On neutral functional differential equations with nonatomic difference operator
A simple prototype of the neutral functional differential equation is considered: \((d/dt)x_ t=Lx_ t+f(t),\) \(t\geq 0\), that is, (1) \(Dx_ t=\eta\), \(t\geq 0\), \(x_ 0=\phi\) is considered, where \(D\phi =\int^{0}_{-r}[d\mu (\theta)]\phi (\theta)\), \(\phi\in C(-r,0)\) or \(\phi \in L^ p\), \(\mu\) is a function of bounded variation on [-r,0] such that \(\lim_{\epsilon \searrow 0}\int^{0}_{-\epsilon}| d\mu (\theta)| =0\) and \(\eta\in R\). For \(\phi\in C\), we give a necessary condition on D for generating semigroups on C by equation (1). In the case \(\phi \in L^ p(-r,0;R)\), we investigate in detail the example of (1): \((2)\quad \int^{0}_{-1}| s|^{-\alpha}\chi (t+s)ds=\eta,\) \(t\geq 0\), \(x_ 0=\phi\), where \(0<\alpha <1\). In case \(p>1/(1-\alpha)\) we prove that (2) with \(\eta =D\phi\) generates a \(C_ 0\)-semigroup on \(L^ p(-1,0;R)\), and we also give a variation of constants formula for the solution of (2). We also prove that (2) with \(\eta =D\phi\) does not generate a \(C_ 0\)-semigroups on \(L^ p(-1,0;R)\) in the limiting case \(p=1/(1-\alpha).\) For the case \(p<1/(1-\alpha)\) we give another representation for the solution of (2) which is valid for any \(\eta\in R\) and \(\phi \in L^ p\) (-1,0;R). And we prove that (2) generates a \(C_ 0\)-semigroup on \(R\times L^ p(-1,0;R)\) for \(1\leq p<1/(1-\alpha)\). Finally we consider the nonhomogeneous problems where \(\eta\) in (2) is replaced by f(t). The method of this work is based on the Laplace-transform techniques.
neutral functional differential equation
scientific article; zbMATH DE number 3957711
scientific article