In this paper we shall study a functional differential equation of second order with mixed type argument. For this problem we give an algorithm based on the step method and the successive approximation method.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
(Babes Bolyai Univ.)
(Univ Med & Pharm Iuliu Hatieganu)
D. Otrocol, V. Ilea, C. Revnic, An iterative method for a functional-differential equation of second order with mixed type argument, Fixed Point Theory, 14(2013), no. 2, pp. 427-434
Fixed Point Theory
Casa Cartii de Stiinta, Cluj-Napoca, Romania
Paper in html format
 G. Belitskii, V. Tkachenko, One-Dimensional Functional Equations, Operator Theory: Advances and Applications 144, Birkh˝auser Verlag, Basel, 2003.
 J.S. Cassell, Z. Hou, Initial value problem of mixed-type differential equations, Monatshefte Math., 124(1997), 133-145.
 V.A. Darzu, Wheeler-Feynman problem on compact interval, Studia Univ. Babe¸s-Bolyai Math., 47(2002), no. 1, 43-46.
 R.D. Driver, A “backwards” two-body problem of classical relativistic electrodynamics, The Physical Review, 178(1969), 2051-2057.
 L.J. Grimm, H. Schmidt, Boundary value problem for differential equations with deviating arguments, Aequationes Math., 4(1970), 176-180.
 Z. Hou, J.S. Cassell, Asymptotic solutions of mixed-type equations with a diagonal matrix, Analysis, 17(1997), 1-12.
 V. Hutson, A note on a boundary value problem for linear differential difference equations of mixed type, J. Math. Anal., 61(1977), 416-425.
 V.A. Ilea, Functional Differential Equations of First Order with Advanced and Retarded Arguments, Cluj University Press, 2006 (in Romanian).
 C.T. Kelley, Solving Nonlinear Equations with Newton’s Method, SIAM, 2003.
 J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Diff. Eq., 11(1999), no. 1, 1-47.
 D. Otrocol, V.A. Ilea, C. Revnic, An iterative method for a functional-differential equations with mixed type argument, Fixed Point Theory, 11(2010), no. 2, 327-336.
 R. Precup, Some existence results for differential equations with both retarded and advanced arguments, Mathematica (Cluj), 44(2002), no. 1, 25-31.
 I.A. Rus, Functional-differential equations of mixed type, via weakly Picard operators, Seminar on Fixed Point Theory Cluj-Napoca, 3(2002), 335-346.
 I.A. Rus, Picard operators and applications, Sciantiae Math. Jpn., 58(2003), no. 1, 191-219.
 I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9(2008), no. 1, 293-307.
 I.A. Rus, M.A. Serban, D. Trif, Step method for some integral equations from biomathematics, Bull. Math. Soc. Sci. Math. Roumanie, 54(102)(2011), no. 2, 167-183.
 I.A. Rus, C. Iancu, Wheeler-Feynman problem for mixed order functional differential equations, Tiberiu Popoviciu Itinerant Seminar of Functional Equations, Approximation and Convexity, Cluj-Napoca, May 23-29, 2000, 197-200.
 L.S. Schulman, Some differential difference equations containing both advance and retardation, J. Math. Phys., 15(1974), 195-198.
 J. Wu, X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Diff. Eq., 135(1997), 315-357.