# Investigating a Coupled Hybrid System of Nonlinear Fractional Differential Equations.

1. Introduction

Fractional calculus is found to be more practical and effective than the classical calculus in the mathematical modeling of several phenomena. Fractional differential equations are very important and significant part of the mathematics and have various applications in viscoelasticity, electroanalytical chemistry, and many physical problems [1-6]. A systematic presentation of the applications of fractional differential equations can be found in the book of Balachandran and Park [7]. In recent years, many works have been devoted to the study of the mathematical aspects of fractional order differential equations [8-12]. There are numerous advanced and efficient methods, which have been focusing on the existence of solution to fractional differential equations. One of the powerful tools for obtaining the existence of solutions to such equations is the fixed point methods. Many authors use fixed point theorems to prove the existence and uniqueness of solution to nonlinear fractional differential equations; see, for example, [13-17].

On the other hand, the study for coupled systems of fractional differential equations is also important as such systems occur in various problems of applied nature, for instance, [18-25]. Additionally, fixed point theory can be used to develop the existence theory for the coupled systems of fractional hybrid differential equations [13,16,17]. Bashiri et al. [17] discussed the existence of solution to the following system of fractional hybrid differential equations of order p [member of] (0,1):

[mathematical expression not reproducible] (1)

where [alpha] > 0, and the functions f:[0,T] x R [right arrow] R, f(0,0) = 0 and g : [0,1] x R x R [right arrow] R satisfy certain conditions. [D.sub.p] is the R-L fractional derivative of order p.

Recently, the existence of solutions for fractional differential equations involving the Caputo fractional derivative was studied in [13, 26-28]. Motivated by the work of Bashiri et al. [17], in this paper we are concerned with the existence of solutions to three-point boundary value problem for a coupled system of hybrid fractional differential equations of order p [member of] [n- 1, n) given by

[mathematical expression not reproducible] (2)

where [alpha] > 0, 0 < [[eta].sub.1] < [[eta].sub.2] < 1, f : [0,1] x R [right arrow] R, f(0, x(0)) = 0,and g :[0,1] x R x R [right arrow] R. [D.sup.P] is the Caputo fractional derivative of order p. Moreover, an example is given to illustrate the validity of the existence result.

2. Preliminaries

Throughout this manuscript [PHI] = {[psi]: [R.sup.+] [right arrow] [R.sub.+] such that [psi](r) < r for r > 0 and [psi](0) = 0}, C([0, T] x R, R) denote the class of continuous functions f:[0,T] x R [right arrow] R, and C([0, T]x R x R, R) denote the class of functions g :[0, T] x R x R [right arrow] R such that

(i) the map t [right arrow] g(t, x, y) is measurable for each x,y [member of] R,

(ii) the map x [right arrow] g(t, x, y) is continuous for each x [member of] R,

(iii) the map y [right arrow] g(t, x, y) is continuous for each y [member of] R.

We need the following definitions which can be found in [9].

Definition 1. The Riemann-Liouville fractional integral of order [alpha] > 0 of function f [member of] [L.sup.1]([R.sup.+]) is defined as

[mathematical expression not reproducible], (3)

provided that the right side is pointwise defined on (0, [infinity]).

Definition 2. Let [alpha] be a positive real number, such that m-1 [less than or equal to] [alpha] < m, m [member of] N, and [f.sup.m](x) exists, a function of class C. Then Caputo fractional derivative of f is defined as

[mathematical expression not reproducible], (4)

provided that the right side is pointwise defined on (0, [infinity]), where m = [[alpha]] + 1 and [[alpha]] represents the integer part of [alpha].

Definition 3 (see [29]). The mapping F : X x X [right arrow] X has a coupledfixedpoint (x, y) [member of] X x X if F(x, y) = x and F(y, x) = y.

Theorem 4 (see [17]). Let S be a nonempty, closed, convex, and bounded subset of the Banach space X and [??] = S x S. Suppose that A: X [right arrow] X and B : S [right arrow] X are two operators such that

([C.sub.1]) there exists [[phi].sub.A] [member of] [PHI] such that, for all x, y [member of] X, one has

[parallel]Ax - Ay[parallel] [less than or equal to] [sigma][[phi].sub.A] ([parallel]x - y[parallel]), for some constant [sigma] > 0, (5)

([C.sub.2]) B is completely continuous,

([C.sub.3]) x = Ax + By [??] x [member of] S, for all y [member of] S.

