.10.8) using the fact that Q is orthogonal, giving t = v and either s = Q · u or u = QT · s (2.10.9) The ... Solves the set of n linear equations R · x = b, where R is an upper triangular matrix stored in a and d a[1 n][1 n] and d[1 n] are input as the output of the routine qrdcmp and are not modified b[1...

Ngày tải lên: 15/12/2013, 04:15

... Gauss-Jordan elimination, each containing one subtraction and one multiplication, are executed N and N M times (where there are N equations and M unknowns) The corresponding loops in Gaussian elimination ... backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution over Gauss-Jordan elimination ... half the matrix is reduced, and the increasing numbers of predictable zeros reduce the count to one-third), and N M times, respectively Each backsubstitution of a right-hand side is N executions...

Ngày tải lên: 24/12/2013, 12:16

... (outside North America) Equations (2.3.6) and (2.3.7) total (for each right-hand side b) N executions of an inner loop containing one multiply and one add If we have N right-hand sides which are ... the α’s and β’s that occur on the right-hand side of equations (2.3.12) and (2.3.13) are already determined by the time they are needed You will also see that every aij is used only once and never ... the right-hand side vector B, and returns with the solution vector X a, n, and indx are not modified by this routine and can be left in place for successive calls with different right-hand sides...

Ngày tải lên: 24/12/2013, 12:16