Research Article
On Some Existence and Uniqueness Results for a Class of
Equations of Order 0<≤1 on Arbitrary Time Scales
Abdourazek Souahi,
1
Assia Guezane-Lakoud,
2
and Rabah Khaldi
2
1
Laboratory of Applied Mathematics and Modeling, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
2
Laboratory of Advanced Materials, University of Badji Mokhtar-Annaba, P.O. Box 12, 23000 Annaba, Algeria
Correspondence should be addressed to Abdourazek Souahi; arsouahi@yahoo.fr
Received 21 April 2016; Accepted 9 June 2016
Academic Editor: Patricia J. Y. Wong
Copyright © 2016 Abdourazek Souahi et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Tis paper investigates the existence and uniqueness of solution for a class of nonlinear fractional diferential equations of fractional
order 0<≤1 in arbitrary time scales. Te results are established using extensions of Krasnoselskii-Krein, Rogers, and Kooi
conditions.
1. Introduction
Tis work concerns the investigation of sufcient conditions
for the existence and uniqueness of the solution of the fol-
lowing initial value problem with fractional derivative up to
the frst order on arbitrary time scales:
T
()=(,()),
∈[
0
,
0
+]
T
,0<≤1,
T
1−
(
0
)=0,
(1)
where
T
is the (lef) Riemann-Liouville fractional deriva-
tive of order on time scales T ,
T
is the Riemann-Liouville
fractional integral on time scales, and [
0
,
0
+]
T
is an interval
on T . We assume that is a right-dense continuous function.
Te theory of time scales calculus allows us to study
the dynamic equations, which include both diference and
diferential equations, both of which are very important in
implementing applications; for further information about the
theoretical and potential applications of the theory of time
scales, we refer the reader to [1–8] and the survey [9].
Te quantitative behaviour of solutions to ordinary dif-
ferential equations on time scales is currently undergoing
active investigations. Many authors studied the existence
and the uniqueness of the solutions of initial and boundary
diferential equations; see [8, 10–20] and the references cited
therein. In the papers [21–25], several authors were interested
by the existence and uniqueness of the frst-order diferential
equations on time scales with initial or boundary conditions
using diverse techniques and conditions. On the other hand,
some existence results for the fractional order diferential
equations were obtained in [10].
Our ideas arise from the papers [26–34], especially [30,
31], where the authors used Nagumo and Krasnoselskii-Krein
conditions on the nonlinear term , without satisfying Lip-
schitz assumption. Motivated greatly by the above works,
under appropriate time scales versions of the Krasnoselskii-
Krein conditions, we obtain the uniqueness and existence of
solution for the following two classes of diferential equations,
namely, the frst-order ODE
Δ
()=(,()), ∈[
0
,
0
+]
T
,
(
0
)=0,
(2)
and the fractional order FDE:
T
()=(,()),
∈[
0
,
0
+]
T
,0<≤1,
T
1−
(
0
)=0.
(3)
Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2016, Article ID 7327319, 8 pages
http://dx.doi.org/10.1155/2016/7327319