Then the operator T(x, y) = Ax + By has at least a coupled fixed point in [??] whenever [sigma] < 1.

Lemma 5. The following result holds for fractional differential equations:

[I.sup.[alpha]] [[sup.c][D.sup.[alpha]]h(t)] = h(t) + [c.sub.0] + [c.sub.1] t + [c.sub.2][t.sub.2] + ... + [c.sub.m- 1][t.sup.m-1], (6)

for arbitrary [c.sub.i] [member of] R, i = 0,1,2, ..., m - 1, where m =[[alpha]] + 1 and [[alpha]] represents the integer part of [alpha].

3. Existence Results

Let us set the following notations for convenience:

[delta] = (1-[[delta].sub.1])(1-[[delta].sub.2][[eta].sup.n-1.sub.2]) + (1- [[delta].sub.2])[[delta].sub.1][[eta].sup.n- 1.sub.1], (7)

[mathematical expression not reproducible] (8)

For the forthcoming analysis, we assume that

([A.sub.1]) the function x [right arrow] x - f(t, x) is increasing in R for all t [member of] [0, 1];

([A.sub.2]) there exists M [greater than or equal to] L > 0 such that

[mathematical expression not reproducible] (9)

([A.sub.3]) there exists a continuous function h [member of] C([0, 1], R) such that

g(t,x(t),y(t)) [less than or equal to] h(t), x, y [member of] R, t [member of] [0,1]. (10)

Lemma 6. If f(0,x(0)) = 0 and ([[partial derivative].sup.i] f(t, x(t))/[[partial derivative] [t.sup.i])|.sub.t=0] = 0 for i = 1,2, ..., (n - 2), then integral representation of the system (2) is given by

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

Proof. Applying the operator [I.sup.p] on the first equation of system (2) and using Lemma 5, we obtain

[mathematical expression not reproducible] (13)

Applying the initial conditions [x.sup.(i)](0) = 0, for i = 1,2,3, ..., (n-2), we conclude that [C.sub.1] = [C.sub.2] = ... = [C.sub.n-2] = 0. Therefore (13) becomes

[mathematical expression not reproducible] (14)

Now, to find the values of [C.sub.0] and [C.sub.n-i], since x(0) = [[delta].sub.1] x([[eta].sub.1]), from (14) we have

[mathematical expression not reproducible] (15)

Also, since x(1) = [[delta].sub.2] x([[eta].sub.2]), from (14), we have

[mathematical expression not reproducible] (16)

Solving (15) and (16) for [C.sub.0], we get

[mathematical expression not reproducible] (17)

and using (7), we can write

[mathematical expression not reproducible] (18)

Similarly, solving (15) and (16) and using (7), we obtained that

[mathematical expression not reproducible] (19)

Substituting the values of [C.sub.0] and [C.sub.1] in (14), we can write

[mathematical expression not reproducible] (20)

which implies that

[mathematical expression not reproducible] (21)

Similarly, repeating the above process with the second equation of system (2), we obtain integral equation (12).

Now, we are in a position to present the existence theorem for the system (2).

Theorem 7. Assume that hypotheses ([A.sub.1])-([A.sub.3]) hold. Then there exists a solution for coupled systems (2) of higher order hybrid FHDEs with three-point boundary conditions.

Proof. Set X = C([0, 1], R) and a subset S of X defined by

[mathematical expression not reproducible]. (22)

Clearly S is a closed, convex, and bounded subset of the Banach space X. Now, since x(t) is a solution of the FHDEs system (2) if and only if x(t) satisfies the system of integral equations in Lemma 6, to show the existence solution of system (2) it is enough to show the existence solution of the integral equations in Lemma 6. For this, define two operators A: X [right arrow] X and B: S [right arrow] X by

[mathematical expression not reproducible] (23)

Then the operators form of system (2) is

[mathematical expression not reproducible] (24)

We have to show that the operators A and B satisfy all the conditions of Theorem 4. For this, let x, y [member of] X; then we have

[mathematical expression not reproducible] (25)

Taking the supremum over t and using (7), we get

[mathematical expression not reproducible] (26)

Thus A satisfies condition [(.sub.C1]) of Theorem 4 with [sigma] = 1/2 and [psi](r) = 3Lr/4(M + r) + L[[delta].sub.2]r/4(M + [[delta].sub.2]r).

