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On the existence of variational principles for differential-difference evolution equations

Abstract

The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.

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References

  1. V. M. Filippov, V. M. Savchin, and S. G. Shorokhov, Variational Principles for Nonpotential Operators (VINITI, Moscow, 1992), Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Noveishie Dostizh. 40; Engl. transl.: J. Math. Sci. 68 (3), 275–398 (1994).

    Google Scholar 

  2. V. M. Savchin, “An operator approach to Birkhoff’s equations,” Vestn. Ross. Univ. Druzhby Narodov, Mat. 2(2), 111–123 (1995).

    MATH  Google Scholar 

  3. V. M. Savchin and S. A. Budochkina, “On the structure of a variational equation of evolution type with the second t-derivative,” Diff. Uravn. 39(1), 118–124 (2003) [Diff. Eqns. 39, 127–134 (2003)].

    MathSciNet  Google Scholar 

  4. V. M. Savchin and S. A. Budochkina, “On the existence of a variational principle for an operator equation with second derivative with respect to ‘time’,” Mat. Zametki 80(1), 87–94 (2006) [Math. Notes 80, 83–90 (2006)].

    Article  MathSciNet  Google Scholar 

  5. S. A. Budotchkina and V. M. Savchin, “On indirect variational formulations for operator equations,” J. Funct. Spaces Appl. 5(3), 231–242 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. D. Myshkis and L. E. El’sgol’ts, “Some results and problems in the theory of differential equations,” Usp. Mat. Nauk 22(2), 21–57 (1967) [Russ. Math. Surv. 22 (2), 19–57 (1967)].

    Google Scholar 

  7. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications (Birkhäuser, Basel, 1997).

    MATH  Google Scholar 

  8. V. M. Filippov and V. M. Savchin, “On the inverse problem of the calculus of variations (IPCV) for functional partial differential equations,” in Function Spaces, Differential Operators, and Problems of Mathematical Education: Abstr. Talks 2nd Int. Conf. Dedicated to the 80th Birthday of L.D. Kudryavtsev (Fizmatlit, Moscow, 2003), pp. 235–236.

    Google Scholar 

  9. A. M. Popov, “Potentiality conditions for differential-difference equations,” Diff. Uravn. 34(3), 422–424 (1998) [Diff. Eqns. 34, 423–426 (1998)].

    Google Scholar 

  10. I. A. Kolesnikova, A. M. Popov, and V. M. Savchin, “On variational formulations for functional differential equations,” J. Funct. Spaces Appl. 5(1), 89–101 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  11. I. A. Kolesnikova and V. M. Savchin, “On the existence of variational principles for a class of the evolutionary differential-difference equations,” J. Funct. Spaces Appl., 780382 (2012).

    Google Scholar 

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Correspondence to V. M. Filippov.

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Original Russian Text © V.M. Filippov, V.M. Savchin, S.A. Budochkina, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 283, pp. 25–39.

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Filippov, V.M., Savchin, V.M. & Budochkina, S.A. On the existence of variational principles for differential-difference evolution equations. Proc. Steklov Inst. Math. 283, 20–34 (2013). https://doi.org/10.1134/S0081543813080038

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  • DOI: https://doi.org/10.1134/S0081543813080038

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