Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations

Abstract

We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs) with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov’s fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.

1. Introduction

In last few decades, FODEs become area of interest for the researcher because of high quality accuracy and usability in various fields of science and technology. A lot of physical and natural phenomena can be modeled through FODEs which provide better result than integer order differential equations. Due to this, FODEs are regarded as a special tool for molding. Numerous applications of FODEs can be studied in various disciplines like chemical technology, viscoelasticity, industrial robotics, mathematical economy, turbulent filtration in porous media, fractals theory, ecology, economics, plasma physics, metallurgy, electromagnetic theory, biology, signal and image processing, control theory, electric technology, chemical reaction design, potential theory, radio physics, aerodynamics, pharmacokinetics, and so on; further details are available in literature [1–7]. In last few decades, the existence theory has been given great attention by the researchers. In the concerned theory, they studied existence, uniqueness, and multiplicity of solutions by using different techniques of nonlinear analysis. Therefore, theory on existence and uniqueness of solutions to nonlinear FODEs has been explored very well; see [8–12]. Systems of FODEs have been considered in large numbers of research articles, because most of physical, biological, and chemical phenomena can be modeled in the form of systems of FODEs. For example, Su [13] studied existence of solutions for coupled system of fractional differential equations with two-point boundary value problems given aswhere , and denote Riemann-Liouville derivatives, , are continuous functions, and satisfy and . Wang et al. [14] investigate existence and uniqueness of positive solutions to a coupled system of fractional differential equations with three-point boundary conditions. The corresponding problem is given as follows: where functions are continuous, , and represent Riemann-Liouville derivatives. Liu et al. [15] studied existence of positive solutions to a coupled system of nonlinear FODEs with integral boundary conditions. The considered problem is given in the following sequel: where are in sense of Riemann-Liouville derivatives, , and are nonlocal functions. The existence and uniqueness of solutions of FODEs are an active area of research for the last few decades. For some remarkable work, we refer the reader to [8, 9, 13, 16–20].

Another qualitative aspect which is very important from the numerical and optimization point of view is devoted to stability analysis of FODEs. The stability of fractional differential equations has gained great attention from the researchers very recently. Different kinds of stability include exponential, Mittag-Leffler, and Lyapunov stabilities; see [21–23]. One of the most relaxed methods for stability for functional equations was introduced by Ulam [24] and Hyers [25] which is known as Hyers-Ulam stability. The aforesaid stability has been very well investigated for ordinary differential and integral equations as well as functional equations; see [26–29]. But for FODEs, the concerned stability is not properly investigated. Very few papers can be found in literature in which some initial and boundary value problems of FODEs have been considered; see [7, 23, 30–32].

Motivated by the aforementioned contributions of researchers, we discuss the existence and uniqueness of solutions for coupled system of nonlinear FODEs with boundary conditions involving fractional integral and derivative. Further, we also investigate the Hyers-Ulam stability for the proposed problem designed bywhere the derivative is in sense of Riemann-Liouville, , and the functions are continuous. We use Perov’s fixed point theorem [33] and Leray-Schauder fixed point theorem to develop some results for existence of at least one solution for our proposed coupled nonlinear FODEs with boundary conditions. Further, we establish some conditions for Hyers-Ulam type stability to the considered problem. The whole analysis is then demonstrated by providing a proper example.

2. Preliminaries

Here we provide some results and definitions for our proposed coupled nonlinear FODEs with boundary conditions from literature [1–3].

Definition 1. The fractional integral of order of a function is defined by provided that the integral on right is converging.

Definition 2. The Riemann-Liouville fractional order derivative of a function is defined by where and represents the integer part of , provided that the right side is pointwise defined on .

Lemma 3. The following result holds for fractional derivative and integral: for arbitrary .

Lemma 4 (see [20]). Let be a Banach space with closed and convex. Let be a relatively open subset of with and be a continuous and compact (completely continuous) mapping. Then either (1)the mapping has a fixed point in or(2)there exist and with

Definition 5. For a nonempty set , a mapping is called a generalized metric on if the following conditions hold: () (), (symmetric property).() (tetrahedral inequality).Note. The properties such as convergent sequence, cauchy sequence, and open/closed subset are the same for generalized metric spaces as held for the usual metric spaces.

Definition 6. For an matrix , the spectral radius is defined by , where are the eigenvalues of matrix .

