Fard and Bidgoli Advances in Diﬀerence Equations (2017) 2017:231
R E S E A R C H Open Access
Existence and uniqueness of solutions to
the second order fuzzy dynamic equations on
Omid S Fard
and TA Bidgoli
School of Mathematics and
Computer Science, Damghan
University, Damghan, Iran
Full list of author information is
available at the end of the article
In this paper, we introduce a new metric space to study the existence and uniqueness
of solutions to second order fuzzy dynamic equations on time scales. In this regard,
we use Banach’s ﬁxed point theorem to prove this result. Also, we see that this metric
guarantees an elegant and easier proof for the existence of solutions to second order
fuzzy dynamic equations on time scales.
MSC: Primary 34A05; secondary 34N07
Keywords: second order fuzzy dynamic equations; Banach’s ﬁxed point theorem;
metric space; time scales
Recently, one of the most interesting and signiﬁcant discussions in the ﬁeld of diﬀerential
equations is dynamic equations on time scales. The valuable applications of these equa-
tions in control theory, mathematical economics, mathematical biology, engineering and
technology have made it more impressive, see [–].
The theory of dynamic equations and the essential concepts in the calculus of time scales
were introduced by Bernd Aulbach and Stefan Hilger . The dynamic systems on time
scales have gained impetus since they demonstrate the interplay of two diﬀerent theories,
namely, the theories of continuous and discrete dynamic systems.
So far, many research papers have been done to investigate the existence of solutions for
ﬁrst and second order, boundary value and other types of dynamic equations, see [–].
Authors in  presented the deﬁnitions of delta derivative and delta integral for fuzzy-
valued functions. So, as the second step, it is natural to investigate the existence and
uniqueness of solutions to fuzzy dynamic equations on time scales. The main aim of this
paper is to prove the existence and uniqueness of solutions to second order fuzzy dynamic
equations on time scales, and we put oﬀ discussing this problem to Section .Beforethat,
we introduce a new metric on the set of the fuzzy continuous functions on time scales and
use it to deﬁne another new metric space.
This work is generalized as follows: Section contains a brief summary of the theory
of fuzzy sets and calculus of time scales. Section deals with the fuzzy calculus on time
scales.TheninSection we restrict our attention to two new metric spaces. Finally in the
last section the main results are stated and proved.
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-
vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.