Russian Mathematics

, Volume 61, Issue 6, pp 72–81 | Cite as

On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operator

  • A. V. ChernovEmail author


We prove some nontrivial corollaries of the Schauder theorem. We use these corollaries to prove a theorem concerning the total preservation of solvability of a controlled functional operator equation of the Hammerstein type with non-isotone and non-majorizable operator component in the right-hand side. We illustrate the application of the abstract theory by the example of the Dirichlet problem associated with a semilinear elliptic equation similar to a stationary diffusionreaction equation.


fixed point Hammerstein type equation semilinear elliptic equation of diffusion-reaction type 


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  1. 1.
    Chernov, A. V. “A Majorant Criterion for the Total Preservation of Global Solvability of Controlled Functional Operator Equation”, RussianMathematics 55, No. 3, 85–95 (2011).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Kalantarov, V. K. and Ladyzhenskaya, O. A. “Blow-up Theorems for Quasilinear Parabolic and Hyperbolic Equations”, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 69, 77–102 (1977) [in Russian].zbMATHGoogle Scholar
  3. 3.
    Sumin, V. I. “The Features of Gradient Methods for Distributed Optimal-Control Problems”, U.S.S.R. Comput.Math. Math. Phys. 30, No. 1, 1–15 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lions, J.-L. Control of Distributed Singular Systems (Nauka, Moscow, 1987) [Russian translation].zbMATHGoogle Scholar
  5. 5.
    Fursikov, A. V. Optimal Control of Distributed Systems. Theory and Applications (Nauchnaya Kniga, Novosibirsk, 1999) [in Russian].zbMATHGoogle Scholar
  6. 6.
    Chernov, A. V. “Smooth Finite-Dimensional Approximations of Distributed Optimization Problems via Control Discretization”, Comput. Math. Math. Phys. 53, No. 12, 1839–1852 (2013).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chernov, A. V. “On Applicability of Control Parametrization Technique to Solving Distributed Optimization Problems”, Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki, No. 1, 102–117 (2014) [in Russian].CrossRefzbMATHGoogle Scholar
  8. 8.
    Sergeev, Ya. D. and Kvasov, D. E. Diagonal Global Optimization Methods (Fizmatlit, Moscow, 2008) [in Russian].zbMATHGoogle Scholar
  9. 9.
    Chernov, A.V. “On Piecewise Constant Approximation inDistributedOptimization Problems”, Tr. Inst.Mat. Mekh. 21, No. 1, 264–279 (2015) [in Russian].MathSciNetGoogle Scholar
  10. 10.
    Chernov, A. V. “Sufficient Conditions for the Controllability of Nonlinear Distributed Systems”, Comput. Math. Math. Phys. 52, No. 8, 1115–1127 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Petrosyan, L. A., Zenkevich, N. A., and Semina, E. A. Game Theory (Vysshaya Shkola, Moscow, 1998) [in Russian].zbMATHGoogle Scholar
  12. 12.
    Chernov, A.V. “On Existence of the Nash Equilibrium in a Differential Game Associated with Elliptic Equations: TheMonotone Case”, Mat. Teor. Igr Prilozh. 7 (3), 48–78 (2015) [in Russian].MathSciNetzbMATHGoogle Scholar
  13. 13.
    Casas E. “Boundary Control of Semilinear Elliptic Equations with Pointwise State Constraints”, SIAMJ. Control and Optim. 31, 993–1006 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chernov, A. V. “On Volterra Type Generalization of Monotonization Method for Nonlinear Functional Operator Equations”, Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki 22, No. 2, 84–99 (2012) [in Russian].CrossRefzbMATHGoogle Scholar
  15. 15.
    Varga, J. Optimal Control of Differential and Functional Equations (Academic Press, New York, 1972; Nauka, Moscow, 1977).Google Scholar
  16. 16.
    Kantorovich, L. V. and Akilov, G. P. Functional Analysis (Nauka, Moscow, 1984) [in Russian].zbMATHGoogle Scholar
  17. 17.
    Ladyzhenskaya, O. A. and Ural’tseva, N. N. Linear and Quasilinear Elliptic Equations (Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar
  18. 18.
    Karchevskii, M. M. and Pavlova, M. F. Equations of Mathematical Physics. Additional Chapters (Kazan Univ. Press, Kazan, 2012) [in Russian].Google Scholar
  19. 19.
    Vorob’ev, A. Kh. Diffusion Problems in Chemical Kinetics (Moscow Univ. Press, Moscow, 2003) [in Russian].Google Scholar
  20. 20.
    Tröltzsch F. Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Stud. in Math. (AmericanMathematical Society, Providence, R. I., 2010), Vol. 112.CrossRefGoogle Scholar
  21. 21.
    Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order (Springer-Verlag, Berlin- Heidelberg-New York 1977; Nauka, Moscow, 1989).zbMATHGoogle Scholar
  22. 22.
    Chernov, A. V. “On Total Preservation of Solvability of Controlled Diriclet Problem for Elliptic Equation”, in Proceedings of International Conference dedicated to the 90th Anniversary of Academician N. N. Krasovski ‘Systems Dynamics and Control Processes’ (SDCP’2014) (IMM UrO RAN, Ekateringurg, 2015), pp. 359–366 (2015).Google Scholar

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© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Nizhny Novgorod State University named after N. I. LobachevskiiNizhny NovgorodRussia
  2. 2.Nizhny Novgorod State Technical University named after R. E. AlekseevNizhny NovgorodRussia

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