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.


34K05 General theory of functional-differential equations
34K20 Stability theory of functional-differential equations
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