A unification theory of Krasnoselskii for differential equations

The theory of differential equations is very broad and contains many seemingly unrelated types of problems with markedly different methods of solution. It is very difficult to discern any unity in the theory. Yet, sixty years ago one of the foremost investigators, Krasnoselskii, suggested the possibility of finding unity. He claimed that the inversion of a perturbed differential operator yields the sum of a contraction and compact map. Accordingly, he proved a general fixed point theorem to cover this situation. In this paper we begin a long study with a view to putting his idea to the test. We begin with fractional differential equations of Caputo type, continue to neutral functional differential equations, and conclude with a study of an old problem of Volterra which continues to describe many important real-world problems. For these problems there is the perfect unity predicted by Krasnoselskii. It is an invitation to continue the study by examining other important real-world problems.