In this paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order . An existence theorem for fractional hybrid differential equations is proved under the φ-Lipschitz condition. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle which will be useful for further study of qualitative behavior of solutions is proved.
MSC: 34A40, 34A12, 34A99.
Keywords:fractional differential inequalities; existence theorem; comparison principle
Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [1-14]. Although the tools of fractional calculus have been available and applicable to various fields of study, there are few papers on the investigation of the theory of fractional differential equations; see [15-19]. The differential equations involving Riemann-Liouville differential operators of fractional order are very important in modeling several physical phenomena [20-22] and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations.
In recent years, quadratic perturbations of nonlinear differential equations have attracted much attention. The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. This class of hybrid differential equations includes the perturbations of original differential equations in different ways. There have been many works on the theory of hybrid differential equations, and we refer the readers to the articles [23-29]. Dhage and Lakshmikantham  discussed the following first-order hybrid differential equation with linear perturbations of first type:
where and . Dhage and Jadhav  discussed the following first-order hybrid differential equation with linear perturbations of second type:
where and . They established the existence and uniqueness results and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proved utilizing the theory of inequalities, its existence of extremal solutions and a comparison result.
From the above works, we develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order . In this paper, we initiate the basic theory of fractional hybrid differential equations of mixed perturbations of second type involving three nonlinearities and prove the basic result such as the strict and nonstrict fractional differential inequalities, an existence theorem and maximal and minimal solutions etc. We claim that the results of this paper are a basic and important contribution to the theory of nonlinear fractional differential equations.
2 Fractional hybrid differential equation
(ii) x satisfies the equations in (2.1).
The theory of strict and nonstrict differential inequalities related to ODEs and hybrid differential equations is available in the literature (see [24,25,28,29]). It is known that differential inequalities are useful for proving the existence of extremal solutions of ODEs and hybrid differential equations defined on J.
3 Existence result
We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage .
(a) Ais nonlinear contraction,
(b) Bis completely continuous,
We consider the following hypotheses in what follows.
(H2) The equality
holds almost everywhere onJ.
The following lemma is useful in what follows.
if and only ifxsatisfies the hybrid integral equation (HIE)
Proof Let x be a solution of the Cauchy problem (3.1) and (3.2). Since the Riemann-Liouville fractional integral is a monotone operator, thus we apply the fractional integral on both sides of (3.1). By Lemma 3.2, we have
then by (3.2), we get
Thus, (3.3) holds.
Now, we are in a position to prove the following existence theorem for FHDE (2.1).
Theorem 3.1Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a solution defined onJ.
Clearly, S is a closed, convex and bounded subset of the Banach algebra X. Now, using the hypotheses (A0)-(A2), it can be shown by an application of Lemma 3.3, FHDE (2.1) is equivalent to the nonlinear HIE
Then HIE (3.5) is transformed into the operator equation as
We will show that the operators A and B satisfy all the conditions of Lemma 3.1.
Next, we show that B is a compact and continuous operator on S into X. First, we show that B is continuous on S. Let be a sequence in S converging to a point . Then, by the Lebesgue dominated convergence theorem,
for all and for all . This shows that is an equicontinuous set in X. Now, the set is a uniformly bounded and equicontinuous set in X, so it is compact by the Arzela-Ascoli theorem. As a result, B is a complete continuous operator on S.
Taking supremum over t,
4 Fractional hybrid differential inequalities
We discuss a fundamental result relative to strict inequalities for FHDE (2.1).
Then it follows that
one of the inequalities being strict. Then
The next result is concerned with nonstrict fractional differential inequalities which require a kind of one-sided φ-Lipshitz condition.
5 Existence of maximal and minimal solutions
Definition 5.1 A solution r of FHDE (2.1) is said to be maximal if for any other solution x to FHDE (2.1), one has for all . Similarly, a solution ρ of FHDE (2.1) is said to be minimal if for all , where x is any solution of FHDE (2.1) on J.
We discuss the case of a maximal solution only, as the case of a minimal solution is similar and can be obtained with the same arguments with appropriate modifications. Given an arbitrary small real number , consider the following initial value problem of FHDE of order ,
An existence theorem for FHDE (5.1) can be stated as follows.
Proof The proof is similar to Theorem 3.1 and we omit the details. □
Our main existence theorem for a maximal solution for FHDE (2.1) is as follows.
Theorem 5.2Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a maximal solution defined onJ.
Then, for any solution u of FHDE (2.1), any solution of auxiliary problem (5.2) satisfies
Therefore, from the above inequality, it follows that
6 Comparison theorems
The main problem of differential inequalities is to estimate a bound for the solution set for the differential inequality related to FHDE (2.1). In this section, we prove that the maximal and minimal solutions serve as bounds for the solutions of the related differential inequality to FHDE (2.1) on .
is uniform on J and the function r is a maximal solution of FHDE (2.1) on J. Hence, we obtain
From the above inequality, it follows that
Note that Theorem 6.1 is useful to prove the boundedness and uniqueness of the solutions for FHDE (2.1) on J. A result in this direction is as follows.
then FHDE (2.1) has a unique solution onJ.
7 Existence of extremal solutions in a vector segment
Sometimes it is desirable to have knowledge of the existence of extremal positive solutions for FHDE (2.1) on J. In this section, we prove the existence of maximal and minimal positive solutions for FHDE (2.1) between the given upper and lower solutions on . We use a hybrid fixed point theorem of Dhage  in ordered Banach spaces for establishing our results. We need the following preliminaries in what follows.
A nonempty closed set K in a Banach algebra X is called a cone with vertex 0 if
We introduce an order relation ‘≤’ in X as follows. Let . Then if and only if . A cone K is called normal if the norm is semi-monotone increasing on K, that is, there is a constant such that for all with . It is known that if the cone K is normal in X, then every order-bounded set in X is norm-bounded. The details of cones and their properties appear in Heikkilä and Lakshmikantham .
We use the following fixed point theorems of Dhage  for proving the existence of extremal solutions for IVP (2.1) under certain monotonicity conditions.
(a) Ais a nonlinear contraction,
(b) Bis completely continuous,
Definition 7.2 A function is called a lower solution of FHDE (2.1) defined on J if it satisfies (4.3). Similarly, a function is called an upper solution of FHDE (2.1) defined on J if it satisfies (4.4). A solution to FHDE (2.1) is a lower as well as an upper solution for FHDE (2.1) defined on J and vice versa.
We consider the following set of assumptions:
Theorem 7.1Suppose that assumptions (A1) and (B0)-(B4) hold. Then FHDE (2.1) has a minimal and a maximal positive solution defined onJ.
Proof Now, FHDE (2.1) is equivalent to integral equation (3.5) defined on J. Let . Define two operators A and B on X by (3.6) and (3.7) respectively. Then the integral equation (3.5) is transformed into an operator equation in the Banach algebra X. Notice that hypothesis (B0) implies . Since the cone K in X is normal, is a norm bounded set in X. Now it is shown, as in the proof of Theorem 3.1, that A is a Lipschitzian with the Lipschitz constant L and B is a completely continuous operator on . Again, hypothesis (B3) implies that A and B are nondecreasing on . To see this, let be such that . Then, by hypothesis (B3),
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Dedicated to Professor Hari M Srivastava.
This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
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