On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
-
Manabu Ito
ABSTRACT
This article discusses differential equations in infinite-dimensional spaces. We introduce a general version of the Lipschitz-type continuity criterion without the notion of a distance and prove an existence and uniqueness result for ordinary differential equations in locally convex topological linear spaces. It is shown that remarkable techniques of classical analysis, such as the method of successive approximations, are still adaptable tools in the study of the abstract models.
- †
The notation | · |, with a vertical bar on each side, will also appear in a somewhat different context: namely, it is also employed to denote the length (or size) of an interval, i.e., the absolute difference between the endpoints of the interval. For example, |I| = |b − a| when I is a (compact) interval [a, b] of ℝ.
Acknowledgements
The author is grateful to the referee for careful comments and suggestions on improving this paper.
REFERENCES
[1] Arnold, V. I.: Ordinary Differential Equations, Translated from the Russian by Roger Cooke, Second printing of the 1992 edition, Universitext, Springer-Verlag, Berlin, 2006.Search in Google Scholar
[2] Coddington, E. A.: An Introduction to Ordinary Differential Equations, Prentice-Hall Mathematics Series Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961.Search in Google Scholar
[3] Dibík, J.—Nowak, C.—Siegmund, S.: A general Lipschitz uniqueness criterion for scalar ordinary differential equations, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Art. No. 34.10.14232/ejqtde.2014.1.34Search in Google Scholar
[4] Dutkiewicz, A.: On the existence of solutions of ordinary differential equations in Banach spaces, Math. Slovaca 65(3) (2015), 573–582.10.1515/ms-2015-0041Search in Google Scholar
[5] Godunov, A. N.—Durkin, A. P.: Differential equations in linear topological spaces, Vestnik Moskov. Univ. Ser. I Mat. Meh. 24(4) (1969), 39–47 (in Russian).Search in Google Scholar
[6] Krasnosel’Skiĭ, M. A.—Kreĭn, S. G.: Nonlocal existence theorems and uniqueness theorems for systems of ordinary differential equations, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 13–16 (in Russian).Search in Google Scholar
[7] Li, T. Y.: Existence of solutions for ordinary differential equations in Banach spaces, J. Differential Equations 18 (1975), 29–40.10.1016/0022-0396(75)90079-0Search in Google Scholar
[8] Lobanov, S. G.: Picard’s theorem for ordinary differential equations in locally convex spaces, translated from Izv. Ross. Akad. Nauk Ser. Mat. 56(6) (1992), 1217–1243 (in Russian), Russian Acad. Sci. Izv. Math. 41(3) (1993), 465–487.10.1070/IM1993v041n03ABEH002272Search in Google Scholar
[9] Millionščikov, V. M.: On the theory of differential equations in locally convex spaces, Mat. Sb. (N.S.) 57 (99) (1962), 385–406 (in Russian).Search in Google Scholar
[10] Polewczak, J.: Ordinary differential equations on closed subsets of Fréchet spaces with applications to fixed point theorems, Proc. Amer. Math. Soc. 107(4) (1989), 1005–1012.10.1090/S0002-9939-1989-1019755-6Search in Google Scholar
[11] Siegmund, S.—Nowak, C.—Dibík, J.: A generalized Picard-Lindelöf theorem, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Art. No. 28.10.14232/ejqtde.2016.1.28Search in Google Scholar
[12] Stanković, B.: Differential equations in locally convex spaces, Mat. Vesnik 7(22) (1970), 551–558 (in Serbo-Croatian).Search in Google Scholar
[13] Stanković, B.: Two theorems on differential equations in locally convex spaces, Acad. Serbe Sci. Arts Glas 283 (1972), Art. Id. 35, 33–42 (in Serbo-Croatian).Search in Google Scholar
[14] Yuasa, T.: Differential equations in a locally convex space via the measure of nonprecompactness, J. Math. Anal. Appl. 84(2) (1981), 534–554.10.1016/0022-247X(81)90186-4Search in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
