RACSAM (2018) 112:407–423
The uniqueness result of solutions to initial value
problems of differential equations of variable-order
Received: 18 October 2015 / Accepted: 9 March 2017 / Published online: 16 March 2017
© Springer-Verlag Italia 2017
Abstract This paper is concerned with the existence and uniqueness of solution to an initial
value problem for a differential equation of variable-order. The results are obtained by means
of ﬁxed point theorem. The obtained results are illustrated with the aid of examples.
Keywords Derivatives and integrals of variable-order · Initial value problem · Piecewise
constant functions · Uniqueness of solution
Mathematics Subject Classiﬁcation 26A33 · 34A08
The study of initial or boundary value problem for fractional differential equations has
become an object of extensive study in view of their extensive applications in various sci-
entiﬁc disciplines, such as ﬂuid mechanics, biomathematics, ecology, visco-elastodynamics,
aerodynamics, control theory, electro-dynamics of complex medium, etc., see . The
researchers have gained many beautiful results for existence of solutions to differential equa-
tions of fractional order by the nonlinear functional analysis methods, such as some ﬁxed
point theorems, monotone iterative method, etc., see [1–4].
In , the author studied the following initial value problem of fractional differential
x(t) = f (t, x ), 0 < t ≤ T < +∞,
x(0) = x
This research is supported by the Natural Science Foundation of China (11371364).
Department of Mathematics, China University of Mining and Technology, Beijing 100083, People’s
Republic of China