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.
Reviewer: Sergei A.Mazanik (Minsk)
MSC:
34K07 | Theoretical approximation of solutions to functional-differential equations |
34K25 | Asymptotic theory of functional-differential equations |
34K05 | General theory of functional-differential equations |
References:
[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 |
[4] | Ke, Wang, On the equation \(x\)′\((t)=f(x(x(t)))\), Funkcialaj Ekvacioj, 33, 405-425 (1990) · Zbl 0714.34026 |
[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) · Zbl 0196.38302 |
[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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.