Abstract

The paper deals with the problem of Γ-convergence of functionals involving nonlocal transformation of argument, in particular, the argument deviation. The property of Γ-convergence of the functionals is related with various types of convergence of inner superposition (composition) operators. The study is conducted in Orlicz spaces. Γ-convergence of functionals with impulsive constraints is also studied as one of the applications.

Author information

Contact details are reproduced from the original publication and may be historical.

Mikhail E. Drakhlin

Research Institute, College of Judea and Samaria, Kedumim-Ariel, D.N. Efraim 44820, Israel.

Eugene Stepanov

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy.

Keywords

  • Calculus of Variations
  • Γ-convergence
  • Functional Differential Equation
  • Orlicz Space

References

  1. E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei, Rendiconti Cl. Sci. Mat. Fis. Natur. 58(8) (1975) 842–850. in Italian.Google Scholar
  2. G. Dal Maso: An Introduction to Γ-convergence, Birkhäuser, Boston, 1993.https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  3. G. Buttazzo, G. Dal Maso: Γ-convergence and optimal control problems, Journal of Optimization Theory and Applications 38 (1982) 385–407.https://doi.org/10.1007/BF00935345Google Scholar
  4. B. Dacorogna: Direct Methods in the Calculus of Variations, volume 78 of Applied Mathematical Sciences, Springer Verlag, Berlin, Heidelberg, 1989.https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  5. J. Appell, P.P. Zabrejko: Nonlinear Superposition Operators, volume 95 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, N.Y., Melbourne, 1990.https://doi.org/10.1017/CBO9780511897450Google Scholar
  6. P. Marcellini, C. Sbordone: Dualità e perturbazione di funzionali integrali, Ricerche di Mathematica 26 (1977) 383–421. in Italian.Google Scholar
  7. D. Azé: Convergence des variables duales dans des problèmes de transmission à travers des couches minces par des methodes d’épi convergence, Ricerche di Mathematica 35 (1986) 125–159. in French.Google Scholar
  8. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, volume 1 of Studies in Mathematics and Its Applications, North Holland Publ. Co. Amsterdam, Oxford, 1976.Google Scholar
  9. N. V. Azbelev, V. P. Maksimov, and L.F. Rakhmatullina: Introduction to the Theory of Functional Differential Equations, “Nauka”, Moscow, 1991. in Russian.Google Scholar
  10. C. Kuratowski: Topologie, volume 1, Państwowe Wydawnictwo Naukowe, Warszawa, 1958. in French.Google Scholar
  11. M.E. Drakhlin: On convergence of sequences of internal superposition operators, Functional Differential Equations. Israel Seminar 1 (1993) 83–94.Google Scholar
  12. M.E. Drakhlin: An inner superposition operator in the spaces of summable functions, Izvestiya VUZ. Matematika 30(5) (1986) 18–24. in Russian.Google Scholar
  13. M.E. Drakhlin: On one linear functional equation, In: Functional Differential Equations, Perm Politechnical Institute, Perm (1985) 91–111. in Russian.Google Scholar
  14. L. Tartar: Nonlocal effects induced by homogenization, In: Partial Differential Equations and the Calculus of Variations (Essays in honour of E. De Giorgi), volume II, Birkhäuser, Boston (1989) 925–938.https://doi.org/10.1007/978-1-4615-9831-2_19Google Scholar
  15. M. L. Mascarenhas: Memory effect phenomena and Γ-convergence, Proc. Royal Soc. Edinburgh 123A (1993) 311–322.https://doi.org/10.1017/S0308210500025701Google Scholar
  16. L. Ambrosio, P. D’Ancona, and S. Mortola: Gamma-convergence and the least squares method, Annali di Matematica Pura ed Applicata CLXVI(IV) (1994) 101–127.https://doi.org/10.1007/BF01765630Google Scholar

Suggested citation

M. E. Drakhlin, E. Stepanov. “Gamma-convergence for a Class of Functionals with Deviating Argument.” Journal of Convex Analysis 4 (1997), No. 1, 69–89. https://doi.org/10.68381/jca04003

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press LLC.