solutions of first order differential equations

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 ... if the number of initial conditions is equal to the number of roots of the characteristic polynomial inside the unit circle and the number of conditions at the end of the interval of integration ... 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

... 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 ... 2003 P J Torres, “Existence of one-signed periodic solutions of some second -order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol 190, no 2, ... 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

... 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 similar ν It is clear that any solution of (2.4) of the form (1, u) ... and L´pez-G´mez [7], the proofs of these o o theorems contain gaps, the original statement of Theorem 1.40 of [4] is not correct, the original statement of Theorem 1.27 of [4] is stronger than what ... 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...

Ngày tải lên: 21/06/2014, 17:20

... Journal 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 ... “Positive solutions of semilinear differential equations with singularities,” Journal of Differential Equations, vol 148, no 2, pp 407–421, 1998 J Wang and D Jiang, “A singular nonlinear second -order ... positive solutions of singular third -order periodic boundary value problem,” Acta Mathematica Scientia Series B, vol 25, no 1, pp 81–88, 2005 14 M A Krasnosel’ski˘, Positive Solutions of Operator Equations, ...

Ngày tải lên: 22/06/2014, 00:20

... 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 for a generalized ... initial and boundary value problems Proof First we construct the lower solution α in (1.5) Let α(k) = cv(k), k ∈ N + , where v ∈ C(N + ,[0, ∞)) is the solution of −∆ ϕ p ∆v(k − 1) k ∈ N, = 1, (2.5) ... Time-mappings and 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...

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 ... the exponential stability of system 1.1 as well as on estimates of the norms of its solutions and their derivatives in the case of exponential stability and in the case of exponential stability...

Ngày tải lên: 21/06/2014, 11:20

... for Higher 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 ... 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 ... problems for 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...

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 ... 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, ... 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, “Asymptotic solutions...

Ngày tải lên: 22/06/2014, 11:20

... ≤ 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 ... 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 ... let N be the natural numbers set First, let us consider the BVP of the form + h(u) = 0, n u(0) = = u(1) − γu(η) u (t) + λa(t) g u + < t < 1, (2.1λ ) n Solutions of three-point nonlinear BVPs Definition...

Ngày tải lên: 22/06/2014, 22:20

... investigation of such asymptotic problems as describing the asymptotic behavior of solutions and the investigation concerning boundedness, convergence, or stability of solutions With the aid of different ... behaviour of solutions of systems of discrete equations via Liapunov type tech[8] nique, Computers & Mathematics with Applications 45 (2003), no 6–9, 1041–1057 , Anti-Lyapunov method for systems of ... Discrete systems of the form (1.1) containing only one delay are often called systems with pure delay The main goal of the present paper is to extend the notion of the delayed exponential of a matrix...

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 ... solutions of discrete Volterra equations, ” Journal of Mathematical Analysis and Applications, vol 314, no 1, pp 174–194, 2006 [16] Y Song and H Tian, “Periodic and almost periodic solutions of...

Ngày tải lên: 22/06/2014, 22:20

... second -order effects in the boundary 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 ... blow-up solutions of elliptic equations with exponential growth, to appear in Proceedings Differential and Difference Equations [3] C Anedda and G Porru, Higher order boundary estimates for blow-up solutions ... boundary estimates for blow-up solutions of elliptic equations, to appear in Differential Integral Equations [4] C Bandle, Asymptotic behaviour of large solutions of quasilinear elliptic problems, Zeitschrift...

Ngày tải lên: 22/06/2014, 22:20

... 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 ... two subspaces Nn−1 ,Nn In the remainder of this paper, for the sake of simplicity, the norm of Rm is assumed to be Euclidean Theorem 3.3 Let (3.1) be of index Moreover, suppose that G−1 (y,x) ... introduce the following notion of index LIDEs, which is quite similar to that of index (transferable) linear DAEs Definition 2.3 The LIDEs (2.2) are said to be of index if, for all n (i) rank...

Ngày tải lên: 23/06/2014, 00:20

... zero convergent solutions of second -order difference equations, J Math Anal Appl 141 (1989), no 2, 463–483 ´ O Doˇl´ and A Elbert, Integral characterization of the principal solution of half-linear ... properties of second order half-linear difference equations, Czechoslovak Math J 51(126) (2001), no 2, 303–321 G Zhang, S S Cheng, and Y Gao, Classification schemes for positive solutions of a second -order ... Recessive solution of half-linear second order difference equations, J Differ Equations Appl (2003), no 1, 49–61 H Hoshino, R Imabayashi, T Kusano, and T Tanigawa, On second -order half-linear oscillations,...

Ngày tải lên: 23/06/2014, 00:20

... 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 matrix A Equations (2.0.4) are called the normal equations ... corresponding x’s are then the columns of the matrix inverse of A (§2.1 and §2.3) • Calculation of the determinant of a square matrix A (§2.3) If M < N , or if M = N but the equations are degenerate, then ... 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 involves • Singular value decomposition of a matrix...

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 ... end of the main loop over columns of the reduction It only remains to unscramble the solution in view of the column interchanges We this by interchanging pairs of columns in the reverse order ... set only if we simultaneously interchange corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in the solution If we this, we will need to unscramble...

Ngày tải lên: 15/12/2013, 04:15

... Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to solve the next set of ... applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) ... 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...

Ngày tải lên: 15/12/2013, 04:15

... (2.2.4) is called backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution ... and then solving What is the advantage of breaking up one linear set into two successive ones? The advantage is that the solution of a triangular set of equations is quite trivial, as we have ... 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

... 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 ... which quite trivially solves the set of N + N equations (2.3.8)–(2.3.11) for all the α’s and β’s by just arranging the equations in a certain order! That order is as follows: • Set αii = 1, i ... (2.3.10) Equations (2.3.8)–(2.3.10) total N equations for the N + N unknown α’s and β’s (the diagonal being represented twice) Since the number of unknowns is greater than the number of equations, ...

Ngày tải lên: 24/12/2013, 12:16