Differentiable solutions of functional equations in Banach spaces
Abstract
We deal with the functional equation
ϕ(F(x)) = g(x,ϕ(x))
where functions F and g are given, defined in open subsets of Banach spaces and taking values in Banach spaces as well. We prove theorems on the existtence and uniqueness of solutions of the equation in classes of differentiable functions. As corollaries we get some results on the conjugacy of diffeomorphisms. Analogous results have been known in the finite dimensional case only.
References
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21. S. Kurepa, Some properties of the spectral radius on a finite set of operators, Glas. Mat. 14 (34) (1979), 283-288.
22. J. Matkowski, On the uniqueness of differentiable solutions of a functional equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 253-255.
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2. G. Belitskii, Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class, (Russian). Funktsional. Anal. i Prilozhen. 7 (1973), 17-28.
3. B. Choczewski, Investigation of the existence and uniqueness of differentiable solutions of a functional equation, Ann. Polon. Math. 15 (1964), 117-141.
4. J. Dieudonné, Foundations of modern analysis, New York-London 1960.
5. L.E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. of the Cambridge Phil. Soc. 83 (1978), 159-165.
6. Ph. Hartman, Ordinary differential equations, New York-London-Sydney 1964.
7. M.W. Hirsch, C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis, Providence, RI 1970, pp. 133-165.
8. A.B. Kharibageshvili, On the solvability of a class of functional equations with degeneration, (Russian). Soobshch. Akad. Nauk Gruzin. SSR 117 (1935) no. 3, 497-499.
9. L.P. Kuchko, On linear functional equations, (Russian). Vest. Khark. Univ. 39 (1974), 3-14.
10. L.P. Kuchko, Local equivalence of functional equations, (Russian). Ukrain. Mat. Zh. 37 (1985) no. 4, 506-509.
11. L.P. Kuchko, Multidimensional functional equations and normalization of local matrix-functions, (Russian). Dokl. Akad. Nauk Ukrain. SSR Ser. A (1986) no. 4, 11-12.
12. L.P. Kuchko, Existence and uniqueness of local solutions of functional equations, (Russian). Funktsional. Anal. i Prilozhen. 20 (1986) no. 4, 83-84.
13. L.P. Kuchko, Local solvability of linear functional equations, (Russian). Ukrain. Mat. Zh. 39 (1987) no. 3, 335-339.
14. L.P. Kuchko, Nontrivial solvability of homogeneous functional equations, (Russian). Teor. Funktsii Funktsional. Anal. i Prilozhen. 40 (1987), 75-79.
15. L.P. Kuchko, Existence and uniqueness of the solutions of multidimensional functional equations, (Russian). Dokl. Akad. Nauk SSSR 297 (1907) no. 4, 788-790.
16. L.P. Kuchko, Functional equations with a degenerate transformation of the argument, (Russian). Vestnik Khar'kov. Gos. Univ. No 298 Mat. Mekh. i Voprosy Upravl. (1987), 98-101.
17. L.P. Kuchko, Multidimensional functional equations and the normalization of local matrix functions, (Russian). Sibirsk. Mat. Zh. 29 (1988), 70-83.
18. M. Kuczma, Functional equations in a single variable, Monografie Matematyczne 46, PWN Warszawa 1968.
19. M. Kuczma, Note on linearization, An. Polon. Math. 29 (1974), 75-81.
20. M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge Univ. Press 1990.
21. S. Kurepa, Some properties of the spectral radius on a finite set of operators, Glas. Mat. 14 (34) (1979), 283-288.
22. J. Matkowski, On the uniqueness of differentiable solutions of a functional equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 253-255.
23. J. Matkowski, On the existence of differentiable solutions of a functional equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 19-21.
24. Z. Nitecki, Differentiable dynamics, Cambridge-Massachusetts-London 1971.
25. J. Palis, On the local structure of hyperbolic points in Banach spaces, Annais da Acad. Brasil. de Ciencas 1969.
26. F. Riesz, B.Sz. Nagy, Leçons d'analyse fonctionnelle, Budapest 1952.
27. S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 309-324.
28. S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space, II, Amer. J. Math. 80 (1958), 623-631.
SablikM. (1993). Differentiable solutions of functional equations in Banach spaces. Annales Mathematicae Silesianae, 7, 17-55. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14244
Maciej Sablik
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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