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.
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
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Prof. RNDr. Ing. Lubomír Kubáček, DrSc., Dr.h.c. –Nonagenarian
- Doc. RNDr. Roman Frič, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete Grüss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, β)-ramification invariants of a number field
Articles in the same Issue
- Regular papers
- Prof. RNDr. Ing. Lubomír Kubáček, DrSc., Dr.h.c. –Nonagenarian
- Doc. RNDr. Roman Frič, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete Grüss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, β)-ramification invariants of a number field