solutions of linear homogeneous differential equations with constant coefficients

Ngày tải lên: 31/03/2014, 15:56

... number of initial conditions should be equal to the number of roots of the characteristic polynomial inside the unit circle and, of course, the number of conditions at the end of the interval of ... satisfy linear difference equations with variable coefficients A complete proof for the stability problem of such methods has been provided by using properties of the polynomials derived from these equations ... location of zeros of certain classes of polynomials with applications to numerical analysis, Journal of the Institute of Mathematics and Its Applications (1971), 397–406 14 Difference equations...

Ngày tải lên: 23/06/2014, 00:20

... structure of the set of positive solutions of parameterized nonlinear operator equations, which is essentially a consequence of Dancer 12, Theorem Suppose that E is a real Banach space with norm ... existence of positive solutions of 3.17 will not be guaranteed in this case Proof of Theorem 3.2 We only prove i ii can be proved by a similar method To study the existence of positive solutions of ... 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...

Ngày tải lên: 21/06/2014, 11:20

... solution of (2.2) is a pair (r, u) ∈ X, which satisfies the equation (2.2) The closure of the set nontrivial solutions of (2.2) is denoted by C, let Σ(T ) denote the set of eigenvalues of linear ... uj ) → (λi , θ) with (µj , uj ) ∈ ν Ck ∩ (R × Sk ) This contradicts to Lemma 2.3 Proof of main results Proof of Theorems 1.1 and 1.2 We only prove Theorem 1.1 since the proof of Theorem 1.2 is ... existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems, u + f (t, u) = 0, t ∈ (0, 1), u(0) = u(1) = The proof of our main results is based...

Ngày tải lên: 21/06/2014, 17:20

... periodic solutions of Hill’s equations with singular nonlinear perturbations,” to appear in Nonlinear Analysis: Theory, Methods & Applications J Chu, P J Torres, and M Zhang, “Periodic solutions of ... periodic solutions of nonlinear differential equations with singularities,” Proceedings of the American Mathematical Society, vol 99, no 1, pp 109–114, 1987 K Lan and J R L Webb, “Positive solutions of ... Differential Equations, vol 103, no 2, pp 260–277, 1993 D Jiang, J Chu, and M Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations, ” Journal of Differential Equations, ...

Ngày tải lên: 22/06/2014, 00:20

... multiplicity of solutions for the onea dimensional p-Laplacian, Nonlinear Anal 21 (1993), no 4, 269–291 M N Nkashama, A generalized upper and lower solutions method and multiplicity results for nonlinear ... lower solution approach, Nonlinear Anal 50 (2002), no 2, 215–222 R P Agarwal, D O’Regan, and P J Y Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, ... initial value problems with sign changing nonlinearities, Math Probl Eng (2001), no 6, 503–524 R P Agarwal, D O’Regan, V Lakshmikantham, and S Leela, Existence of positive solutions for singular...

Ngày tải lên: 23/06/2014, 00:20

... suggested to the study of stability and estimates of solutions of 1.1 Exponential Stability and Estimates of the Convergence of Solutions to Stable Systems First we give two definitions of stability to ... solution of 1.1 is exponentially stable in the metric C1 Estimates of Solutions in a General Case Now we will estimate the norms of solutions of 1.1 and the norms of their derivatives in the case of ... solution of the system of equations of neutral type 1.1 is called exponentially stable in the metric C0 if there exist constants Ni > 0, i 1, and μ > such that, for an arbitrary solution x x t of 1.1...

Ngày tải lên: 21/06/2014, 11:20

... nth-order nonlinear differential equations, ” Journal of Differential Equations, vol 29, no 2, pp 205–213, 1978 16 K N Murty and Y S Rao, “A theory for existence and uniqueness of solutions to three-point ... and lower solutions in reversed order,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 6, pp 1109–1121, 2005 Z Du, W Ge, and X Lin, “Existence of solutions for a class of third-order ... Order Differential Equations, World Scientific, Teaneck, NJ, USA, 1986 R P Agarwal and F.-H Wong, “Existence of positive solutions for non-positive higher-order BVPs,” Journal of Computational and...

Ngày tải lên: 21/06/2014, 20:20

... of solutions of difference Volterra equations, ” Applied Mathematics Letters, vol 16, no 6, pp 857–862, 2003 18 Z H Li, “The asymptotic estimates of solutions of difference equations, ” Journal of ... of solutions of linear Volterra difference equations with infinite delay,” Computers & Mathematics with Applications, vol 47, no 10-11, pp 1555–1563, 2004 22 Ch G Philos and I K Purnaras, “On linear ... ri, and D W Reynolds, “On exact convergence rates for solutions of linear o systems of Volterra difference equations, ” Journal of Difference Equations and Applications, vol 12, no 12, pp 1257–1275,...

