Theory of fractional hybrid differential equations with linear perturbations of second type
© Lu et al.; licensee Springer. 2013
Received: 30 November 2012
Accepted: 22 January 2013
Published: 11 February 2013
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.
Keywordsfractional 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.
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
Let ℝ be a real line and be a bounded interval in ℝ for some with . Let denote the class of continuous functions .
Definition 2.1 
where , denotes the integer part of number α, provided that the right-hand side is pointwise defined on .
Definition 2.2 
provided that the right-hand side is pointwise defined on .
the function is continuous for each , and
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
In this section, we prove the existence results for FHDE (2.1) on the closed and bounded interval under mixed Lipschitz and compactness conditions on the nonlinearities involved in it.
We place FHDE (2.1) in the space of continuous real-valued functions defined on J. Define a supremum norm in by . Clearly, is a Banach algebra with respect to the above norm.
We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage .
for all , where .
Further, if φ satisfies the condition , , then T is called a nonlinear contraction with a control function φ.
Lemma 3.1 
A is nonlinear contraction,
B is completely continuous,
for all .
Then the operator equation has a solution in S.
We consider the following hypotheses in what follows.
(A0) The function is increasing in ℝ for all .
for all and .
for all .
Lemma 3.2 
Let and .
(H1) The equality holds.
holds almost everywhere on J.
The following lemma is useful in what follows.
Thus, (3.3) holds.
The map is increasing in ℝ for all , the map is injective in ℝ, hence . The proof is completed. □
Now, we are in a position to prove the following existence theorem for FHDE (2.1).
Theorem 3.1 Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a solution defined on J.
where and .
We will show that the operators A and B satisfy all the conditions of Lemma 3.1.
for all . This shows that A is a nonlinear contraction on X with a control function φ defined by .
for all . This shows that B is a continuous operator on S.
for all . This shows that B is uniformly bounded on S.
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.
Thus, all the conditions of Lemma 3.1 are satisfied and hence the operator equation has a solution in S. As a result, FHDE (2.1) has a solution defined on J. This completes the proof. □
4 Fractional hybrid differential inequalities
We discuss a fundamental result relative to strict inequalities for FHDE (2.1).
Lemma 4.1 
for all .
Proof Suppose that inequality (4.4) is strict. Assume that the claim is false. Then there exists a , , such that and for .
This is a contradiction to . Hence, the conclusion (4.6) is valid and the proof is complete. □
The next result is concerned with nonstrict fractional differential inequalities which require a kind of one-sided φ-Lipshitz condition.
for all .
Also, we have . Hence, by an application of Theorem 4.1 with yields that for all . By the arbitrariness of , taking the limits as , we have for all . This completes the proof. □
Remark 4.1 Let and . We can easily verify that f and g satisfy the condition (4.7).
5 Existence of maximal and minimal solutions
In this section, we prove the existence of maximal and minimal solutions for FHDE (2.1) on . We need the following definition in what follows.
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.
An existence theorem for FHDE (5.1) can be stated as follows.
Theorem 5.1 Assume that hypotheses (A0)-(A2) hold. Then, for every small number , FHDE (5.1) has a solution defined on J.
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 5.2 Assume that hypotheses (A0)-(A2) hold. Then FHDE (2.1) has a maximal solution defined on J.
for all and .
uniformly for all .
for all .
for all . Thus, the function r is a solution of FHDE (2.1) on J. Finally, from inequality (5.3), it follows that for all . Hence, FHDE (2.1) has a maximal solution on J. This completes the proof. □
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 .
for all , where r is a maximal solution of FHDE (2.1) on J.
Now, we apply Theorem 4.2 to inequalities (6.1) and (6.4) and conclude that for all . This further, in view of limit (6.3), implies that inequality (6.2) holds on J. This completes the proof. □
for all , where ρ is a minimal solution of FHDE (2.1) on J.
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 on J.
for almost everywhere , and .
for all . Then we can get in view of hypothesis (A0). This completes the proof. □
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.
for , ,
, where 0 is the zero element of X,
A cone K is called positive if , where ∘ is a multiplication composition in X.
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 .
Lemma 7.1 
Let K be a positive cone in a real Banach algebra X and let be such that and . Then .
Definition 7.1 A mapping is said to be nondecreasing or monotone increasing if implies for all .
We use the following fixed point theorems of Dhage  for proving the existence of extremal solutions for IVP (2.1) under certain monotonicity conditions.
Lemma 7.2 
A is a nonlinear contraction,
B is completely continuous,
for each .
Further, if the cone K is positive and normal, then the operator equation has a least and a greatest positive solution in .
It is well known that the cone K is positive and normal in . We need the following definitions in what follows.
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:
(B0) , .
(B1) FHDE (2.1) has a lower solution a and an upper solution b defined on J with .
(B2) The function is increasing in the interval almost everywhere for .
(B3) The functions and are nondecreasing in x almost everywhere for .
(B4) There exists a function such that .
We remark that hypothesis (B4) holds in particular if f is continuous and g is -Carathéodory on .
Theorem 7.1 Suppose that assumptions (A1) and (B0)-(B4) hold. Then FHDE (2.1) has a minimal and a maximal positive solution defined on J.
for all and . As a result, for all and . Hence, for all .
Now, we apply Lemma 7.2 to the operator equation to yield that FHDE (2.1) has a minimal and a maximal positive solution in defined on J. This completes the proof. □
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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