Ordinary differential equations on closed subsets of Fréchet spaces with applications to fixed point theorems

Author:
Jacek Polewczak

Journal:
Proc. Amer. Math. Soc. **107** (1989), 1005-1012

MSC:
Primary 47H15; Secondary 34A05, 34G20, 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1989-1019755-6

MathSciNet review:
1019755

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The construction and convergence of an approximate solution to the initial value problem , defined on closed subsets of a Fréchet space is given. Sufficient conditions that guarantee the existence of an approximate solution are analyzed in a relation to the Nagumo boundary condition used in the Banach space case. It is indicated that the Nagumo boundary condition does not guarantee the existence of an approximate solution. Applications to fixed points are given.

**[1]**Jacek Polewczak,*Ordinary differential equations on closed subsets of locally convex spaces with applications to fixed point theorems*, J. Math. Anal. Appl.**151**(1990), no. 1, 208–225. MR**1069457**, https://doi.org/10.1016/0022-247X(90)90252-B**[2]**R. S. Phillips,*Integration in a convex linear topological space*, Trans. Amer. Math. Soc.**47**(1940), 114–145. MR**0002707**, https://doi.org/10.1090/S0002-9947-1940-0002707-3**[3]**E. Dubinsky,*Differential equations and differential calculus in Montel spaces*, Trans. Amer. Math. Soc.**110**(1964), 1–21. MR**0163191**, https://doi.org/10.1090/S0002-9947-1964-0163191-9**[4]**V. M. Millionščikov,*A contribution to the theory of differential equations 𝑑𝑥/𝑑𝑡=𝑓(𝑥,𝑡) in locally convex spaces*, Soviet Math. Dokl.**1**(1960), 288–291. MR**0118931****[5]**Toshio Yuasa,*Differential equations in a locally convex space via the measure of nonprecompactness*, J. Math. Anal. Appl.**84**(1981), no. 2, 534–554. MR**639682**, https://doi.org/10.1016/0022-247X(81)90186-4**[6]**Richard S. Hamilton,*The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 1, 65–222. MR**656198**, https://doi.org/10.1090/S0273-0979-1982-15004-2**[7]**M. DeWilde,*Closed graph theorems and webbed spaces*, Pitmann, London, 1978.**[8]**Robert H. Martin Jr.,*Nonlinear operators and differential equations in Banach spaces*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR**0492671****[9]**B. N. Sadovskiĭ,*Limit-compact and condensing operators*, Uspehi Mat. Nauk**27**(1972), no. 1(163), 81–146 (Russian). MR**0428132****[10]**V. Lakshmikantham and S. Leela,*Nonlinear differential equations in abstract spaces*, International Series in Nonlinear Mathematics: Theory, Methods and Applications, vol. 2, Pergamon Press, Oxford-New York, 1981. MR**616449****[11]**Klaus Deimling,*Fixed points of condensing maps*, Volterra equations (Proc. Helsinki Sympos. Integral Equations, Otaniemi, 1978) Lecture Notes in Math., vol. 737, Springer, Berlin, 1979, pp. 67–82. MR**551029****[12]**Benjamin R. Halpern and George M. Bergman,*A fixed-point theorem for inward and outward maps*, Trans. Amer. Math. Soc.**130**(1968), 353–358. MR**0221345**, https://doi.org/10.1090/S0002-9947-1968-0221345-0**[13]**Simeon Reich,*Fixed points in locally convex spaces*, Math. Z.**125**(1972), 17–31. MR**0306989**, https://doi.org/10.1007/BF01111112**[14]**Simeon Reich,*A fixed point theorem for Fréchet spaces*, J. Math. Anal. Appl.**78**(1980), no. 1, 33–35. MR**595760**, https://doi.org/10.1016/0022-247X(80)90206-1

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H15,
34A05,
34G20,
47H10

Retrieve articles in all journals with MSC: 47H15, 34A05, 34G20, 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-1019755-6

Keywords:
Fréchet space,
approximate solution to initial value problem,
solution of IVP,
fixed point theory

Article copyright:
© Copyright 1989
American Mathematical Society