Solution existence for non-autonomous variable-order fractional differential equations

Abstract

In this paper, we discuss the existence of the solution for a generalized fractional differential equation with non-autonomous variable order operators. In contrast to constant order fractional calculus, some standard relations including composition and sequential derivative rules do not remain correct under this generalization. Therefore, solving such a generalized fractional differential equation requires a different methodology, essential modifications, and generalizations for the basic concepts such as existence and uniqueness of the solution. The main goal of this paper is the proof of existence for the solution of a variable order fractional differential equation which is achieved by presenting four theorems. It is shown that if Lebesgue measurability, the continuity of the nonlinear term, and the conditions of differintegration operation are satisfied, then a solution for the variable order fractional differential equation exists.