Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients.(English)Zbl 0917.34055

The authors consider the nonhomogeneous iterative functional-differential equation with variable coefficients $x'(t)= \sum^m_{j=1} a_j(t) x^{[j]} (t)+F(t)$ with certain side conditions $$x^{(i)}(\tau)= \tau_i$$, $$i=0,1, \dots,m$$, and with $$a_j$$, $$F$$ given real functions. Here, $$x^{[0]}(t)=t$$, $$x^{[ 1]}= x(t)$$; $$x^{[k]}(t)= x(x^{[k-1]}(t))$$; $$k=2,3, \dots,m$$. Under assumptions, far too technical to be included here, and applying classical fixed point theorems, the authors obtain the existence of a solution.

MSC:

 34K05 General theory of functional-differential equations 34K20 Stability theory of functional-differential equations
Full Text:

References:

 [1] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag New York [2] Eder, E., The functional differential equationxtxxt, J. differential equations, 54, 390-400, (1984) · Zbl 0497.34050 [3] Feck ǎ, E., On certain type of functional differential equations, Math. slovaca, 43, 39-43, (1993) [4] Wang, Ke, On the equationxtfxxt, Funkcial. ekvac., 33, 405-425, (1990) · Zbl 0714.34026 [5] Staněk, S., On global properties of solutions of functional differential equationxtxxtxt, Dynam. systems appl., 4, 263-278, (1995) · Zbl 0830.34064 [6] 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 [7] J. G. Si, S. S. Cheng, Smooth solutions of a nonhomogeneous iterative functional differential equation · Zbl 0912.34057
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.