Analytic solutions of a second-order iterative functional differential equation. (English) Zbl 0983.34056

Authors’ abstract: This paper is concerned with the second-order iterative functional-differential equation \[ x''(x^{[r]}(z))=c_{0}z+c_{1}x(z)+\dots +c_{m}x^{[m]}(z), \] where \(r\) and \(m\) are nonnegative integers, \(x^{[0]}(z)=z,\) \(x^{[1]}(z)=x(z),\) \(x^{[2]}(z)=x(x(z)),\) etc., are the iterates of the function \(x(z).\) By constructing a convergent power series solution \(y(z)\) to a companion equation of the form \[ \alpha^{2}y''(\alpha^{r+1}z)y'(\alpha^{r}z)=\alpha y'(\alpha^{r+1}z)y''(\alpha^{r} z)+[y'(\alpha^{r} z)]^{3}\left[\sum_{i=0}^{m}c_{i}y(\alpha^{i}z)\right], \] analytic solutions of the form \(y(\alpha y^{-1}(z))\) to the original differential equation are obtained.


34K07 Theoretical approximation of solutions to functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K05 General theory of functional-differential equations
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[1] Eder, E., The functional differential equation x′(t)=x(x(t)), J. differential equations, 54, 390-400, (1984) · Zbl 0497.34050
[2] Feckan, E., On certain type of functional differential equations, Math. slovaca, 43, 39-43, (1993) · Zbl 0789.34036
[3] S.A. Gusarenko, Solvability of equations of neutral type with a locally Volterra operator, Functional-Differential Equations, ii – iii, Perm. Politekhn. Inst. Perm, 1985, pp. 26-29 (in Russian).
[4] Ke, Wang, On the equation x′(t)=f(x(x(t))), Funkcialaj ekvacioj, 33, 405-425, (1990) · Zbl 0714.34026
[5] M. Kuczma, Functional equations in a single variable, Polish Scientific Publishers, Warszawa, 1968. · Zbl 0196.16403
[6] Petahov, V.R., On a boundary value problem. trudy sem teor diff uravnenii otklon argument, Univ. druzby narodov patrisa lumumby, 3, 252-255, (1965)
[7] A.N. Sarkovskii, Functional and functional-differential equations in which the deviation of argument depends on the unknown function, Functional and Differential-Difference Equations, Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 148-155 (in Russian).
[8] Si, J.G.; Cheng, S.S., Analytic solutions of a functional differential equation with state dependent argument, Taiwanese J. math., 1, 4, 471-480, (1997) · Zbl 0892.30023
[9] Si, J.G.; Cheng, S.S., Note on an iterative functional differential equation, Demonstratio math., 31, 3, 609-614, (1998) · Zbl 0919.34064
[10] Si, J.G.; Li, W.R.; Cheng, S.S., Analytic solutions of an iterative functional differential equation, Comput. math. appl., 33, 6, 47-51, (1997) · Zbl 0872.34042
[11] Stanek, S., On global properties of solutions of functional differential equation x′(t)=x(x(t))+x(t), Dyn. systems appl., 4, 263-278, (1995) · Zbl 0830.34064
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