Ngày tải lên: 22/06/2014, 11:20

... solutions of singular nonlinear m-point boundary value problems, Journal of Mathematical Analysis and Applications 284 (2003), no 2, 576–590 [7] R Ma and N Castaneda, Existence of solutions of ... ≤ un,λ (t) for t ∈ [0,1] Thus, un,λ (t) is a positive solution of (2.1λ ) The proof is complete n Proof of the main results Proof of Theorem 2.1 Let Λ = {λ ∈ (0,+∞)|(1.1λ ) has at least one positive ... results on the existence of at least one positive solution for some m-point boundary value problems if the nonlinearity f is either superlinear or sublinear The purpose of this paper is to X Xian...

Ngày tải lên: 22/06/2014, 22:20

... Description of the problem considered The motivation of our investigation goes back to [10] dealing with the linear system of differential equations with constant coefficients and constant delay One of the ... accordance with the theory of linear equations, as the sum of the solution of adjoint homogeneous problem (3.1), (3.2) (satisfying the same initial data) and a particular solution of (3.15) being ... results of the paper With the aid of discrete matrix delayed exponential we give formulas for the solution of the homogeneous and nonhomogeneous problems (1.1), (1.2) 3.1 Representation of the...

Ngày tải lên: 22/06/2014, 22:20

... periodic solutions of nonlinear functional difference equations with finite delay via uniform stability properties of a bounded solution This motivates us to investigate almost periodic solutions of ... of (1.1) In Section 3, we discuss the existence of periodic solutions of (1.1) In Section 4, we discuss the existence of almost periodic solutions of (1.1) Preliminaries We formalize our notation ... Journal of Difference Equations and Applications, vol 10, no 4, pp 379–397, 2004 [12] R P Agarwal, D O’Regan, and P J Y Wong, Constant- sign periodic and almost periodic solutions of a system of difference...

Ngày tải lên: 22/06/2014, 22:20

... behaviour of large solutions of semilinear elliptic [7] problems, Differential and Integral Equations 11 (1998), no 1, 23–34 , Dependence of blowup rate of large solutions of semilinear elliptic equations, ... for semilinear elliptic equations with nonlinear gradient terms, Advances in Differential Equations (1996), no 1, 133–150 [6] C Bandle and M Marcus, “Large” solutions of semilinear elliptic equations: ... K(x) is the mean curvature of the surface {x ∈ Ω : δ(x) = constant} The effect of the geometry of the domain in the behaviour of boundary blowup solutions for special equations has been observed...

Ngày tải lên: 22/06/2014, 22:20

... 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, ... difference equations Acknowledgment The first author thanks Prof Shao Xiumin and Prof Liang Guoping for their hospitality during his visit to the Institute of Mathematics, Chinese Academy of Sciences ... problems for linear implicit nonautonomous systems of difference equations, Vietnam J Math 29 (2001), no 3, 281–286 M S Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems...

Ngày tải lên: 23/06/2014, 00:20

... superlinearity are required For the sake of completeness, a sketch of the proof is provided here 196 Recessive solutions for half -linear equations Proof (sketch) Choose n0 large so that ∞ bk ... Recessive solutions for half -linear equations Theorem 3.2 (see [11]) If (1.1) is nonoscillatory, recessive solutions of (1.1) exist and they are determined up to a constant factor Analogously to the linear ... m → ∞, we obtain (4.3) (4.5) 202 Recessive solutions for half -linear equations We show that also (4.4) holds In view of Proposition 3.3, without loss of generality, we can assume that un > 0,...

Ngày tải lên: 23/06/2014, 00:20

... Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... consists of a particular solution xp added to any linear combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding the solution space of A ... the • Linear least-squares problem The reduced set of equations to be solved can be written as the N ×N set of equations (AT · A) · x = (AT · b) (2.0.4) where AT denotes the transpose of the...

Ngày tải lên: 15/12/2013, 04:15

... two 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 ... row in A by a linear combination of itself and any other row, as long as we the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • Interchanging ... a very good choice A curiosity of this procedure, however, is that the choice of pivot will depend on the original scaling of the equations If we take the third linear equation in our original...

Ngày tải lên: 15/12/2013, 04:15

... simply the product of Q with the 2(N − 1) Jacobi rotations In applications we usually want QT , and the algorithm can easily be rearranged to work with this matrix instead of with Q Sample page ... d[], float b[]) 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 ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numerical algorithms involve solving a succession of linear systems each of which differs...

Ngày tải lên: 15/12/2013, 04:15

... Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) ... set of equations The advantage of Gaussian elimination and backsubstitution over Gauss-Jordan elimination is simply that the former is faster in raw operations count: The innermost loops of Gauss-Jordan ... matrix as part of the Gauss-Jordan scheme.) For computing the inverse matrix (which we can view as the case of M = N right-hand sides, namely the N unit vectors which are the columns of the identity...

Ngày tải lên: 24/12/2013, 12:16

... reduction free_vector(vv,1,n); 48 Chapter Solution of Linear Algebraic Equations To summarize, this is the preferred way to solve the linear set of equations A · x = b: float **a,*b,d; int n,*indx; ... subsequently want to solve a set of equations with the same A but a different right-hand side b, you repeat only lubksb(a,n,indx,b); not, of course, with the original matrix A, but with a and indx as were ... modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors...

Ngày tải lên: 24/12/2013, 12:16