Geodesic
Analysis of Solvability for Weak Nonlinear Differential Algebraic Systems
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 55-62 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we consider the system of differential algebraic equations with a linear part and a small nonlinear term. We refer to such systems as weak nonlinear. Coefficient matrices of the linear part might be rectangular. Additionally, it is assumed that the solution meets some boundary conditions of a general kind. Basic assumption for the linear part is that it can be reduced to canonic form introduced by V. F. Chistyakov. By applying a special technique, analysis of the boundary problem is reduced to mastering of an operator which becomes a compression at a sufficiently small parameter. Under assumptions mentioned, we obtain sufficient and necessary existence conditions for weak nonlinear differential algebraic systems.
Keywords: differential algebraic equations; index; implicit; weakly nonlinear. 
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M. A. Perepelitsa; A. A. Pokutnyi. Analysis of Solvability for Weak Nonlinear Differential Algebraic Systems. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 6 (2013) no. 4, pp. 55-62. http://geodesic.mathdoc.fr/item/VYURU_2013_6_4_a5/

[1] S. L. Campbell, L. R. Petzold, “Canonical Forms and Solvable Singular Systems of Differential Equations”, SIAM J. Alg. Discrete Methods, 1983, no. 4, 517–521 | DOI | MR | Zbl

[2] A. M. Samoilenko, V. P. Yakovets, “On the Reducibility of a Singular Linear System to Central Canonical Form”, Dokl. Akad. Nauk Ukrainy, 1993, no. 4, 10–15 | MR

[3] Bojarintsev Yu. Ye., Danilov V. A., Loginov A. A., Chistyakov V. F., Numerical Methods for Solving Singular Systems, Nauka, Novosibirsk, 1989 | MR | Zbl

[4] Bojarintsev Yu. Ye., Chistyakov V. F., Algebro-Differential Systems. Methods for Solution and Investigation, Siberian Publishing House Nauka, Novosibirsk, 1998 | MR | Zbl

[5] Samoilenko A. M., Shkil M. I., Yakovets V. P., Linear Systems of Differential Equations with Singularities, Vishca Scola, Kiev, 2000

[6] Chistyakov V. F., Shcheglova A. A., Selected Chapters of Theory of Algebro-Differential Systems, Siberian Publishing House Nauka, Novosibirsk, 2003 | MR | Zbl

[7] P. Kunkel, V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, 2006 | MR | Zbl

[8] Zhuk S. M., “Closedness and Normal Solvability of the Operator Generated by Linear Differential Equation with Variable Coefficients”, Nonlinear Oscillations, 10:4 (2007), 464–479

[9] A. A. Boichuk, L. M. Shegda, “Singular Fredholm boundary value problems”, Nelin. Koliv., 10:3 (2007), 303–312 | MR | Zbl

[10] Boichuk A. A., Pokutnyi A. A., Chistyakov V. F., “Application of Perturbation Theory to the Solvability Analysis of Differential Algebraic Equations”, Computational Mathematics and Mathematical Physics, 53:6 (2013), 777–788 | DOI | DOI | MR | Zbl

[11] A. A. Boichuk, A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht–Boston, 2004 | MR | Zbl

[12] A. A. Boichuk, L. M. Shegda, “Bifurcation of Solutions of Singular Fredholm Boundary Value Problems”, Differential equations, 47:4 (2011), 459–467 | DOI | MR | Zbl

[13] A. A. Pokutnyi, “Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with unbounded operator in the linear part”, Differential equations, 48:6 (2012), 803–813 | DOI | MR | Zbl