Let \(C\) denote the Banach space of continuous functions \(\Phi:I \to\mathbb{R}^n\), \(I=[-h,0]\), \(h>0\), endowed with sup norm. The norms considered in \(\mathbb{R}^n\) and in the space of \(n\times n\) matrices are the Euclidean norms.
The paper deals with the functional equation \[ X(t)= F(t,X_t), \quad X\in \mathbb{R}^n,\tag{1} \] where \(X_t\in C\) is defined by \(X_t(\theta)= X(t+ \theta)\), \(\theta\in I\), and \(F:\mathbb{R}\times C\to\mathbb{R}^n\) is a continuous function.
The author defines uniform boundedness and ultimate uniform boundedness of solutions \(X\) of equation (1) and investigates the existence of periodic solutions of equation (1) using Horn’s asymptotic fixed point theorem.
As an application he also considers the integral equation \[ X(t)=a(t)+ \int^t_{t-\alpha} G\bigl(t,s,X(s) \bigr)ds, \quad X\in\mathbb{R}^n, \] where \(a:\mathbb{R} \to\mathbb{R}^n\) is a continuous function and \(G:\mathbb{R} \times\mathbb{R} \times\mathbb{R}^n \to\mathbb{R}^n\) is a continuous function for \(-\infty <s\leq t<+\infty\) and \(X\in\mathbb{R}^n\).
Using Lyapunov’s second method, he offers sufficient conditions for uniform boundedness and ultimate uniform boundedness of solutions \(X\) and the existence of periodic solutions \(X\) of the integral equation (2).