Nonlinear functional differential equations of arbitrary orders. (English) Zbl 0934.34055

The initial value problem is considered for the fractional delay differential equation \[ D^\alpha x(t)= f(t,x(t), D^{\alpha_1} x(t- r),\dots, D^{\alpha_n} x(t- nr)),\tag{1} \]
\[ D^jx(t)= 0,\quad\text{for }t\leq 0,\quad j= 0,1,\dots, n,\tag{2} \] with \(\alpha\in (n,n+1]\), \(\alpha_k\in (k- 1,k]\), \(k= 1,\dots, n\). A sufficient condition for the existence of at least one (nondecreasing) solution to (1), (2) is established.


34K05 General theory of functional-differential equations
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