Lemma 7 (see [33]). Let be a complete generalized metric space and let be an operator such that there exists a matrix with . If , then has a fixed point ; further for any the iterative sequence converges to .

Definition 8. Consider a Banach space such that be two operators. Then the operatorial system provided byis called Hyers-Ulam stable if we can find , such that, for each and for each solution of the inequalities given bythere exists a solution of system (8) which satisfies

Theorem 9 (see [26]). Considering a Banach space with being two operators such thatand if the matrix converges to zero, then the operatorial system (8) is Hyers-Ulam stable.

Lemma 10. An equivalent Fredholm integral representation of the system of boundary value problems (4) is given bywhere are Green’s functions given by

Proof. Applying the operator on the first equation of (4) and using Lemma 3, we haveThe boundary conditions and they yield due to singularity and . Hence, (14) takes the formSimilarly, by the same process with the second equation of the system, we obtain the second part of (12).

Lemma 11 (see [18]). Green’s function of system (12) has the following properties: () is continuous function on the unit square for all .() for all and for all .().() There exists a constant such that

3. Existence and Hyers-Ulam Stability

Define endowed with the Chebyshev norm . Further, define the norms and . Then, the product spaces are Banach spaces. Define the cones by and , where .

Lemma 12. Assume that are continuous. Then is a solution of (12), if and only if is a solution of system of Fredholm integral equations (4).

Proof. The proof of Lemma 12 is similar to proof of Lemma in [18].

Define byBy Lemma 12 the problem of existence of solutions of the integral equations (12) coincides with the problem of existence of fixed points of .

Lemma 13. Assume that are continuous. Then and , where is defined by (17).

Proof. The relation easily follows from the properties and of Lemma 11 and all we need to show is that holds. For , we have and in view of property of Lemma 11, for all , we obtainHence, it follows that Similarly, we obtain It follows that which implies that .

Lemma 14. Assume that are continuous; then is completely continuous.

Proof. We omit the proof, because it is similar to the proof of Lemma in [18].

Lemma 15. Assume that and are continuous on and there exist such that the following hold: (), for ;(), for (), where the matrix is defined by Then system (12) has a unique positive solution .

Proof. In view of Definition 5, we define the generalized metric by Obviously is a generalized complete metric space. For any using properties and , we obtain which implies thatSimilarly, we obtainHence, it follows thatwhereBy , . Hence by Lemma 7, system (12) has a unique positive solution.

Lemma 16. Let and be continuous on and there exist satisfying();();();().Then system (12) has at least one positive solution in

Proof. Choose and define . By Lemma 14, the operator is completely continuous. Choose and such that . Then, by properties , , and , we obtain for all Similarly, we obtain ; hence , which shows that . Thus, by Schauder fixed point theorem, has at least one fixed point in .

4. Hyers-Ulam Stability of (12)

In this section, we obtain some appropriate conditions under which the toppled system under our consideration is Hyers-Ulam stable.

Let the following assumption hold:() There exist constants , such that

Theorem 17. Assume that hypothesis holds and such thatfor all solutions , with converges to zero. Then the solution of coupled system (4) is Hyers-Ulam stable.

Proof. Considerwhich implies that
According to the previous sentence, . Similarly, we can also getwhere Therefore from (34) and (35), we have the following system of inequalities:where which converges to zero. Thus, in view of Theorem 9, the solution of coupled system (4) is Hyers-Ulam stable.

5. Examples

Example 1. Consider the following coupled nonlinear FODEs of boundary conditions: Here and . Moreover Here, ; hence by Lemma 15 the BVP (37) has a unique solution. For and , we have and, by simple calculation, we obtainHence, by using Lemma 16, BVP (37) has at least one positive solution. Further, it is easy to compute the matrix which converges to zero and so the solution is Hyers-Ulam stable by using Theorem 17.

6. Conclusion

In this paper, we investigate existence and uniqueness of solutions for the nonlinear FODEs with boundary conditions and also investigate Hyers-Ulam stability for the mentioned problem. We use Perov’s fixed point theorem [33] and Leray-Schauder fixed point theorem to develop some results for existence of at least one solution for our proposed coupled nonlinear FODEs with boundary conditions. Further, we establish some conditions for Hyers-Ulam type stability to the considered problem. The whole paper is very easy because of relaxed methods and conditions.

Conflicts of Interest

The authors declare that no conflicts of interest exist regarding this manuscript.

Authors’ Contributions

All authors equally contributed to this paper and approved the final version.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (11571378).