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Second-order linear differential equations in a Banach space and splitting operators

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Abstract

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.

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Authors and Affiliations

  1. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia

    A. G. Baskakov, T. K. Katsaran & T. I. Smagina

Authors
  1. A. G. Baskakov
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  2. T. K. Katsaran
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  3. T. I. Smagina
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Correspondence to A. G. Baskakov.

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Original Russian Text © A.G. Baskakov, T.K. Katsaran, T.I. Smagina, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 10, pp. 38–50.

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Baskakov, A.G., Katsaran, T.K. & Smagina, T.I. Second-order linear differential equations in a Banach space and splitting operators. Russ Math. 61, 32–43 (2017). https://doi.org/10.3103/S1066369X1710005X

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  • DOI: https://doi.org/10.3103/S1066369X1710005X

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