Existence theorems for a neutral functional differential equation whose leading part contains a difference operator of higher degree
This paper considers the problem of existence of solutions for a neutral functional differential equation of the form (A) \(D^n \Delta^m_\lambda x(t)+ f(t, x(g(t)))= 0\), \(t\geq t_0\), where \(D^n\) and \(\Delta^m_\lambda\) stand, respectively, for the \(n\)th iterate of the differential operator \(D\) and the \(m\)th iterate of the difference operator \(\Delta_\lambda\) defined by \(Dx(t)= {d\over dt} x(t)\) and \(\Delta_\lambda x(t)= x(t)- \lambda x(t- \tau)\). In case \(\lambda= 1\) use is made of the symbol \(\Delta\) instead of \(\Delta_1\), i.e., \(\Delta x(t)= x(t)- x(t- \tau)\). The conditions always assumed for (A) are as follows: (a) \(m\geq 1\), \(n\geq 1\), \(\lambda> 0\), \(\tau> 0\) and \(t_0> 0\); (b) \(g\in C[t_0, \infty)\), and \(\lim_{t\to \infty} g(t)= \infty\); (c) \(f\in C([t_0, \infty)\times \mathbb{R})\), and \(|f(t, x)|\leq F(t, |x|)\), \((t, x)\in [t_0, \infty)\times \mathbb{R}\), for some continuous function \(F(t, u)\) on \([t_0, \infty)\times \mathbb{R}_+\), \(\mathbb{R}_+= [0, \infty)\), which is nondecreasing in \(u\) for each fixed \(t\geq t_0\).
neutral functional differential equation
scientific article; zbMATH DE number 774755