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Factorization of the linear differential operator. (English) Zbl 1375.47036

Summary: The paper deals with the problem of factorization of a linear differential operator with matrix-valued coefficients into a product of lower order operators of the same type. Necessary and sufficient conditions for the factorization of the considered operator are given. These conditions are obtained by using the integral manifolds approach. Some consequences of the obtained results are also considered.

MSC:

47E05 General theory of ordinary differential operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
34A30 Linear ordinary differential equations and systems
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[1] Bronstein M: On the factorization of linear ordinary differential operators.Math. Comput. Simul. 1996,42(4-6):386-389. · Zbl 1037.34501 · doi:10.1016/S0378-4754(96)00013-4
[2] Coppel WA: Disconjugacy. Springer, Berlin; 1971. [Lecture Notes in Mathematics 220] · Zbl 0934.39006
[3] Pòlya G: On the mean-value theorem corresponding to a given linear homogeneous differential equation.Trans. Am. Math. Soc. 1922, 24:312-324. · JFM 50.0299.02 · doi:10.2307/1988819
[4] Zettl A: General theory of the factorizations of ordinary linear differential operators.Trans. Am. Math. Soc. 1974, 197:341-353. · Zbl 0302.34007 · doi:10.1090/S0002-9947-1974-0364724-6
[5] Zettl A: Explicit conditions for the factorization ofnth order linear differential operators.Proc. Am. Math. Soc. 1973,41(1):137-145. · Zbl 0247.47041
[6] Etgen GJ, Jones GD, Taylor WE Jr.: On the factorizations of ordinary linear differential operators.Trans. Am. Math. Soc. 1986, 297:717-728. · Zbl 0605.34033 · doi:10.1090/S0002-9947-1986-0854095-9
[7] Janglajew K, Valeev K: The factorization of the difference operator.Comput. Math. Appl. 2001, 42:729-733. · Zbl 1008.39010 · doi:10.1016/S0898-1221(01)00192-4
[8] Valeev K, Janglajew K: The Factorization of the Differential Expression. Srednevolgskoye Matematicheskoye Obshchestvo, Saransk; 2003. [SVMO Preprint Series 53] · Zbl 1008.39010
[9] Berkovich L: Method of factorization of ordinary differential operators and some of its applications.Appl. Anal. Discrete Math. 2007, 1:122-149. · Zbl 1199.34003 · doi:10.2298/AADM0701122B
[10] Dobrogowska A, Janglajew K: The factorization of the [InlineEquation not available: see fulltext.]-difference operators. J. Differ. Equ. Appl. 2007, 13:1171-1177. · Zbl 1144.39015 · doi:10.1080/10236190701465027
[11] Littlejohn LL, López JL: Variation of parameters and solutions of composite products of linear differential equations.J. Math. Anal. Appl. 2010, 369:658-670. · Zbl 1197.34008 · doi:10.1016/j.jmaa.2010.03.064
[12] Janglajew KR, Valeev KG: Conditions for factorization of linear differential-difference equations.Tatra Mt. Math. Publ. 2013, 54:93-99. · Zbl 1389.34196
[13] Valeev KG: Splitting of Matrix Spectra. ‘Vishcha Shkola’, Kiev; 1986. (in Russian)
[14] Abazari R: Solution of Riccati types matrix differential equations using matrix differential transform method.J. Appl. Math. Inf. 2009,27(5-6):1133-1143.
[15] Janglajew K: On the reduction principle of difference equations.Dyn. Contin. Discrete Impuls. Syst. 1999, 6:381-388. · Zbl 0934.39006
[16] Reinfelds A, Janglajew K: Reduction principle in the theory of stability of difference equations.Discrete Contin. Dyn. Syst. 2007, 2007:864-874. supplement · Zbl 1163.39303
[17] Janglajew, K: Construction of an integral manifold for linear systems. Paper presented at the Israeli-Polish mathematical meeting, PTM, Isr. Math. Union (org.), Lodz, Poland, 11-15 Sept. 2011
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