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 . An existence theorem for fractional hybrid differential equations is proved under the φLipschitz 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 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 are very important in modeling several physical phenomena [2022] and therefore seem to deserve an independent study of their theory parallel to the wellknown 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 [2329]. Dhage and Lakshmikantham [24] discussed the following firstorder hybrid differential equation with linear perturbations of first type:
where and . Dhage and Jadhav [25] discussed the following firstorder 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 RiemannLiouville 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[19]
The RiemannLiouville fractional derivative of order of a continuous function is given by
where , denotes the integer part of number α, provided that the righthand side is pointwise defined on .
Definition 2.2[19]
The RiemannLiouville fractional integral of order of a function is given by
provided that the righthand side is pointwise defined on .
We consider fractional hybrid differential equations (in short FHDE) involving RiemannLiouville differential operators of order ,
By a solution of FHDE (2.1), we mean a function such that
(i) the function is continuous for each , and
(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 under mixed Lipschitz and compactness conditions on the nonlinearities involved in it.
We place FHDE (2.1) in the space of continuous realvalued 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 [30].
Definition 3.1 Let X be a Banach space. A mapping is called φLipschitzian if there exists a continuous and nondecreasing function such that
Further, if φ satisfies the condition , , then T is called a nonlinear contraction with a control function φ.
Lemma 3.1[30]
LetSbe a nonempty, closed convex and bounded subset of the Banach algebraXand letandbe two operators such that
(a) Ais nonlinear contraction,
(b) Bis completely continuous,
Then the operator equationhas a solution inS.
We consider the following hypotheses in what follows.
(A_{0}) The function is increasing in ℝ for all .
(A_{1}) There exist constants such that
(A_{3}) There exists a continuous function such that
Lemma 3.2[19]
(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 anyand, the functionis a solution of the FHDE
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 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
i.e.,
Thus, (3.3) holds.
Conversely, assume that x satisfies HIE (3.3). Then applying on both sides of (3.3), (3.1) is satisfied. Again, substituting in (3.3) yields
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 3.1Assume that hypotheses (A_{0})(A_{2}) hold. Then FHDE (2.1) has a solution defined onJ.
Proof Set and define a subset S of X defined by
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
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 . Then by hypothesis (A_{1}),
for all . Taking supremum over t, we obtain
for all . This shows that A is a nonlinear contraction on X with a control function φ defined by .
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 . This shows that B is a continuous operator on S.
Now, we show that B is a compact operator on S. It is enough to show that is a uniformly bounded and equicontinuous set in X. On the one hand, let be arbitrary. Then by hypothesis (A_{2}),
for all . Taking supremum over t,
for all . This shows that B is uniformly bounded on S.
On the other hand, let with . Then, for any , one has
Hence, for , there exists a such that
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 ArzelaAscoli theorem. As a result, B is a complete continuous operator on S.
Next, we show that hypothesis (c) of Lemma 3.1 is satisfied. Let . Then, by assumption (A_{1}), we have
Taking supremum over t,
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[17]
Letbe locally Hölder continuous such that for any, we have
Then it follows that
Theorem 4.1Assume that hypothesis (A_{0}) holds. Suppose that there exist functionsthat are locally Hölder continuous such that
and
one of the inequalities being strict. Then
implies
Proof Suppose that inequality (4.4) is strict. Assume that the claim is false. Then there exists a , , such that and for .
Define
Then we have and by virtue of hypothesis (A_{0}), we get for all . Setting , , we find that , and . Then by Lemma 4.1, we have . By (4.3) and (4.4), we obtain
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 onesided φLipshitz condition.
Theorem 4.2Assume that the conditions of Theorem 4.1 hold. Suppose that there exists a real numbersuch that
for allwith. Thenimplies, provided,
Proof
We set
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.
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.
Theorem 5.1Assume that hypotheses (A_{0})(A_{2}) hold. Then, for every small number, FHDE (5.1) has a solution defined onJ.
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 be a decreasing sequence of positive real numbers such that . By Theorem 5.1, then there exists a solution of the FHDE defined on J
Then, for any solution u of FHDE (2.1), any solution of auxiliary problem (5.2) satisfies
where . By Theorem 4.2, we infer that
Since , then by Theorem 4.2, we infer that . Therefore, is a decreasing sequence of positive real numbers, the limit
exists. We show that the convergence in (5.4) is uniform on J. To finish, it is enough to prove that the sequence is equicontinuous in . Let with be arbitrary. Then
Since f is continuous on a compact set , it is uniformly continuous there. Hence,
Therefore, from the above inequality, it follows that
uniformly for all . Therefore,
Next, we show that the function is a solution of FHDE (2.1) defined on J. Now, since is a solution of FHDE (5.2), we have
for all . Taking the limit as in above Eq. (5.5) yields
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 .
Theorem 6.1Assume that hypotheses (A_{0})(A_{2}) hold. Suppose that there exists a real numbersuch that
for allwith, where. Furthermore, if there exists a functionsuch that
Then
for all, whereris a maximal solution of FHDE (2.1) onJ.
Proof Let be arbitrary small. By Theorem 5.2, is a maximal solution of FHDE (5.1) and the limit
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 for all . This further, in view of limit (6.3), implies that inequality (6.2) holds on J. This completes the proof. □
Theorem 6.2Assume that hypotheses (A_{0})(A_{2}) hold. Suppose that there exists a real numbersuch that
for allwith, where. Furthermore, if there exists a functionsuch that
Then
for all, whereρis a minimal solution of FHDE (2.1) onJ.
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 functionsuch that
for all. If an identically zero function is the only solution of the differential equation
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 and of FHDE (2.1) existing on J. Define a function by
Now, we apply Theorem 6.1 with to get that for all . This gives
for all . Then we can get in view of hypothesis (A_{0}). 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 [26] 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
(iii) , where 0 is the zero element of X,
(iv) 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 semimonotone 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 orderbounded set in X is normbounded. The details of cones and their properties appear in Heikkilä and Lakshmikantham [31].
Lemma 7.1[26]
LetKbe a positive cone in a real Banach algebraXand letbe such thatand. Then.
For any , the order interval is a set in X given by
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 [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 letbe such that. Suppose thatare two nondecreasing operators such that
(a) Ais a nonlinear contraction,
(b) Bis completely continuous,
Further, if the coneKis positive and normal, then the operator equationhas a least and a greatest positive solution in.
We equip the space with the order relation ≤ with the help of a cone K defined by
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:
(B_{1}) FHDE (2.1) has a lower solution a and an upper solution b defined on J with .
(B_{2}) The function is increasing in the interval almost everywhere for .
(B_{3}) The functions and are nondecreasing in x almost everywhere for .
(B_{4}) There exists a function such that .
We remark that hypothesis (B_{4}) holds in particular if f is continuous and g is Carathéodory on .
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 . 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 (B_{0}) 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 (B_{3}) implies that A and B are nondecreasing on . To see this, let be such that . Then, by hypothesis (B_{3}),
for all . So, A and B are nondecreasing operators on . Lemma 7.1 and hypothesis (B_{3}) together imply that
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. □
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)