LINEAR DIFFERENCE EQUATIONS PPT

,yn= A +yn−1xn−ryn−s. Appl Math Comput. 176, 403–408 (2006). doi:10.1016/j.amc.2005.09.03910. Yalcinkaya, I: On the global asymptotic stability of a second-order system of difference equations. Discrete Dyn Nat Soc2008, 12 (2008). (Article ID 860152)11. Irićanin, B, Stević, S: Some sys[r]

Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2010, Article ID 594783, 19 pagesdoi:10.1155/2010/594783Research ArticleError Bounds for Asymptotic Solutions ofSecond-Order Linear Difference Equations II:The First CaseL. H. Cao1, 2and J. M. Zhang31Department[r]

be difficult to prove the limit characterization and the summation properties of recessivesolutions using only the knowledge of the asymptotic behavior of solutions and theirquasi-differences. This problem, when (1.10) fails, jointly with a discussion about relatedsummation criteria, is considered in t[r]

Moti vated by the work mentioned above, in this paper, we will establish some moregeneralized finite difference inequalities, which provide new bounds for unknown func-tions lying in these inequalities. We will illustrate the usefulness of the establishedresults by applying them to study the[r]

≡ Ᏸk1k2.Remark 2.3. The terminology used in Numerical Analysis is often different from the oneused in the Difference Equations setting. In order to avoid confusion, it is worth to notethat the terms such as “stable methods” refer to the well-conditioning of the error equa-tion and not necessari[r]

Reh´ak 1–5and it is summarized in 6, Chapter 3. It was shown there that the oscillation theory of 1.1is very similar to that of the linear equationΔrkΔxk ckxk1 0, 1.2which is the special case p  2in1.1. We will recall basic facts of the oscillation theory of1.1 in the follow[r]

+ μM(r, T),(3.10)which implies (2.3), since under (2.2) νM(r,T) < 1. The proof is complete.Assertion of Theorem 2.1 follows from the previous lemma and the periodicity of F(·,t)and B(·,t)int.M. I. Gil’ 54. Systems with linear majorantsIn this section and the next one it is assumed[r]

mentioned one can be investigated using properties of this associated linear equation. The main toolwe use is a linearization technique applied to a certain Riccati-type difference equation correspond-ing to the above-mentioned one.Copyright q 2008 O. Doˇsl´y and S. Fiˇsnarov´a. This is an ope[r]

singular perturbation method, indeed it offers considerable reduction and simplicity incomputation since it does not require to compute boundary layer correction solutions. Thismethod can be easily extended for initial value problems.References[1] C.ComstockandG.C.Hsiao,Singular perturbations for diff[r]

× S with similar properties to those of ᏸ describedin Lemmas 2.4 and 2.5.(2) To generalize results of this paper to the linear nth-order difference equationsinvolving quasi-differences.Z. Doˇsl´a and A. Kobza 13AcknowledgmentThis work was supported by the Czech Grant Agency, Grant 201/04/0586.R[r]

m+1. Further, taking n = 2 in (17.1.7.6) and using the initial values y2, , ym–1and the calculated values ym, ym+1,wefind ym+2. In a similar way, we consecutively findall subsequent values ym+3, ym+4, The above method of solving difference equations is called the step method.17.2. Lin[r]

kk + 1Bk+1(x),where Bk(x) are Bernoulli polynomials (see Paragraph 17.2.1-5, Item 1◦).Below we state a more general result that requires no conditions of the type (17.2.1.21).17.2. LINEAR DIFFERENCE EQUATIONS WITH A SINGLE CONTINUOUS VARIABLE 8932◦. Suppose that the right-hand s[r]

sufficient conditions for global exponential stability of the impulsive difference equa-tions with distributed delays are obtained. The conditions (A1)-(A5)areconservative.For example, we get the absolute value of all coefficients of (2). We will combinedelay-partitioning approach with dif[r]

sition can be used to solve systems of linear equations. To solveA · x = b (2.10.3)first form QT· b and then solveR · x = QT· b (2.10.4)by backsubstitution. Since QR decomposition involves about twice as many operations asLU decomposition, it is not used for typical systems of linear[r]

icol=k;}} else if (ipiv[k] > 1) nrerror("gaussj: Singular Matrix-1");}++(ipiv[icol]);We now have the pivot element, so we interchange rows, if needed, to put the pivotelement on the diagonal. The columns are not physically interchanged, only relabeled:indxc[i], the column of the ith pivot ele[r]

n∈Z, or, in otherwords, to the invertibility of the operator T on suitable sequence spaces defined onZ.This means that one can drop the above priori condition in the case that the differenceequations are defined onZ (see [7, Theorem 3.3] for the original result and see also [2,3, 11, 15] for recent res[r]

(x)+ak−1f ◦ Φk−1(x)+···+ a1f ◦ Φ1(x)+a0f ◦ Φ0(x) = 0 (2.1)in S is called a linear homogeneous functional equation of kth order with constant coef-ficients.Thecoefficients are the constants aj, j = 0,1,2, ,k. It is assumed that ak= 0.A solution of (2.1) is a function f∈ C0(᏶) that satisfies the e[r]

Hindawi Publishing CorporationAdvances in Difference EquationsVolume 2010, Article ID 693867, 12 pagesdoi:10.1155/2010/693867Research ArticleOscillation of Solutions of a Linear Second-OrderDiscrete-Delayed EquationJ. Baˇstinec,1J. Dibl´ık,1, 2and Z.ˇSmarda11Department of Mathematics, Faculty[r]

[7] G. M. Lee and N. D. Yen, Fr´echet and normal coderivatives of implicit multifunctions, Appl. Anal., 90 (2011), pp. 1011–1027.[8] S. Lu and S. M. Robinson, Variational inequalities over perturbed polyhedral convex sets, Math. Oper. Res., 33 (2008), pp. 689–711.[9] S. Lucidi, L. Palagi, and M. Rom[r]

, i =1,2,...,M.Nonsingular versus Singular Sets of EquationsIf N = M then there are as many equations as unknowns, and there is a goodchance of solving for a unique solution set of xj’s. Analytically, there can fail tobe a unique solution if one or more of the M equations is a linea[r]

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