Journal of Mathematical Sciences

, Volume 201, Issue 5, pp 566–580 | Cite as

Approximate Solution of Nonlinear Discrete Equations of Convolution Type

  • S. N. Askhabov


By the method of potential monotone operators we prove global theorems on existence, uniqueness, and ways to find a solution for different classes of nonlinear discrete equations of convolution type with kernels of special form both in weighted and in weightless real spaces p . Using the property of potentiality of the operators under consideration, in the case of space 2 and in the case of a weighted space p (ϱ) with a generic weight ϱ, we prove that a discrete equation of convolution type with an odd power nonlinearity has a unique solution and it (the main result) can be found by gradient method.


Convolution Monotone Operator Nonlinear Operator Weighted Space Convolution Operator 
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  1. 1.
    J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Univ. Press, Cambridge (1990).CrossRefzbMATHGoogle Scholar
  2. 2.
    S. N. Askhabov, “Application of the method of monotone operators to some nonlinear equations of convolution type and singular integral equations,” Russ. Math. (Iz. VUZ), 9, 64–66 (1981).MathSciNetGoogle Scholar
  3. 3.
    S. N. Askhabov, Singular Integral Equations and Equations of Convolution Type with a Monotone Nonlinearity [in Russian], Maikop State Technological University, Maikop, 2004.Google Scholar
  4. 4.
    S. N. Askhabov, Nonlinear Equations of Convolution Type [in Russian], Fizmatlit, Moscow (2009).Google Scholar
  5. 5.
    S. N. Askhabov and N. K. Karapetyanz, “Discrete equations of convolution type with a monotone nonlinearity,” Differ. Uravn., 25, No. 10, 1777–1784 (1989).zbMATHGoogle Scholar
  6. 6.
    S. N. Askhabov and N. K. Karapetyanz, “Discrete equations of convolution type with a monotone nonlinearity in complex spaces,” Dokl. RAN, 322, No. 6, 1015–1018 (1992).Google Scholar
  7. 7.
    M. R. Crisci, V. B. Kolmanovskii, E. Russo, and A. Vecchio, “A priori bounds on the solution of a nonlinear Volterra discrete equation,” Stability and Control: Theory and Appl., 3, No. 1, 38–47 (2000).MathSciNetGoogle Scholar
  8. 8.
    F. Dedagich and P. P. Zabreiko, “On superposition operators in spaces p ,Sib. Math. J., 28, No. 1, 86–98 (1987).CrossRefMathSciNetGoogle Scholar
  9. 9.
    H. Gajewski, K. Greger, and K. Zacharias, Nonlinear Operator Equations and Operator Differential Equations, Mir, Moscow (1978).Google Scholar
  10. 10.
    F. D. Gakhov and Yu. I. Tcherskii, Equations of Convolution Type, Nauka, Moscow, (1978).zbMATHGoogle Scholar
  11. 11.
    R. I. Kachurovskii, “Nonlinear monotone operators in Banach spaces,” Usp. Mat. Nauk, 23, No. 2, 121–168 (1968).Google Scholar
  12. 12.
    A. I. Kalandiya, Mathematical Methods of Two-Dimensional Elasticity, Nauka, Moscow (1978).Google Scholar
  13. 13.
    V. Koppenfels and F. Stahlman, Practice of Conformal Mappings, IL, Moscow (1963).Google Scholar
  14. 14.
    N. N. Luzin, Integral and Trigonometric Series, GITTL, Moscow (1951).Google Scholar
  15. 15.
    D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht–Boston–London, 61 (1993).Google Scholar
  16. 16.
    M. M. Vainberg, Variation Methods of Study of Nonlinear Operators, GITTL, Moscow (1956).Google Scholar
  17. 17.
    M. M. Vainberg, Variation Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Nauka, Moscow (1972).Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Chechen State UniversityGroznyRussia

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