We discuss the existence and uniqueness of solution for two types of fractional order ordinary and delay differential equations. Fixed point theorems are the main tool used here to establish the existence and uniqueness results. First we use Banach contraction principle to prove the uniqueness of solution and then Krasnoselskii's fixed point theorem to show the existence of the solution under certain conditions in a Banach space.
In mathematics delay differential equations are a type of differential equation in which the derivative of unknown function at a certain time is given in terms of the values of the function at previous times.
While physical events such as acceleration and deceleration take little time compared to the times needed to travel most distances, times involved in biological processes such as gestation and maturation can be substantial when compared to the data-collection times in most population studies. Therefore, it is often imperative to explicitly incorporate these process times into mathematical models of population dynamics. These process times are often called delay times, and the models that incorporate such delay times are referred as delay differential equation models [
Recently theory of fractional differential equations attracted many scientists and mathematicians to work on them [
Recently Benchoohra et al. [
First, in this paper we consider nonlinear delayed fractional differential equations:
For investigating to establish an existence theorem, we also consider a class of nonlinear delayed fractional differential equations of the form
For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory and functional analysis. These definitions and results can be found in the literature [
Let
For a function
We denote
A two-parameter function of the Mittag-Leffler type is defined by
The beta function is usually defined by
A subset of
Consider a metric space
Let for any
Then there exists
In this section we prove (
Let
Assume that there exists a constant
We prove that
In this section, by using Krasnoselskii’s theorem, we discuss the existence solution of (
Consider the following nonlinear fractional differential equation of the form
We consider the following fractional differential equation:
Now we prove our main result using Lemma We assume that Let
Furthermore we assume that
If the assumptions (H1) and (H2) satisfied, then the problem (
(i) Note that by Lemma
(ii) We will prove that
(iii) Finally we prove that
On the other hand for
Then
We considered two types of nonlinear delay fractional differential equations (FDE) with periodic boundary conditions involving Remann-Liouville fractional derivative possessing with a lower terminal at 0. In order to obtain the results in this paper, we have shown the existence and the uniqueness of solution for a class of nonlinear delayed FDE by Banach contraction principle. Then using Krasnoselskii's fixed point theorem we established an existence theorem for a different type of the equation that we have proven its uniqueness theorem.
The authors would like thank to the anonymous referees for their valuable comments and suggestions.