Home Mathematics Successive approximations for a differential equation in a Banach space via Constantin condition
Article
Licensed
Unlicensed Requires Authentication

Successive approximations for a differential equation in a Banach space via Constantin condition

  • Aldona Dutkiewicz and Mirosława Zima EMAIL logo
Published/Copyright: January 29, 2021
Become an author with De Gruyter Brill

Abstract

We give some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space. Our approach is based on a generalized Nagumo condition due to A. Constantin and the properties of the Kuratowski measure of noncompactness.


M. Zima was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów.


  1. Communicated by Michal Fečkan

Acknowledgement

The authors warmly thank an anonymous referee for careful reading of the manuscript and constructive remarks that helped to improve the presentation of the results.

References

[1] Akhmerov, R. R.—Kamenskii, M. I.—Potapov, A. S.—Rodkina, A. E.—Sadovskii, B. N.: Measures of Noncompactness and Condensing Operators. Oper. Theory Adv. Appl. 55, Birkhäuser, Basel, 1992.10.1007/978-3-0348-5727-7Search in Google Scholar

[2] Ambrosetti, A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349–360.Search in Google Scholar

[3] Athanassov, Z. S.: Uniqueness and convergence of succeessive approximations for ordinary differential equations, Math. Japon. 35 (1990), 351–367.Search in Google Scholar

[4] Banaś, J.: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math. 10 (2012), 2003–2011.10.2478/s11533-012-0120-9Search in Google Scholar

[5] Banaś, J.—Goebel, K.: Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York-Basel, 1980.Search in Google Scholar

[6] Banaś, J.—Krajewska, M.: Existence of solutions for infinite systems of differential equations in spaces of tempered sequences, Electron. J. Differential Equations 2017 (2017), # 60.Search in Google Scholar

[7] Banaś, J.—Lecko, M.: Solvability of infinite systems of differential equations in Banach sequence spaces, J. Comput. Appl. Math. 137 (2001), 363–375.10.1016/S0377-0427(00)00708-1Search in Google Scholar

[8] Banaś, J.—Rzepka, B.: On solutions of infinite systems of integral equations of Hammerstein type, J. Nonlinear Convex Anal. 18 (2017), 261–276.Search in Google Scholar

[9] Constantin, A.: On Nagumo’s theorem, Proc. Japan Acad. 86, Ser. A (2010), 41–44.10.3792/pjaa.86.41Search in Google Scholar

[10] Deimling, K.: Nonlinear Functional Analysis, Springer, Berlin, 1985.10.1007/978-3-662-00547-7Search in Google Scholar

[11] Dutkiewicz, A.: On the convergence of successive approximations for a fractional differential equation in Banach spaces, Zeitschrift Anal. Anwend. 32 (2013), 301–307.10.4171/ZAA/1513Search in Google Scholar

[12] Dutkiewicz, A: On the existence of solutions of ordinary differential equations in Banach spaces, Math. Slovaca 65 (2015), 573–582.10.1515/ms-2015-0041Search in Google Scholar

[13] Ferreira, R. A. C.: A uniqueness result for a fractional differential equation, Fract. Calc. Appl. Anal. 15 (2012), 611–615.10.2478/s13540-012-0042-zSearch in Google Scholar

[14] Ferreira, R. A. C.: A Nagumo-type uniqueness result for an nth order differential equation, Bull. London Math. Soc. 45 (2013), 930–934.10.1112/blms/bdt022Search in Google Scholar

[15] Geyer, A.: A note on uniqueness and compact support of solutions in a recent model for tsunami background flows, Comm. Pure Appl. Anal. 11 (2012), 1431–1438.10.3934/cpaa.2012.11.1431Search in Google Scholar

[16] Guo, D.—Lakshmikantham, V.—Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Math. Appl. 373, Kluwer, Dordrecht, 1996.10.1007/978-1-4613-1281-9Search in Google Scholar

[17] Hartman, P.: Ordinary Differential Equations, Wiley, New York-London-Sydney, 1964.Search in Google Scholar

[18] Hazarika, B.—Srivastava, H. M.—Arab, R.—Rabbani, M.: Existence of solution for an infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it, J. Comput. Appl. Math. 343 (2018), 341–352.10.1016/j.cam.2018.05.011Search in Google Scholar

[19] Heinz, H. P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 6 (1983), 1351–1371.10.1016/0362-546X(83)90006-8Search in Google Scholar

[20] Mejstrik, T.: Some remarks on Nagumo's theorem, Czechoslovak Math. J. 62 (137) (2012), 235–242.10.1007/s10587-012-0008-7Search in Google Scholar

[21] Mursaleen, M.: Application of measure of noncompactness to infinite systems of differential equations, Canad. Math. Bull. 56 (2013), 388–394.10.4153/CMB-2011-170-7Search in Google Scholar

[22] Mustafa, O. G.: On the uniqueness of flow in a recent tsunami model, Appl. Anal. 91 (2011), 1375–1378.10.1080/00036811.2011.569499Search in Google Scholar

[23] Mustafa, O. G.—O’Regan, D.: On the Nagumo uniqueness theorem, Nonlinear Anal. 74 (2011), 6383–6386.10.1016/j.na.2011.06.019Search in Google Scholar

[24] Pianigiani, G.: Existence of solutions of ordinary differential equations in Banach spaces, Bull. Acad. Polon. Sci. Math. 23 (1975), 853–857.Search in Google Scholar

[25] Rozhan, J. R.: Existence of solutions for a class of system of functional integral equation via measure of noncompactness, J. Comput. Appl. Math. 313 (2017), 129–141.10.1016/j.cam.2016.09.011Search in Google Scholar

[26] Szufla, S.: On Volterra integral equations in Banach spaces, Funkcial. Ekvac. 20 (1977), 247–258.Search in Google Scholar

[27] Szufla, S.: On the existence of solutions of differential equations in Banach spaces, Bull. Acad. Polon. Sci. Math. 30 (1982), 507–515.10.7151/dmdico.1107Search in Google Scholar

[28] Szufla, S.—SzukaŁa, A.: An existence theorem for the equation x(m) = f(t, x) in Banach spaces, Functiones et Approximatio 25 (1997), 181–188.Search in Google Scholar

[29] Xu, J.—Zhu, Y. M.—Liu, J. C.: Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion, Acta Math. Sinica 28 (2012), 2407–2416.10.1007/s10114-012-1003-5Search in Google Scholar

[30] Zeidler, E.: Nonlinear Functional Analysis and its Applications, Springer, New York, 1993.Search in Google Scholar

Received: 2019-07-26
Accepted: 2020-06-21
Published Online: 2021-01-29
Published in Print: 2021-02-23

© 2021 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0455/pdf
Scroll to top button