In this paper we deal with a class of inverse problems for an operator-valued mapping in a Banach space, which is usually associated with a compact operator equation. Our aim is to remove the restriction of compactness. The existence, uniqueness and stability of the solution of the inverse problem are shown. The results obtained here are applied to an inverse problem for the transport equation in an inhomogeneous medium. An example is given to illustrate the relationship between stability and topology.