Abstract
In this paper, we develop the theory of fractional hybrid differential equations with
linear perturbations of second type involving RiemannLiouville differential operators
of order
MSC: 34A40, 34A12, 34A99.
Keywords:
fractional differential inequalities; existence theorem; comparison principle1 Introduction
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 [114]. 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 [1519]. The differential equations involving RiemannLiouville differential operators of
fractional order
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 [2329]. Dhage and Lakshmikantham [24] discussed the following firstorder hybrid differential equation with linear perturbations of first type:
where
where
From the above works, we develop the theory of fractional hybrid differential equations
involving RiemannLiouville differential operators of order
2 Fractional hybrid differential equation
Let ℝ be a real line and
Definition 2.1[19]
The RiemannLiouville fractional derivative of order
where
Definition 2.2[19]
The RiemannLiouville fractional integral of order
provided that the righthand side is pointwise defined on
We consider fractional hybrid differential equations (in short FHDE) involving RiemannLiouville
differential operators of order
where
By a solution of FHDE (2.1), we mean a function
(i) the function
(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
In this section, we prove the existence results for FHDE (2.1) on the closed and bounded
interval
We place FHDE (2.1) in the space
We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage [30].
Definition 3.1 Let X be a Banach space. A mapping
for all
Further, if φ satisfies the condition
Lemma 3.1[30]
LetSbe a nonempty, closed convex and bounded subset of the Banach algebraXand let
(a) Ais nonlinear contraction,
(b) Bis completely continuous,
(c)
Then the operator equation
We consider the following hypotheses in what follows.
(A_{0}) The function
(A_{1}) There exist constants
for all
(A_{3}) There exists a continuous function
for all
Lemma 3.2[19]
Let
(H_{1}) The equality
(H_{2}) The equality
holds almost everywhere onJ.
The following lemma is useful in what follows.
Lemma 3.3Assume that hypothesis (A_{0}) holds. Then, for any
and
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 RiemannLiouville
fractional integral
then by (3.2), we get
i.e.,
Thus, (3.3) holds.
Conversely, assume that x satisfies HIE (3.3). Then applying
The map
Now, we are in a position to prove the following existence theorem for FHDE (2.1).
Theorem 3.1Assume that hypotheses (A_{0})(A_{2}) hold. Then FHDE (2.1) has a solution defined onJ.
Proof Set
where
Clearly, S is a closed, convex and bounded subset of the Banach algebra X. Now, using the hypotheses (A_{0})(A_{2}), it can be shown by an application of Lemma 3.3, FHDE (2.1) is equivalent to the nonlinear HIE
Define two operators
and
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.
First, we show that A is a Lipschitz operator on X with the Lipschitz constant L. Let
for all
for all
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
for all
Now, we show that B is a compact operator on S. It is enough to show that
for all
for all
On the other hand, let
Hence, for
for all
Next, we show that hypothesis (c) of Lemma 3.1 is satisfied. Let
Taking supremum over t,
Thus, all the conditions of Lemma 3.1 are satisfied and hence the operator equation
4 Fractional hybrid differential inequalities
We discuss a fundamental result relative to strict inequalities for FHDE (2.1).
Lemma 4.1[17]
Let
Then it follows that
Theorem 4.1Assume that hypothesis (A_{0}) holds. Suppose that there exist functions
and
one of the inequalities being strict. Then
implies
for all
Proof Suppose that inequality (4.4) is strict. Assume that the claim is false. Then there
exists a
Define
Then we have
This is a contradiction to
The next result is concerned with nonstrict fractional differential inequalities which require a kind of onesided φLipshitz condition.
Theorem 4.2Assume that the conditions of Theorem 4.1 hold. Suppose that there exists a real number
for all
for all
Proof
We set
for small
Define
for all
Also, we have
Remark 4.1 Let
5 Existence of maximal and minimal solutions
In this section, we prove the existence of maximal and minimal solutions for FHDE
(2.1) on
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
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
where
An existence theorem for FHDE (5.1) can be stated as follows.
Theorem 5.1Assume that hypotheses (A_{0})(A_{2}) hold. Then, for every small number
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 (A_{0})(A_{2}) hold. Then FHDE (2.1) has a maximal solution defined onJ.
Proof Let
Then, for any solution u of FHDE (2.1), any solution of auxiliary problem (5.2) satisfies
where
for all
Since
exists. We show that the convergence in (5.4) is uniform on J. To finish, it is enough to prove that the sequence
Since f is continuous on a compact set
uniformly for all
Therefore, from the above inequality, it follows that
uniformly for all
for all
Next, we show that the function
for all
for all
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
Theorem 6.1Assume that hypotheses (A_{0})(A_{2}) hold. Suppose that there exists a real number
for all
Then
for all
Proof Let
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
Now, we apply Theorem 4.2 to inequalities (6.1) and (6.4) and conclude that
Theorem 6.2Assume that hypotheses (A_{0})(A_{2}) hold. Suppose that there exists a real number
for all
Then
for all
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.
Theorem 6.3Assume that hypotheses (A_{0})(A_{2}) hold. Suppose that there exists a function
for all
then FHDE (2.1) has a unique solution onJ.
Proof By Theorem 3.1, FHDE (2.1) has a solution defined on J. Suppose that there are two solutions
As
for almost everywhere
Now, we apply Theorem 6.1 with
for all
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
A nonempty closed set K in a Banach algebra X is called a cone with vertex 0 if
(i)
(ii)
(iii)
(iv) A cone K is called positive if
We introduce an order relation ‘≤’ in X as follows. Let
Lemma 7.1[26]
LetKbe a positive cone in a real Banach algebraXand let
For any
Definition 7.1 A mapping
We use the following fixed point theorems of Dhage [27] for proving the existence of extremal solutions for IVP (2.1) under certain monotonicity conditions.
Lemma 7.2[27]
LetKbe a cone in a Banach algebraXand let
(a) Ais a nonlinear contraction,
(b) Bis completely continuous,
(c)
Further, if the coneKis positive and normal, then the operator equation
We equip the space
It is well known that the cone K is positive and normal in
Definition 7.2 A function
We consider the following set of assumptions:
(B_{0})
(B_{1}) FHDE (2.1) has a lower solution a and an upper solution b defined on J with
(B_{2}) The function
(B_{3}) The functions
(B_{4}) There exists a function
We remark that hypothesis (B_{4}) holds in particular if f is continuous and g is
Theorem 7.1Suppose that assumptions (A_{1}) and (B_{0})(B_{4}) 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
for all
for all
for all
Now, we apply Lemma 7.2 to the operator equation
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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).
References

Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York (1993)

Oldham, KB, Spanier, J: The Fractional Calculus, Academic Press, New York (1974)

Podlubny, I: Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York (1999)

Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integral and Derivative, Theory and Applications, Gordon & Breach, New York (1993)

Zhao, Y, Sun, S, Han, Z, Li, Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul.. 16, 2086–2097 (2011). Publisher Full Text

Zhao, Y, Sun, S, Han, Z, Li, Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abstr. Appl. Anal.. 2011, 1–16 (2011)

Zhou, Y, Jiao, F, Li, J: Existence and uniqueness for ptype fractional neutral differential equations. Nonlinear Anal. TMA. 71, 2724–2733 (2009). Publisher Full Text

Zhou, Y, Jiao, F, Li, J: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA. 71, 3249–3256 (2009). Publisher Full Text

Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl.. 11, 4465–4475 (2010). Publisher Full Text

Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl.. 12, 262–272 (2011). Publisher Full Text

Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl.. 59, 1095–1100 (2010). Publisher Full Text

Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of boundary value problem for fractional differential equations. Comput. Math. Appl.. 59, 1363–1375 (2010). Publisher Full Text

Diethelm, K: The Analysis of Fractional Differential Equations, Springer, Berlin (2010)

Sun, S, Zhao, Y, Han, Z, Xu, M: Uniqueness of positive solutions for boundary value problems of singular fractional differential equations. Inverse Probl. Sci. Eng.. 20, 299–309 (2012). Publisher Full Text

Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. TMA. 69, 2677–2682 (2008). Publisher Full Text

Lakshmikantham, V, Vatsala, AS: Theory of fractional differential inequalities and applications. Commun. Appl. Anal.. 11, 395–402 (2007)

Lakshmikantham, V: Theory of fractional functional differential equations. Nonlinear Anal. TMA. 69, 3337–3343 (2008). Publisher Full Text

Lakshmikantham, V, Devi, JV: Theory of fractional differential equations in Banach space. Eur. J. Pure Appl. Math.. 1, 38–45 (2008)

Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

Caputo, M: Linear models of dissipation whose Q is almost independent II. Geophys. J. R. Astron. Soc.. 13, 529–539 (1967). Publisher Full Text

Diethelm, K, Ford, NJ: Analysis of fractional differential equations. J. Math. Anal. Appl.. 265, 229–248 (2002). Publisher Full Text

Diethelm, K, Ford, NJ: Multiorder fractional differential equations and their numerical solution. Appl. Math. Comput.. 154, 621–640 (2004). Publisher Full Text

Dhage, BC: On αcondensing mappings in Banach algebras. Math. Stud.. 63, 146–152 (1994)

Dhage, BC, Lakshmikantham, V: Basic results on hybrid differential equations. Nonlinear Anal., Real World Appl.. 4, 414–424 (2010)

Dhage, BC, Jadhav, NS: Basic results in the theory of hybrid differential equations with linear perturbations of second type (submitted)

Dhage, BC: A nonlinear alternative in Banach algebras with applications to functional differential equations. Nonlinear Funct. Anal. Appl.. 8, 563–575 (2004)

Dhage, BC: Fixed point theorems in ordered Banach algebras and applications. Panam. Math. J.. 9, 93–102 (1999)

Lakshmikantham, V, Leela, S: Differential and Integral Inequalities, Academic Press, New York (1969)

Zhao, Y, Sun, S, Han, Z, Li, Q: Theory of fractional hybrid differential equations. Comput. Math. Appl.. 62, 1312–1324 (2011). Publisher Full Text

Dhage, BC: A fixed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J.. 44, 145–155 (2004)

Heikkilä, S, Lakshmikantham, V: Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Dekker, New York (1994)