Research Article | Open Access

# A -Dimensional System of Fractional Neutral Functional Differential Equations with Bounded Delay

**Academic Editor:**M. Mursaleen

#### Abstract

In 2010, Agarwal et al. studied the existence of a one-dimensional fractional neutral functional differential equation. In this paper, we study an initial value problem for a class of -dimensional systems of fractional neutral functional differential equations by using Krasnoselskii’s fixed point theorem. In fact, our main result generalizes their main result in a sense.

#### 1. Introduction

As you know, many researchers are interested in developing the theoretical analysis and numerical methods of fractional equations, because different applications of this area have been foundedin various fields of sciences and engineering (see, e.g., [1–37]). In this paper, we investigate the initial value problem of a -dimensional system of fractional neutral functional differential equations with bounded delay: where , , and are constants, , , for , is the standard Caputo’s fractional derivative, are given functions satisfying some assumptions that will be specified later,, and for . If , then for each define by for all . One-dimensional version of the problem has been studied by Agarwal et al. (see [4]). We show that the problem (1) is equivalent to an integral equation and by using Krasnoselskii's fixed point theorem, we conclude that the equivalent operator has (at least) a fixed point. This implies that the problem (1) has at least one solution. One can find the following lemma in [38].

Lemma 1 (Krasnoselskii’s fixed point theorem). *Let be a Banach space and a closed convex subset of . Suppose that and are two maps of into such that for all . If is a contraction and is completely continuous, then the equation has a solution on .*

Let be an interval in and with the norm , where denotes a suitable complete norm on . Consider the product Banach space with the norm . The fractional integral of order with the lower limit for a function is defined by for and , provided the right-hand side is pointwise defined on . Here, is the gamma function. Also, Caputo’s derivative of order with the lower limit for a function is defined by for and ([34]).

#### 2. Main Results

Consider the problem (1). Let and be positive constants, , and
where . For obtaining our results, we need the following conditions:(H_{1}) is measurable with respect to on for all ,(H_{2}) is continuous with respect to on for all ,(H_{3})there exist and a real-valued function such that
for all , , and ,(H_{4}) for all ,(H_{5}) is continuous and
for all , and , where is a constant, for all ,(H_{6}) is completely continuous and the family is equicontinuous on for all bounded set in and .

Lemma 2. *Suppose that there exist and such that hold. Then the problem (1) for is equivalent to the equation
**
with conditions for and .*

*Proof. *It is easy to see that is Lebesgue measurable on by using conditions (H_{1}) and (H_{2}) for all . Also, a direct calculation shows that for . By using Holder's inequality and condition (H_{3}), we get that is Lebesgue integrable with respect to for all , , and , and
It is easy to see that if is a solution of the problem (1), then is a solution of . Now, suppose that is a solution of the equation and . Then and for all and . Thus, is a solution of the problem (1). This completes the proof.

Theorem 3. *Suppose that there exist and such that hold. Then the problem (1) has at least one solution on for some positive number .*

*Proof. *Since condition (H_{4}) holds, the equation is equivalent to the equation
and for all and . Let be defined by and for all and . If is a solution of problem (1) and for and , then for and . Thus,
for and . Since , are continuous and is continuous in for all , there exists such that and for and . Put , where and for all . Define
In fact, is a closed, bounded, and convex subset of . Define the operators and on by
where
for . It is easy to check that the operator equation has a solution if and only if is a solution for for all . In this case, will be a solution of the problem (1) on . Thus, the existence of a solution of the problem (1) is equivalent to the existence of a fixed point for the operator on . Hence, it is sufficient that we show that has a fixed point in . We prove it in three steps.*Step I*. for all .

Let be given. Then, for all . It is easy to check that for all . Also, we have
for all and . Thus, for all . Hence, for all .*Step II*. is a contraction on .

Let . Then,
and so
for all . This implies that , where . Since , is a contraction on .*Step III*. is a completely continuous operator.

Suppose that
for . It is clear that

Since is completely continuous for all , is continuous and also is uniformly bounded. By using condition (H_{6}), it is easy to check that is equicontinuous. On the other hand,
for all and . This implies that is uniformly bounded. Now, we prove that is equicontinuous. Let and be given. Then, we have
for all . Thus, is equicontinuous. Moreover, it is clear that is continuous for all . This implies that is a completely continuous operator. Now, by using Krasnoselskii's fixed point theorem we get that has a fixed point on and so the problem (1) has a solution , where for all and .

If we put for all , then we obtain next result.

Corollary 4. *Suppose that there exist and such that conditions hold, is continuous for all , and for all , , and , where is a constant for all . Then the problem (1) has at least one solution on for some positive number .*

If we put for all , then we obtain next result.

Corollary 5. *Suppose that there exist and such that conditions hold, is completely continuous for all , and the family is equicontinuous on for all bounded set in . Then the problem (1) has at least one solution on for some positive number .*

#### 3. Conclusions

In this work, we study an initial value problem for a class of -dimensional systems of fractional neutral functional differential equations by using Krasnoselskii’s fixed point theorem. Our result generalizes some old related results in a sense.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The research of the second and third authors was supported by Azarbaidjan Shahid Madani University.

