A new index theory for linear self-adjoint operator equations and its applications (Q898572)

Consider a self-adjoint operator equation \[ Au=F'(u) \tag{OD} \] where \(H\) is an infinite-dimentional separable Hilbert space, \(A\) a self-adjoint operator on \(H\) with its domain \(D(A)\), \(F\) is a nonlinear functional on \(H\). In this paper, the authors develop a new index theory for \((OD)\) where the essential spectrum of the operator \(A\) may not be empty and they show that this index theory is a generalization of some other index theories. As applications, by assuming some twisted condition, they obtain the existence and multiplicity of periodic solutions for wave equations and beam equations.