Abstract
We consider an integro-differential equation with two time lags and we prove the existence, uniqueness and convergence of the sequence of the successive approximation by using contraction principle and step method with a weaker Lipschitz condition.
Also, we propose a new algorithm of successive approximation sequence generated by the step method and we give an example to illustrate the applications of the abstract results.
Authors
V.Ilea
Department of Mathematics, Babes-Bolyai University Cluj-Napoca, Romania
Diana Otrocol
Tiberiu Popoviciu Institutue of Numerical Analysis
M.A. Serban
Department of Mathematics, Babes-Bolyai University Cluj-Napoca, Romania
D.Trif
Department of Mathematics, Babes-Bolyai University Cluj-Napoca, Romania
Keywords
Integro-differential equation; two time lags, step method; Picard operators; fibre contraction principle.
Paper coordinates
V.Ilea, D. Otrocol, M.A. Serban, D. Trif, Integro-Differential Equation with two time lags, Fixed Point Theory, 13 (2012) no. 1, pp. 85-97.
About this paper
Journal
Fixed Point Theory
Publisher Name
House of the Book of Science Cluj-Napoca
DOI
Print ISSN
1583-5022
Online ISSN
2066-9208
google scholar link
[1] M. Dobrit¸oiu, I.A. Rus, M.A. S¸erban, An integral equation arising from infectious diseases, via Picard operator, Studia Univ. “Babes-Bolyai”, Mathematica, 52(2007), No. 3, 81-83.
[2] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Acad. Publ., Dordrecht, 1996.
[3] V.A. Ilea, Functional Differential Equations of First Order with Advanced and Retarded Arguments, Cluj University Press, 2006, (in Romanian).
[4] V.A. Ilea, D. Otrocol, Integro-differential equation with two times modifications, Carpathian J. Math., 27(2011), No. 2, 209-216.
[5] V. Kolmanovskii, A. Mishkis, Applied Theory of Functional Differential Equations, Kluwer Acad. Publ., 1992.
[6] D. Otrocol, Lotka-Volterra Systems with Retarded Argument, Cluj University Press, 2007, (in Romanian).
[7] R. Precup, Positive solution of initial value problem for an integral equation modelling infectious diseases, Seminar of Fixed Point Theory, Cluj-Napoca, 1991, 25-30.
[8] R. Precup, E. Kirr, Analysis of nonlinear integral equation modelling infectious diseases, Proc. Conf. West. Univ. of Timi¸soara, 1997, 178-195.
[9] I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9(2008), No. 1, 293-307.
[10] I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58(2003), No. 1, 191-219.
[11] I.A. Rus, Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, 2(2001), 41-58.
[12] I.A. Rus, Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey, Carpathian J. Math., 26(2010), No. 2, 230-258.
[13] I.A. Rus, M.A. S¸erban, D. Trif, Step method for some integral equations from biomathematics, Bull. Math. Soc. Sci. Math. Roumanie, 54(102)(2011), No. 2, 167-183.
[14] S. Sakata, T. Hara, Stability regions for linear differential equations with two kinds of time lags, Funkcialaj Ekvacioj, 47(2004), 129-144.
[15] N.L. Trefethen, An extension of Matlab to continuous functions and operators, SIAM J. Sci. Comput., 25(2004), No. 5, 1743-1770.
[16] D. Trif, LibScEig 1.0, > Mathematics > Differential Equations > LibScEig 1.0, http://www.mathworks.com/matlabcentral/fileexchange, 2005.