Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces
The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.
Keywordslocally convex space order and type of an operator operator-differential equation equicontinuous bornology bornological convergence vector-valued function
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- 2.S. G. Krein, Linear Differential Equations in a Banach Space (Nauka, Moscow, 1967) [in Russian].Google Scholar
- 12.N. A. Aksenov, Application of the Theory of the Order and Type of an Operator in Locally Convex Spaces to the Study of Analytic Problems for Operator-Differential Equations, Cand. Sci. (Phys.–Math.) Dissertation (Orlovsk. Gos. Univ., Orel, 2011) [in Russian].Google Scholar
- 13.N. A. Aksenov, “The boundary-value problemfor differential-operator equations of the first order in a locally convex space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 3–15 (2011) [RussianMath. (Iz. VUZ) 55 (2), 1–12 (2011)].Google Scholar
- 14.S. H. Mishin, “On a particular form of operator-differential equations with variable coefficients,” Vestnik RUDN Ser. Mat. Inform. Phys., No. 1, 3–14 (2015).Google Scholar
- 16.V. P. Gromov, S. N. Mishin, and S. V. Panyushkin, Operators of Finite Order and Operator-Differential Equations (Orlovsk. Gos. Univ., Orel, 2009) [in Russian].Google Scholar
- 18.S. H. Mishin, “Relation between the characteristics of a sequence of operators with bornological convergence,” Vestnik RUDN Ser. Mat. Inform. Phys., No. 4, 26–34 (2010).Google Scholar
- 22.L. V. Kantopovich and G. P. Akilov, Functional Analysis in Normed Spaces (Fizmatgiz, Moscow, 1959) [in Russian].Google Scholar