#### References

- R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1200–1214, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,”
*Advances in Difference Equations*, vol. 2009, Article ID 981728, 47 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,”
*Acta Applicandae Mathematicae*, vol. 109, no. 3, pp. 973–1033, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1095–1100, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Ahmad, “Existence of solutions for fractional differential equations of order
*q**∈*(2,3] with anti-periodic boundary conditions,”*Journal of Applied Mathematics and Computing*, vol. 34, no. 1-2, pp. 385–391, 2010. View at: Publisher Site | Google Scholar - B. Ahmad, “New results for boundary value problems of nonlinear fractional differential equations with non-separated boundary conditions,”
*Acta Mathematica Vietnamica*, vol. 36, no. 3, pp. 659–668, 2011. View at: Google Scholar | MathSciNet - B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,”
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838–1843, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-Z. Bai and J.-X. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 150, no. 3, pp. 611–621, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495–505, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Bai and W. Sun, “Existence and multiplicity of positive solutions for singular fractional boundary value problems,”
*Computers & Mathematics with Applications*, vol. 63, no. 9, pp. 1369–1381, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Baleanu, H. Mohammadi, and Sh. Rezapour, “Positive solutions of an initial value problem for nonlinear fractional differential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 837437, 7 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Baleanu, R. P. Agarwal, H. Mohammadi, and Sh. Rezapour, “Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces,”
*Boundary Value Problems*, vol. 2013, article 112, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu, Sh. Rezapour, and H. Mohammadi, “Some existence results on nonlinear fractional differential equations,”
*Philosophical Transactions of the Royal Society of London A*, vol. 371, no. 1990, Article ID 20120144, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu, H. Mohammadi, and Sh. Rezapour, “On a nonlinear fractional differential equation on partially ordered metric spaces,”
*Advances in Difference Equations*, vol. 2013, article 83, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “The existence of positive solutions for a new coupled system of multiterm singular fractional integrodifferential boundary value problems,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 368659, 15 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations,”
*Advances in Differential Equations*, vol. 2013, article 368, 2013. View at: Google Scholar - D. Baleanu, S. Z. Nazemi, and Sh. Rezapour, “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*, vol. 4014, p. 31, 2014. View at: Google Scholar - M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 2391–2396, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, “Fractional telegraph equations,”
*Journal of Mathematical Analysis and Applications*, vol. 276, no. 1, pp. 145–159, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Cuevas and J. César de Souza, “Existence of $S$-asymptotically $\omega $-periodic solutions for fractional order functional integro-differential equations with infinite delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1683–1689, 2010. View at: Publisher Site | Google Scholar | MathSciNet - C. Cuevas and J. C. de Souza, “$S$-asymptotically $\omega $-periodic solutions of semilinear fractional integro-differential equations,”
*Applied Mathematics Letters of Rapid Publication*, vol. 22, no. 6, pp. 865–870, 2009. View at: Publisher Site | Google Scholar | MathSciNet - C. Cuevas, M. Rabelo, and H. Soto, “Pseudo-almost automorphic solutions to a class of semilinear fractional differential equations,”
*Communications on Applied Nonlinear Analysis*, vol. 17, no. 1, pp. 33–48, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. A. Darwish and S. K. Ntouyas, “On initial and boundary value problems for fractional order mixed type functional differential inclusions,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1253–1265, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. P. C. dos Santos and C. Cuevas, “A symptotically almost automorphic solutions of abstract fractional integro-differential neutral equations,”
*Applied Mathematics Letters of Rapid Publication*, vol. 23, no. 9, pp. 960–965, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,”
*Journal of Differential Equations*, vol. 199, no. 2, pp. 211–255, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, vol. 378, pp. 223–276, Springer, New York, NY, USA, 1997. View at: Google Scholar | MathSciNet - H. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, Singapore, 2000. View at: Publisher Site | MathSciNet - V. Lakshmikantham, “Theory of fractional functional differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 10, pp. 3337–3343, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,”
*European Journal of Pure and Applied Mathematics*, vol. 1, no. 1, pp. 38–45, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. A. Tenreiro MacHado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 639801, 34 pages, 2010. View at: Publisher Site | Google Scholar - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at: MathSciNet - H. Mohammadi and Sh. Rezapour, “Existence results for nonlinear fractional differential equations on ordered gauge spaces,”
*Journal of Advanced Mathematical Studies*, vol. 6, no. 2, pp. 154–158, 2013. View at: Google Scholar | MathSciNet - S. K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary conditions,”
*Advances in Difference Equations*, vol. 2012, article 130, 2012. View at: Publisher Site | Google Scholar | MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, New York, NY, USA, 1999. View at: MathSciNet - X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”
*Applied Mathematics Letters of Rapid Publication*, vol. 22, no. 1, pp. 64–69, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for $p$-type fractional neutral differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 2724–2733, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 3249–3256, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. R. Smart,
*Fixed Point Theorems*, Cambridge University Press, New York, NY, USA, 1974. View at: MathSciNet

#### Copyright

Copyright © 2014 Dumitru Baleanu et al. This 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.