Nonlinear partial functional differential equations: Existence and stability (Q5952257)

The author investigates the existence and uniqueness of the solutions of a class of nonlinear functional differential equations in Hilbert spaces with finite delay, NEWLINE\[NEWLINEdx(t)/dt= A\bigl(t,x(t) \bigr)+ f(t, x_t)\;(t>0),\quad x(t)= \psi(t)\;(t\in [-h,0]),NEWLINE\]NEWLINE where \(A(t,\cdot): V\to V'\) is a family of nonlinear monotone and coercive operators and \(f(t,\cdot):X\to H\) is Lipschitz continuous, \(V\) is a separable Banach space and \(H\) is a real separable Hilbert space such that \(V\hookrightarrow H\equiv H'\hookrightarrow V'\). Here \(V'\) is the dual of \(V\) and the injections are continuous and dense, \(x_t(s)= x(t+s)\) \((s\in [-h,0])\).NEWLINENEWLINENEWLINEThe main purpose of the paper is to establish existence and uniqueness results using a variational approach. Some stability criteria for the problem under consideration are studied by using a Razumikhin type argument. Several examples illustrating the theory are given.