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About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

Annotation

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt; sinBt; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied.

Keywords

complex operator; real operator; pure imaginary operator; characteristic operator polynomial; family of solutions; Cauchy problem; operator determinant

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DOI

10.20310/1810-0198-2019-24-126-211-217

UDC

517.937

Pages

211-217

References

[1] V. I. Fomin, “On the general solution of a linear n th-order differential equation with constant bounded operator coefficients in a Banach space”, Differential Equations, 41:5 (2005), 687–692. [2] V. I. Fomin, “On the n th-order linear differential equation in a Banach space with special right part”, Differential Equations, 45:10 (2009), 1554–1556. [3] V. I. Fomin, “About a complex operator Banach algebra”, Tambov University Reports. Series: Natural and Technical Sciences, 23:124 (2018), 813–823, DOI: 10.20310/1810-0198-2018-23-124-813-823 (In Russian). [4] V. I. Fomin, “On the case of multiple roots of the characteristic operator polynomial of an n th-order linear homogeneous differential equation in a Banach space”, Differential Equations, 43:5 (2007), 732–735. [5] V. I. Fomin, “About a solutions family of a linear homogeneous differential equation of the n-th order in a Banach space”, Actual Areas of Research of the 21th Century: Theory and Practice, 6:42 (2018), 382–384 (In Russian). [6] V. I. Fomin, “About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators”, Modern Methods of the Theory of Functions and Related Problems, Voronezh Winter Mathematical School (Voronezh, January 28 – February 2, 2019), Materials of the International Conference, VSU Publishing House, Voronezh, 2019, 271–273 (In Russian).

Received

2019-02-25

Section of issue

Scientific articles

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