Perturbation method for abstract second-order inverse problems

We are concerned with an abstract second-order inverse problem. We restrict ourselves here to abstract inverse problems in which, in order to solve the differential equation for u(t)u(t), it is necessary to know the value of some operator or functional ϕ[u(t)]=g(t) as a function of the argument tt. A new approach based on the perturbation theory of linear operators is introduced. Among the results obtained we point out sufficient conditions of existence and uniqueness of a solution. As a possible application of the abstract theorems, some applications from elasticity theory are given.