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Existence and Extendability of Solutions of the Equation g(t, x) = 0

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Abstract

We consider the problem of extendability and existence of solutions of the equation g(t, x) = 0 on the maximum interval of their definition.

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REFERENCES

  1. V. V. Stepanov, A Course in Differential Equations [in Russian], Mir, Moscow (1953).

    Google Scholar 

  2. L. S. Pontryagin, Ordinary Differential Equations [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  3. P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  4. E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameter and Relaxation Oscillations [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  5. Encyclopaedia of Mathematics [in Russian], Vol. 5, Sovetskaya Éntsiklopediya, Moscow (1985).

  6. N. P. Erugin, Implicit Functions [in Russian], Leningrad University, Leningrad (1956).

    Google Scholar 

  7. É. Goursat, Cours D'Analyse Mathématique [Russian translation], Vol. 1, ONTI NKTP, Moscow (1936).

    Google Scholar 

  8. V. I. Zubov, “On the problem of existence and approximate representation of implicit functions,” Vestn. Leningrad. Univ. No. 19, 48–54 (1956).

    Google Scholar 

  9. S. M. Lozinskii, “Inverse functions, implicit functions, and the solution of equations,” Vestn. Leningrad. Univ. No. 7, 131–142 (1957).

    Google Scholar 

  10. L. Nirenberg, Topics in Nonlinear Functional Analysis [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  11. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  12. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  13. A. F. Izmailov, “Theorems on representation of families of nonlinear mappings and implicit-function theorems,” Mat. Zametki 67, Issue 1, 57–68 (2000).

    Google Scholar 

  14. A. V. Arutyunov, “ Implicit-function theorems and abnormal points,” Dokl. Ros. Akad. Nauk 368, No. 5, 586–589 (1999).

    Google Scholar 

  15. S. Reich, V. Khatskevich, and D. Shoikhet, “A global implicit-function theorem and fixed-point theorems for holomorphic mappings and semigroups,” Dokl. Ros. Akad. Nauk 347, No. 6, 743–745 (1996).

    Google Scholar 

  16. G. E. Shilov, Mathematical Analysis. Functions of Several Real Variables [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  17. A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations [in Ukrainian], Lybid', Kiev (1994).

    Google Scholar 

  18. B. V. Shabat, Introduction of Complex Analysis [in Russian], Vol. 1, Nauka, Moscow (1985).

    Google Scholar 

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Samoilenko, V.H., Kaplun, Y.I. Existence and Extendability of Solutions of the Equation g(t, x) = 0. Ukrainian Mathematical Journal 53, 427–437 (2001). https://doi.org/10.1023/A:1012344305548

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