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Attractivity for a k-dimensional system of fractional functional differential equations and global attractivity for a k-dimensional system of nonlinear fractional differential equations
Journal of Inequalities and Applications volume 2014, Article number: 31 (2014)
Abstract
In this paper, we present some results for the attractivity of solutions for a k-dimensional system of fractional functional differential equations involving the Caputo fractional derivative by using the classical Schauder’s fixed-point theorem. Also, the global attractivity of solutions for a k-dimensional system of fractional differential equations involving Riemann-Liouville fractional derivative are obtained by using Krasnoselskii’s fixed-point theorem. We give two examples to illustrate our main results.
1 Introduction
In recent years, many researchers have been focused on investigation of fractional differential equations which has played an important role in different areas of science (see for example, [1–22] and the references therein). As you know, there are many practical applications of fractional differential equations in different fields of science such as economy, biology, and the study of forced van der Pol oscillators (see for example, [23–25] and the references therein). On the other hand, there are a few papers on the attractivity of solutions for fractional differential equations and fractional functional differential equations (see for example, [15] and [16]). For the details of basic notions of this paper such the standard Caputo fractional derivative, the standard Riemann-Liouville fractional derivative, and the fractional integral of order q for a function f see [18]. In 2011, Chen and Zhou reviewed the attractivity of solutions for the fractional functional differential equation for via the boundary value condition for , where , , , cD is the standard Caputo fractional derivative, and a function with some properties [16]. In 2012, Chen et al. reviewed the global attractivity of solutions for the nonlinear fractional differential equation for via the boundary value problem , where , , is a constant, D is the standard Riemann-Liouville fractional derivative, and a function with some properties [15]. Also, they investigated the global attractivity of solutions for the nonlinear fractional differential equation for via the boundary value problem , where , , is a constant, cD is the standard Caputo fractional derivative, and a function with some properties [15].
In this paper, we investigate the attractivity of solutions for a k-dimensional system of fractional differential equations. Also, we investigate the global attractivity of solutions for another k-dimensional system of nonlinear fractional differential equations.
2 Preliminaries
In this paper, we investigate the attractivity of solutions for k-dimensional system of fractional functional differential equations
via the boundary value problems for , where for , , , cD is the standard Caputo fractional derivative, , , for and is a function satisfying some assumptions that will be specified later for . If , then is defined by for and . Also, we investigate the global attractivity of solutions for the k-dimensional system of nonlinear fractional differential equations
via the boundary value problems , where for , , , D is the Riemann-Liouville fractional derivative, , are constants, and is an integrable function satisfying some assumptions that will be specified later for . In fact, we say that the solution of the problem (2.1) is attractive whenever if there exists a constant such that for all and , then . Also, the zero solution of the problem (2.2) is said to be globally attractive whenever each solution tends to zero as . Let be the Banach space of all continuous functions from J into with the norm , where denotes a suitable complete norm on . It is clear that the product space is also a Banach space, where . We need the following Schauder fixed-point theorem and improvement of a fixed-point theorem of Krasnoselskii due to Burton, which one can find in [14, 17] and [19].
Theorem 2.1 If U is a nonempty, closed, bounded, and convex subset of the Banach space X and is completely continuous, then T has a fixed point.
Theorem 2.2 Let S be a nonempty, closed, convex, and bounded subset of the Banach space X, a contraction with constant , a continuous map which resides in a compact subset of X and and implies . Then the operator equation has a solution in S.
3 Main results
First, we investigate attractive solutions of the problem (2.1). In this way, suppose that for . We assume that is Lebesgue measurable with respect to t on and is continuous with respect to on for . Note that the problem (2.1) is equivalent to the system of equations
or
for . Define the operator by
where
for . It is easy to check that is a solution of the problem (2.1) if and only if is a fixed point of the operator T.
Theorem 3.1 Suppose that for each there exist and such that
for all and . Then the problem (2.1) has at least one attractive solution such that for all .
Proof Consider the set
where is a constant. It is easy to check that is a closed, bounded, and convex subset of . We show that the operator T has a fixed point in . This implies that the problem (2.1) has a solution. Note that for all and so . Now, we show that T is continuous. Let for all and for all . Then, we have for all and . Let be given. Choose such that implies that . Let and note that for . Also, we have
for . Thus, for all . Also, we have
for . Hence, for . This implies that is continuous for and so T is continuous. Now, we show that the set is equi-continuous. Let . Since for , there is a such that for all and . Let and . If , then
and so . If , then
Now, let . Since
we get in all cases. This implies that the set is equi-continuous. Since is uniformly bounded, is relatively compact. Now by using Theorem 2.1, T has a fixed point in which is a solution of the problem (2.1). Since , . Thus, is an attractive solution for the problem (2.1). □
Theorem 3.2 Suppose that for each there exist , and such that and
for all , and . Then the problem (2.1) has at least one attractive solution such that for all .
