Dissipativity of \(\theta \)-methods for nonlinear delay differential equations of neutral type
This paper is concerned with the dissipativity properties of nonlinear delay differential equations of neutral type with the form \[ y'(t) = F (t, y(t), G(t, y(t-\tau ), y'(t- \tau ))),\quad t \geq 0\tag{1} \] where \( \tau \) is a positive constant, \( y: \mathbb{R} \to H \) (real or complex Hilbert space) and \( y(t) = g(t)\), \(t\leq 0\) is the given initial function. Here the evolutionary functional equation (1) is said to be dissipative in \(H\) if there is an absorbing bounded set \( B \subset H \) such that for all bounded set \(D\) and initial function \( g(t) \in D\) there is \( t_0 = t_0(D)\) such that the corresponding solution of (1) satisfies \( y(t) \in B \) for all \( t \geq t_0\). In this context the author gives a set of sufficient conditions on functions \(F\) and \(G\) that ensure the dissipativity in the above mentioned context. These sufficient conditions extend, in some sense, the ideas of \textit{A. M. Stuart} and \textit{A. R. Humphries} [SIAM Rev. 36, No.~2, 226--257 (1994; Zbl 0807.65091)] for standard ordinary differential equations to nonlinear delay differential equations of neutral type. In the second part of the paper the dissipativity of linear \(\theta\)-methods in finite dimensional spaces is studied. Now under the theoretical sufficient conditions of dissipativity it is proved that the above \(\theta\)--methods are dissipative for \( 1/2 < \theta \leq 1\). The paper ends with an interesting 2D example that confirms the previous analysis and also shows that the conditions of the above analysis are only sufficient but not necessary for dissipativity.
scientific article; zbMATH DE number 5550384