On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems

In this pages, we discuss the problem of equivalence between fractional differential and integral problems. Although the said problem was studied for ordinary derivatives, it makes some troubles in the case of fractional derivatives. This paper dealing with the question revealing the equivalence between the boundary value problems for Caputo-type fractional differential equations and the corresponding integral form. One of our main part is to determine the scope of equivalence of such problems. However, with the help of appropriate examples, we will show that even for the Hölderian functions, but being off the space of absolutely continuous functions, the equivalence between the Caputo-type fractional differential problems and the corresponding integral forms can be lost. As a pursuit of this, we will show here that, the main results obtained by many researchers contained a common mathematical error in the proof of the equivalence of the boundary value problems for Caputo-type fractional differential equations and the corresponding integral forms. In this connection, we are going to slightly modify the definition of the Caputo-type fractional differential operator into a more suitable one. We will show that the modified definition is more convenient in studying the boundary value problems for Caputo-type fractional differential equations. As an application, after recalling some properties of the said (new) operator, we summarize our discussion by presenting an equivalence result for differential and integral problems for Caputo-type fractional derivatives and for the weak topology. In order to cover the full scope of this paper, we investigate the equivalence problem in case of multivalued fractional problems.