Proof It is sufficient we consider the set
where is a constant. By using a similar techniques and proof in Theorem 3.1, one can show that , T is continuous and is relatively compact. Now by using Theorem 2.1, T has a fixed point in which is a solution of the problem (2.1). Since , . Thus, is an attractive solution for the problem (2.1). □
Theorem 3.3 Suppose that for each there exists such that
for all . Then the problem (2.1) has at least one attractive solution such that for all .
Proof It is sufficient we consider the set
where is a constant. By using a similar techniques and proof in Theorem 3.1, one can show that , T is continuous and is relatively compact. Now, by using Theorem 2.1, T has a fixed point in which is a solution of the problem (2.1). Since , . Thus, is an attractive solution for the problem (2.1). □
Here, we are going to investigate global attractivity of solutions of the problem (2.2). We assume that is Lebesgue measurable with respect to t on and there exists a constant such that and is continuous with respect to on for all . Note that the problem (2.2) is equivalent to the system of equations
for all and . Define the operator by
where
for all . Now, define
where for all . Finally, define
where for all . It is easy to check that is a solution of the problem (2.2) if and only if it is a fixed point of the operator T. Note that A is a contraction with constant 0.
Theorem 3.4 Suppose that for each there exist and such that for all and . Then the zero solution of the problem (2.2) is globally attractive.
Proof Consider the set
where and is chosen such that for all . First, we show that B maps into . It is easy to check that is a closed, bounded, and convex subset of . Note that
and for all and . Thus,
for all and . Hence, . Now, we show that B is continuous on . Let for all and . Then, one can get for all . Let be given. Choose such that for all . Let for . Then, we have
for all . Hence, for all . Also,
for all . Thus, for all . This implies that is continuous on for and so B is continuous on . Now, we show that is equi-continuous. Let be given. Since , there exists such that for . Let and . If , then we have
and so . If , then
If , then we have
and so . Thus, is equi-continuous. Since is uniformly bounded, is relatively compact. Now, suppose that , and . Then,
for all . Since for , we get
Thus, for all and . This implies that for all . Therefore, by using Theorem 2.2 T has a fixed point in which is a solution of the problem (2.2). Since all elements of the set tend to 0 as , the zero solution of the problem (2.2) is globally attractive. □
Theorem 3.5 Suppose that for each there exist and such that for all and . Then the zero solution of the problem (2.2) is globally attractive.
Proof It is sufficient to consider the set
where and is chosen such that for all . Similar to the proof of Theorem 3.4, one can show that is a closed, bounded, and convex set, B maps into , is relatively compact, and B is continuous on . Now, suppose that , and . Then,
for all . Since for , we get
Thus, for all and . This implies that , for . Since all elements of the set tend to 0 as , the zero solution of the problem (2.2) is globally attractive. □
4 Examples
Here, we give an example to illustrate our results.
Example 4.1 Consider the 3-dimensional system of fractional functional differential equations
Define the maps
and put , and . It is easy to check that , and . Since
and
we get , and . Now, let , and . Then,
and
Thus, all conditions of Theorem 3.1 hold and so this system of fractional functional differential equations has an attractive solution.
Example 4.2 Let , , and be a constant for . Consider the 3-dimensional system of fractional differential equations
Define the maps
and
Thus, one can check that all conditions of Theorem 3.4 hold and so this system of fractional differential equations has a globally attractive solution.
5 Conclusions
Investigating the attractive solutions of the problems is an interesting topic within the fractional calculus. In this manuscript, we focus on the attractivity of solutions for two k-dimensional systems of fractional differential equations. Two illustrative examples show the applicability of the proposed methods. The techniques of the reported results can be applied for investigating the attractivity and global attractivity of solutions of different systems of (singular) fractional differential equations. Also, it is an interesting issue to investigate the attractivity and global attractivity of solutions of some systems of fractional differential inclusions.
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Acknowledgements
Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved the final version of this paper.
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Baleanu, D., Nazemi, S.Z. & Rezapour, S. Attractivity for a k-dimensional system of fractional functional differential equations and global attractivity for a k-dimensional system of nonlinear fractional differential equations. J Inequal Appl 2014, 31 (2014). https://doi.org/10.1186/1029-242X-2014-31
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DOI: https://doi.org/10.1186/1029-242X-2014-31