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Russian Mathematics

, Volume 62, Issue 11, pp 28–44 | Cite as

On a Class of Operator Equations in Locally Convex Spaces

  • S. N. MishinEmail author
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Abstract

We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.

Keywords

locally convex space order and type of an operator characteristic function of an operator 

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References

  1. 1.
    Gel’fond, A. O. The Calculus of Finite Differences (GITTL,Moscow–Leningrad, 1952).Google Scholar
  2. 2.
    Gel’fond, A. O. “Linear Differential Equations of Infinite Order with Constant Coefficients and Asymptotic Periods of Entire Functions”, TrudyMat. Inst. Steklov. 38, 42–67 (1951).[in Russian].MathSciNetGoogle Scholar
  3. 3.
    Gel’fond, A.O., Leont’ev, A. F. “On aGeneralization of Fourier Series”, Mat. SbornikN.S. 29 (71), 477–507 (1951).[in Russian].Google Scholar
  4. 4.
    Demčenko, T. I. “An Investigation of the Solvability of an Equation of InfiniteOrder in GeneralizedGel’fond–Leont’ev Derivatives in a Certain Class of Entire Functions”, Litovsk. Mat. Sb. 7, 611–618 (1967).[in Russian].MathSciNetGoogle Scholar
  5. 5.
    Demčenko, T. I. “The Solvability of a Certain Class of Generalized Differential Equations of Infinite Order”, Soviet Mathematics, No. 8, 35–42 (1973).[in Russian].MathSciNetGoogle Scholar
  6. 6.
    Korobeiňik, Yu. F. On the Solvability in a Complex Domain of Certain General Classes of Linear Operator Equations (Rostov–on–Don, 2005) [in Russian].Google Scholar
  7. 7.
    Korobeiňik, Yu. F. “On a Class of Equations of Infinite Order in Generalized Derivatives”, Litovsk. Mat. Sb. 4, 497–515 (1964).[in Russian].MathSciNetGoogle Scholar
  8. 8.
    Korobeiňik, Yu. F. “On Equations of Infinite Order in Generalized Derivatives”, Sibirsk. Mat. Z. 5, 1259–1281 (1964).[in Russian].MathSciNetGoogle Scholar
  9. 9.
    Leont’ev, A. F. Generalization of Series of Exponentials (Nauka, Moscow, 1981) [in Russian].zbMATHGoogle Scholar
  10. 10.
    Leont’ev, A. F. “Differential Equations of Infinite Order and their Applications”, in Proc. of the IVth All–Union Mathematical Congress, II (Leningrad, 1964), 648–660 [in Russian].Google Scholar
  11. 11.
    Frolov, Yu. N. “On Nonhomogeneous Equations of InfiniteOrder in Generalized Derivatives”, VestnikMosk. Univ. Ser. I Mat. Mekh. No. 4, 3–13 (1960).[in Russian].Google Scholar
  12. 12.
    Frolov, Yu. N. “Solutions of an Equation of Infinite Order in Generalized Derivatives”, Trudy Mat. Inst. Steklo. 64, 294–315 (1961).[in Russian].MathSciNetGoogle Scholar
  13. 13.
    Radyno, Ya. V. Linear Equations and Bornology (Beloruss. Gos. Univ.,Minsk, 1982) [in Russian].zbMATHGoogle Scholar
  14. 14.
    Kantorovič, L. V., Akilov, G. P. Functional Analysis in Normed Spaces (GIFML, Moscow 1959) [in Russian].Google Scholar
  15. 15.
    Gromov, V. P., Mishin, S. N., Panyushkin, S. V. Operators of Finite Order and Differential–Operator Equations (OGU, Orel, 2009) [in Russian].Google Scholar
  16. 16.
    Mishin, S. N. “Homogeneous Differential–Operator Equations in locally convex Spaces”, Russian Mathematic. 61, No. 1, 22–38 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leont’ev, A. F. Entire Functions. Series of Exponentials (Nauka, Moscow, 1983) [in Russian].zbMATHGoogle Scholar
  18. 18.
    Mishin, S. N. “Connection of Characteristics of Operator Sequences with Bornological Convergence”, Vestnik RUDN. Ser.: Matem., Informatika, Fiz., No. 4, 26–34 (2010).Google Scholar
  19. 19.
    Radyno, Ya.V. “Linear Differential Equations in Locally Convex Spaces. I. Regular Operators and Their Properties”, Differencial’nye Uravnenij. 13 (8), 1402–1410 (1977).[in Russian].MathSciNetGoogle Scholar
  20. 20.
    Kondakov, V. P., Runov, L. V., Koval’chuk, V. E. “Bornologies and a Natural Extension of Classes of Regular Elements in Algebras of Operators”, Vladikavkaz.Mat. Zh. 8, No. 3, 29–39 (2006).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Nelson, E. “Analytic Vectors”, Ann. ofMath. 70 (3), 572–615 (1959).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goodman R. “Analytic and Entire Vectors for Representations of Lie Groups”, Trans. Amer.Math. Soc. 143Google Scholar
  23. 23.
    Radyno, Ya. V. “The Space ofVectors of Exponential Type”, Dokl. Akad.Nauk BSS. 27 (9), 791–793 (1983).[in Russian].Google Scholar
  24. 24.
    Mishin, S. N. “Invariance of the Order and Type of a Sequence of Operators”, Math. Notes 100, No. 3–4, 429–437 (2016).Google Scholar
  25. 25.
    Robertson, A. P., Robertson, V. Topological Vector Spaces (Cambridge University Press, 1966; Mir, Moscow, 1967).zbMATHGoogle Scholar
  26. 26.
    Krasičkov, I. F. “Closed Ideals in Locally Convex Algebras of Entire Functions”, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1), 37–60 (1967).[in Russian].MathSciNetGoogle Scholar
  27. 27.
    Krasičkov, I. F. “Completeness in a Space of Complex Valued Functions Determined by the Behavior of the Modulus”, Mat. Sb. (N. S.. 68 (1), 26–57 (1965).[in Russian].MathSciNetGoogle Scholar
  28. 28.
    Samko, St. G., Kilbas, A. A., Marichev, O. I. Fractional Integrals and Derivatives: Theory and Applications (Nauka i Tekhnika,Minsk, 1987; Gordon and Breach, New York, 1993).Google Scholar

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© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Orel State University named after I. S. TurgenevOryolRussia

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