Using the step method, we study a system of delay differential equations and we prove the existence and uniqueness of the solution and the convergence of the successive approximation sequence using Perov’s contraction principle and the step method. Also, we propose a new algorithm of successive apprthe oximation sequence generated by the step method and, as an example, we consider some second order delay differential equations with initial conditions.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy,
Technical University of Cluj-Napoca)
D. Otrocol; M.A. Serban, An efficient step method for a system of differential equations with delay, J. Appl. Anal. Comput. Vol. 8 (2018) no. 2, pp. 498-508
Journal of Applied Analysis and Computation
Wilmington Scientific Publisher, USA
 C. T. H. Baker, C. A. H. Paul and D. R. Wille, Issues in the numerical solution of evolutionary delay differential equations, Advances in Comput. Math., 1995, 3, 171–196.
 M. Dobritoiu and M.-A. Serban, Step method for a system of integral equations from biomathematics, Appl. Math. Comput., 2014, 227, 412–421.
 R. D. Driver, Ordinary and Delay Differential Equations, Vol. 20 of Applied Mathematical Sciences, Springer-Verlag, New York, 1977.
 V. Ilea, D. Otrocol, M. -A. Serban and D. Trif, Integro-differential equations with two times lags, Fixed Point Theory, 2012, 13(1), 85–97. 508 D. Otrocol & M.-A. Serban
 S. Micula, An iterative numerical method for Fredholm–Volterra integral equations of the second kind, Appl. Math. Comput., 2015, 270, 935–942.
 S. Micula, A fast converging iterative method for Volterra integral equations of the second kind with delayed arguments, Fixed Point Theory, 2015, 16(2), 371–380.
 D. Otrocol, A numerical method for approximating the solution of a LotkaVolterra system with two delays, Studia Univ. “Babes–Bolyai”, Mathematica, 2005, 50(1), 99–110.
 D. Otrocol, V.A. Ilea and C. Revnic, An iterative method for a functionaldifferential equations with mixed type argument, Fixed Point Theory, 2010, 11(2), 327–336.
 A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Priblijen. Metod Res. Dif. Urav Kiev, 1964 (in Russian).
 R. Precup, The role of the matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modelling, 2009, 49(3–4), 703– 708.
 I. A. Rus, Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, 2001, 2, 41–58.
 I. A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, ClujNapoca, 2001.
 I. A. Rus, Picard operators and applications, Sci. Math. Jpn., 2003, 58(1), 191–219.
 I. A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 2008, 9(1), 293–307.
 I. A. Rus, M. A. S¸erban and D. Trif, Step method for some integral equations from biomathematics, Bull. Math. Soc. Sci. Math. Roumanie, 2011, 54(102)(2), 167–183.
 L. F. Shampine, Solving delay differential equations with dde23, www.radford.edu/ thompson/webddes/tutorial.html.
 N. L. Trefethen, An extension of Matlab to continuous functions and operators, SIAM J. Sci. Comput., 2004, 25(5), 1743–1770.
 D. Trif, LibScEig 1.0, > Mathematics > Differential Equations > LibScEig 1.0, http://www.mathworks.com/matlabcentral/fileexchange, 2005.
 D. Trif, Chebpack, MATLAB Central, URL: http://www.mathworks.com/ matlabcentral/fileexchange/32227-chebpack, 2011.
 D. Trif, Matrix based operatorial approach to differential and integral problems, in MATLAB, AUbiquitous Tool for the Practical Engineer, Ed. Clara Ionescu, InTech, 2011, 37–62.
 D. Trif, Operatorial tau method for some delay equations, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity, 2012, 10, 169–189.