ONE PARAMETER FAMILY OF LINEAR DIFFERENCE EQUATIONS AND THE STABILITY PROBLEM FOR THE NUMERICAL pdf

... 5. 2. The problem of stability for boundary value methods As mentioned in the Introduction, the study of the stability for dissipative nonlinear problems is made by linearizing the nonlinearity ... around the asymptotically stable critical point. This is equivalent to examine the behavior of the solutions of a numerical method ONE PARAMETER FAMILY OF...

Ngày tải lên: 23/06/2014, 00:20

... Introduction Difference equations appear naturally as discrete analogues and in the numerical solutions of differential and delay differential equations, and they have applications in biology, ecology, physics, and ... ecology, physics, and so forth 1. The study of properties of nonlinear difference equations has been an area of intense interest in recent years. The...

Ngày tải lên: 21/06/2014, 11:20

... for the global attraction of (u ∗ ,v ∗ )relativetothe whole space R 2 (or the positive cone R 2 + ) for any of the three types of quasimonotone functions. Theorem 4.5. Let the conditions in Theorems ... initial functions for each of the above three types of quasimonotone functions. The comparison results for the first two types of quasimonotone functions are b...

Ngày tải lên: 23/06/2014, 00:20

... (1.9) is, for large n, a solution of the generalized Riccati equation (1.6)andsoproperty(1.7) coincides with (1.5), stated in the linear case. In [11], the open problems whether analogous results as the ... \{0}, ∆u n u n < ∆x n x n for large n. (3.1) The following theorem holds. 198 Recessive solutions for half -linear equations Theorem 3.2 (see [11]). If (1.1) is n...

Ngày tải lên: 23/06/2014, 00:20

... completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: ... still open. The purpose of this and the next paper Error bounds for asymptotic solutions of second-order linear difference equations II: the second case i...

Ngày tải lên: 21/06/2014, 11:20

... 4.9 then h is not the recessive solution. Proof. Similarly, as in the proof of Theorem 4.1, denote w k  r k ΦΔh k /h k  and let w k be a solution of 2.1 generated by another solution linearly ... Differential Equations, vol. 202 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2005. 17 M. Cecchi, Z. Do ˇ sl ´ a, and M. Marini, “On recessiv...

Ngày tải lên: 21/06/2014, 20:20

... that {u(n)} n≥0 is ᐁᏭ᏿. Then the solution {u k (n)} of (4.4)isalso ᐁᏭ᏿ with the same pair (δ 0 ,ε,N(ε)) as the one for {u(n)} n≥0 .Let(δ ∗ (ε),ε) be the pair for uniform stability of {η(n)}. For any given ... Functions and Ergodicity, Science Press, Beijing, China; Kluwer Academic, Dordrecht, The Netherlands, 2003. [19] T. Yoshizawa, Stability Theory and the Ex...

Ngày tải lên: 22/06/2014, 22:20

... by the Taylor formula for the function  Fk,v :R k  vHk, v at the center v  0 see the com- putations in the proof of Theorem 4.1 and observe that 4.29 is still sufficient for the uniform convergence ... distinguish between the cases p ≥ 2andp ∈ 1, 2. The last section of the paper is devoted to an application of the results of the previous sectio...

Ngày tải lên: 22/06/2014, 22:20

... ρ(K 2 ) < 1.Then(1.3) has at least one pe riodic solution. Proof. Note that Ω 0 × Ω 0 is a normal solid cone of X × X.LetA 1 , A 2 ,andA be the same operators in the proof of Theorem 2.1.Set g n = n+ω−1  s=n G(n,s)h s b s , ... that (A 2 (u,v)) ˇ n  σ(A 2 (u,v)) n for any n, ˇ n ∈ Z.Thus,A : Ω × Ω → Ω × Ω. Furthermore, in view of the boundedness of G and  G, and t...

Ngày tải lên: 23/06/2014, 00:20

... and the proof of Theorem 4.4, we get the fact that C is an oscillatory solution of (4.8). The opposite statement follows from Theorem 4.6.  Open problems. (1) It is an open problem whether the ... third-order differential equations. Moreover, the oscillation of (E )and( E A ) is characterized by means of second-order linear difference equations and the problem...

Ngày tải lên: 23/06/2014, 00:20