The exact analytic solutions of a nonlinear differential iterative equation. (English) Zbl 1155.34339

Liu, Hanze; Li, Wenrong

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 8, 2466-2478 (2008).

Summary: This paper is concerned with a second-order nonlinear iterated differential equation of the form \(c_{0}x{^{\prime\prime}}(z)+c_{1}x{^{\prime}}(z)+c_{2}x(z)=x(p(z)+bx(z))+h(z)\). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case \(|\beta |\neq 1\), but also the critical case \(|\beta |=1\), especially when \(\beta\) is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.

MSC:

34K05	General theory of functional-differential equations
34A05	Explicit solutions, first integrals of ordinary differential equations

Keywords:

iterative functional differential equation; exact analytic solution; Diophantine condition; majorant series

References:

[1]	Eder, E., The functional differential equation \(x^\prime(t) = x(x(t))\), J. differential equations, 54, 390-400, (1984) · Zbl 0497.34050
[2]	Si, J.G.; Li, W.R.; Cheng, S.S., Analytic solutions of an iterative functional differential equation, Comput. math. appl., 33, 47-51, (1997) · Zbl 0872.34042
[3]	Li, W.R., Analytic solutions for a class second order iterative functional differential equation, Acta math. sinica., 41, 1, 167-176, (1998), (in Chinese) · Zbl 1030.34058
[4]	Liu, H.Z.; Li, W.R., Analytic solutions of an iterative equation with second order derivatives, DCDIS proc., 3, 1030-1037, (2005)
[5]	Liu, Han Ze; Li, Wen Rong, Discussion on the analytic solutions of the second-order iterative differential equation, Bull. Korean math. soc., 43, 4, 791-804, (2006) · Zbl 1131.34048
[6]	Si, J.G.; Wang, X.P., Analytic solutions of a second-order functional differential equation with a state dependent delay, Results math., 39, 345-352, (2001) · Zbl 1017.34074
[7]	Petuhov, V.R., On a boundary value problem, Trudy sem. teor. diff. uravnenii otklon. argumentom univ. druzby narodov patrisa lumnmby, 3, 252-255, (1965), (in Russian)
[8]	Si, J.G.; Zhang, W.N., Analytic solutions of a second-order nonautonomous iterative functional differential equation, J. math. anal. appl., 306, 398-412, (2005) · Zbl 1083.34060
[9]	Li, W.R.; Cheng, S.S., Analytic solutions of an iterative functional equation, Aequationes math., 68, 21-27, (2004) · Zbl 1063.39018
[10]	Siegel, C.L., Iteration of analytic functions, Ann. of math., 43, 4, 607-612, (1942) · Zbl 0061.14904
[11]	Fichtenholz, G.M., Functional series, (1970), Gordon & Breach New York, London, Paris · Zbl 0213.35001