Existence results on iterative functional differential equations (Q2782941)

The author considers an iterative functional-differential equation of the form NEWLINE\[NEWLINEx'(t)= H\bigl(t,x^{[n_1]} (t),\dots,x^{[n_m]} (t)\bigr), \tag{1}NEWLINE\]NEWLINE where \(m\) and \(n_1,\dots, n_m\) are positive integers and where, for each \(k\in \mathbb{N}\), \(x^{[k]}\) denotes the \(k\)th iterate of \(x\). Under the assumption that the function \(H\) satisfies a Lipschitz condition, the local existence and uniqueness of a solution \(x\) to (1) satisfying the condition (2) \(x(a)=a\) is proved. Next, conditions imposed upon \(H\) guarantee the existence of a minimal solution and a maximal solution to problem (1), (2) and the fact that any solution to this problem is increasing, convex and its second derivative is continuous. The author generalizes some results by \textit{J.-G. Si}, \textit{X.-P. Wang} and \textit{S. S. Cheng} [Aequationes Math. 60, No. 1-2, 38-56 (2000; Zbl 0968.34044)].