Unified stability theorems in RFDE and NFDE
This work deals with the neutral functional differential equations (NFDE) with infinite delay of the form \[ (*)\quad d/dt(D(t,x(s));\quad \alpha \leq s\leq t)=F(t,x(s);\quad \alpha \leq s\leq t),\quad t\geq t_*, \] where \(-\infty \leq \alpha \leq t_*\), \(\alpha\) could be -\(\infty\), F and D are Volterra functionals determined by t and the values of x(s) for \(\alpha\leq s\leq t\) and taking values in \({\mathbb{R}}^ n\). Three stability theorems for (*) are established, which correspond to the basic types of the ones for retarded functional differential equations (RFDE) with finite delay. It is shown that all the relevant results no matter whether for RFDE with finite delay, for RFDE with infinite delay, or for NFDE with finite delay can be deduced as corollaries of the obtained results. In this way, one arrives at stability theorems in unified forms for different types of FDE. Moreover, the Razumikhin techniques are employed in all the obtained results.
neutral functional differential equations
scientific article; zbMATH DE number 4122413