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Abstract
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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.
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