Next, we show that B is compact and continuous operator on S. Let {[x.sub.m]} be a sequence in S such that {[x.sub.m]} [right arrow] x [member of] S. Then for all t [member of] [0,1], we have

[mathematical expression not reproducible] (27)

Thus the map B is continuous on S.

Let x [member of] S; then we have

[mathematical expression not reproducible] (28)

Taking the supremum over t, we get

[mathematical expression not reproducible]. (29)

Thus is uniformly bounded on S.

Now, let [t.sub.1], [t.sub.2] [member of] [0,1] such that [t.sub.1] [not equal to] [t.sub.2]; then for any x [member of] S, we have

[mathematical expression not reproducible] (30)

Since [t.sup.p] is uniformly continuous on [0, 1] for n-1 < p < n, for any [epsilon] > 0 we can find [[delta].sup.*.sub.1] > 0 such that

[mathematical expression not reproducible] (31)

Let [mathematical expression not reproducible]; then, we have

[mathematical expression not reproducible] (32)

Thus B(S) is equicontinuous and hence B is completely continuous on S.

To prove hypothesis ([C.sub.3]) of Theorem 4, let x [member of] X and y [member of] S such that x = Ax + By; using (7) and (8), we have

[mathematical expression not reproducible] (33)

This implies

[mathematical expression not reproducible] (34)

Taking the supremum over t on [0, 1], we can write

[mathematical expression not reproducible] (35)

That is, x [member of] S. Thus condition ([C.sub.3]) of Theorem 4 holds. Therefore, all the conditions of Theorem 4 are satisfied; hence the operator T(x, y) = Ax+By has a coupled fixed point on [??]. Consequently, system (2) has a solution defined on [0, 1].

To illustrate Theorem 7, we construct the following example.

Example 8. We discuss the following hybrid fractional differential equations with three-point boundary conditions:

[mathematical expression not reproducible] (36)

where t [member of] [0,1], f : [0,1] x R [right arrow] R, f(0, x(0)) = 0, and g :[0,1] x R x R [right arrow] R. [D.sup.5/2] is the Caputo fractional derivative of order 5/2.

Here

[mathematical expression not reproducible], (37)

Therefore,

[mathematical expression not reproducible] (38)

Now, for M = 11 and L = 512/19e < M, we have

[mathematical expression not reproducible] (39)

Similarly,

[mathematical expression not reproducible] (40)

Next,

[mathematical expression not reproducible] (41)

That is, there exists a continuous function h [member of] C([0,1], R) such that

g(t,x(t), y(t)) [less than or equal to] h(t), x,y [member of] R, t [member of] [0,1]. (42)

Finally, since

[mathematical expression not reproducible] (43)

we can write

[mathematical expression not reproducible] (44)

Thus N [greater than or equal to] 70. It follows that assumptions ([A.sub.1]) and ([A.sub.2]) are satisfied. Therefore, by Theorem 7 we conclude that problem (36) has a solution.

4. Conclusion

We have successfully developed appropriate conditions for existence of at least one solution to a complicated higher order coupled system of nonlinear hybrid fractional differential equations. The respective conditions have been derived by using coupled fixed point theorem of Krasnoselskii type. The obtained results were also demonstrated by a suitable example.

https://doi.org/10.1155/2018/5937572

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

All the authors have equal contribution. The first two authors designed the problem and the last two authors read and corrected its style and language and prepared the final version.

Acknowledgments

This work was financially supported by the Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, Thailand.

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Wiyada Kumam (iD), (1) Mian Bahadur Zada (iD), (2) Kamal Shah (iD), (2) and Rahmat Ali Khan (iD) (2)

(1) Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathum Thani 12110, Thailand

(2) Department of Mathematics, University of Malakand, Chakdara, Khyber Pakhtunkhwa, Pakistan