Abstract
The purpose of this paper is to present a differential equation with ”maxima”. Existence, uniqueness, inequalities of Caplygin type and data dependence (monotony, continuity) results for the solution of the Cauchy problem of this equation are obtained using weakly Picard operators theory.
Authors
Diana Otrocol
Tiberiu Popoviciu” Institute of Numerical Analysis,Cluj-Napoca, Romania Academy
I.A. Rus
Department of Applied Mathematics, “Babes-Bolyai” University, Cluj-Napoca, Romania
Keywords
Picard operators; weakly Picard operators; functionaldifferential equations with ”maxima”; fixed points; data dependence.
Paper coordinates
D. Otrocol, I.A. Rus, Functional-differential equations with “maxima” via weakly Picard operators theory, Bull. Math. Soc. Sci. Math. Roumanie, 51(99) 2008, no. 3, 253-261
About this paper
Journal
Bull. Math. Soc. Sci. Math. Roumanie
Publisher Name
Bucuresti, Societatea de Ştiinţe Matematice din România,
DOI
Print ISSN
Online ISSN
1220-3874
google scholar link
[1] N.V. Azbelev (ed), Functional-differential equations (Russian), Perm. Politekh. Inst., Perm, 1985.
[2] D. Bainov and D. Mishev, Oscillation theory of operator-differential equations, World Scientific, Singapore, 1995.
[3] L. Georgiev, V.G. Angelov, On the existence and uniqueness of solutions for maximum equations, Glasnik Matematicki, 37 (2002), no. 2, 275–281.
[4] P. Gonzales and M. Pinto, Convergent solutions of certain nonlinear differential equations with maxima, Math. Comput. Modelling, 45 (2007), nos. 1–2, 1–10.
[5] J. Hale, Theory of functional differential equations, Springer, 1977.
[6] A. Ivanov, E. Liz, S. Trofimchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Mathematical Journal, Vol. 54 (2002), 277–295.
[7] A. Ivanov, E. Liz, S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations and Dynamical Systems, Vol. 11 (2003), 33–54.
[8] V. Kolmanovskii and A. Myshkis, Applied theory of functional-differential equations, Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992.
[9] M. Ma lgorzata and G. Zhang, On unstable neutral difference equations with ”maxima”, Math. Slovaca, 56 (2006), no. 4, 451–463.
[10] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003), No.1, 191–219.
[11] I.A. Rus, Generalized contractions, Cluj University Press, 2001.
[12] I.A. Rus, Weakly Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, 2 (2001), 41–58.
[13] E. Stepanov, On solvability of same boundary value problems for differential equations with ”maxima”, Topological Methods in Nonlinear Analysis, 8 (1996), 315–326.
[14] M. Zima, The abstract Gronwall lemma for some nonlinear operators, Demonstratio Matematica, 31 (1998), no. 2, 325–332.