The purpose of this paper is to study the solvability of the following nonlinear sequential fractional dynamical systems which involve fractional Riemann- Liouville derivatives with damping:
In dimensional Euclidean space, where, , , , is the standard Riemann-Liouville fractional derivative with the lower limit zero, and denotes the sequential Riemann-Liouville fractional derivative presented by Miller and Ross in . is a vector, a sufficiently order differentiable function; is an any matrix and ; is a continuous function.
In this connection, the so-called time-fractional diffusion equation that is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order with has to be especially mentioned. As a consequence, the time-fractional diffusion equation appeared to be a suitable mathematical model for the so-called sub-diffusion processes and thus became important and useful for different applications. For more details on this topics one can see for instance (see   ) and the reference therein.
Significant progresses have been made for the integer and fractional order differential equations (see   ). However, to the best of our knowledge, there is still little information known for the solvability of the nonlinear sequential fractional dynamical systems with damping and this fact is the motivation of the present work. Our aim in this paper is to provide some suitable sufficient conditions for the existence and uniqueness of solutions of the nonlinear sequential fractional dynamical systems which involve fractional Riemann- Liouville derivatives with damping.
The rest of this paper is organized as follows: In Section 2, we will present some basic definitions and preliminary facts which will be used throughout the following sections. In Section 3, we establish a suitable concept of solutions for problem (1.1) and present the existence and uniqueness of solutions under some appropriate conditions.
In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. For the -dimensional Euclidean space, let denote the Banach space of all continuous functions from into equipped with the norm for and we also introduce the space with the norm. Obviously, the space is a Banach space.
Next, for the convenience of the readers, we first present some useful definitions and fundamental facts of fractional calculus theory, which can be found in  .
Definition 2.1. The integral
is called Riemann-Liouville fractional integral of order, where is the gamma function.
Definition 2.2. For a function given in the interval the expression
is called the Riemann-Liouville fractional derivative of order q, where, denotes the integer part of number q.
Definition 2.3. 1) The Mittag-Leffler function in two parameters is defined as
where and denotes the complex plane.
In particularly, for we obtain the Mittag-Leffler function in one parameter as:
In addition, the Laplace transform of the Mittag-Leffler function is
where denotes the real parts of.
2) For an matrix, we define the Mittag-Leffler matrix function as follows:
and the Laplace transform of the Mittag-Leffler matrix function is
where is the identity matrix.
In order to study the solutions of problem (1.1), we need:
Lemma 2.4. () Let and let be the fractional integral of order If and then we have the following equality
Next, the Laplace transform formula for the Riemann-Liouville fractional integral is defined by
where is the Laplace of defined by
is a constant.
Lemma 2.5 Let and if and is a solution of the problem
then, satisfies the following equation
Proof. Apply Riemann-Liouville fractional integral operator on both sides of the equation (2.1), we get
Then by Lemma 2.4, we obtain
It follows from (2.2) that
Next, let taking the Laplace transformations
to the Equation (2.3), one can obtain
Taking inverse Laplace transform to both sides of the expression (2.4), then
Finally substituting Laplace transformation of Mittag-Leffler function and Laplace convolution operator, we get the solution of the given system as
This completes the proof of the lemma.
According to Lemma 2.5, we give the following definition:
Definition 2.6. A function is called a generalized solution of (1.1) if it satisfies the following fractional integral equation
3. Existence of Solutions
In this section, we present the existence and uniqueness of solutions for problem (1.1) under some appropriate conditions by a well known fixed point theorem.
To obtain the global existence of mild solutions of problem (1.1), we suppose:
H(f): The function satisfied is continuous for all and there exists a constant such that
Now, we are in the position to present the main result of this section.
Theorem 3.1. Assume that the condition H(f) holds. Then the problem (1.1) has a unique solution on
Proof. Define the operator as
Clearly, the problem of finding solutions for system (1.1) is reduced to find the fixed point of F. Firstly, under the assumption of our theorem, it is easy to check that F maps into itself. So it is only need to show that is a contraction operator on.
Let and then for any and we have
Using (3.1) and induction on, it follows easily that
Therefore, we obtain
Since is the general term of the Mittag-Leffler series and this series is uniformly convergent on real axis, then for large enough, one can obtain
Hence, is a contraction operator for large integer and hence. By applying the well-known Banach’s contraction mapping principle, we know that the operator and also has a unique fixed point on So problem (1.1) has a unique solution on. This completes the proof.
The authors would like to thank the anonymous reviewers for their careful reviewing and valuable suggestions. This work is supported by National Science Foundation of China (Grant No. 11371125), is also supported by the Natural Science Foundation of the Department of Education of Hunan Province (Grant No. 13A013). The second author is also supported by open fund of Guangxi Key laboratory of hybrid computation, NNSF of China Grants No. 11661001, the Project of Guangxi Education Department grant No. KY2016YB417.
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