- Open Access
Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions
© Zhao and Gong; licensee Springer. 2014
Received: 20 June 2014
Accepted: 9 September 2014
Published: 29 September 2014
By constructing Green’s function, we give the natural formulae of solutions forthe following nonlinear impulsive fractional differential equation with generalizedperiodic boundary value conditions:
where is a real number, is the standard Caputo differentiation. We present theproperties of Green’s function. Some sufficient conditions for the existence ofsingle or multiple positive solutions of the above nonlinear fractional differentialequation are established. Our analysis relies on a nonlinear alternative of theSchauder and Guo-Krasnosel’skii fixed point theorem on cones. As applications,some interesting examples are provided to illustrate the main results.
MSC: 34B10, 34B15, 34B37.
In recent years, the fractional order differential equation has aroused great attentiondue to both the further development of fractional order calculus theory and theimportant applications for the theory of fractional order calculus in the fields ofscience and engineering such as physics, chemistry, aerodynamics, electrodynamics ofcomplex medium, polymer rheology, Bode’s analysis of feedback amplifiers,capacitor theory, electrical circuits, electron-analytical chemistry, biology, controltheory, fitting of experimental data, and so forth. Many papers and books on fractionalcalculus differential equation have appeared recently. One can see [1–17] and the references therein.
In order to describe the dynamics of populations subject to abrupt changes as well asother phenomena such as harvesting, diseases, and so on, some authors have used animpulsive differential system to describe these kinds of phenomena since the lastcentury. For the basic theory on impulsive differential equations, the reader can referto the monographs of Bainov and Simeonov , Lakshmikantham et al. and Benchohra et al..
where a, b are real constants with . is the Caputo fractional derivative of order. is jointly continuous. , . The impulsive point set satisfies . and represent the right and left limits of at the impulsive point . Let us set , , . The goal of this paper is to study the existence ofsingle or multiple positive solutions for the impulsive BVPs (1.1) by a nonlinearalternative of the Schauder and Guo-Krasnosel’skii fixed point theorem oncones.
The rest of the paper is organized as follows. In Section 2, we present some usefuldefinitions, lemmas and the properties of Green’s function. In Section 3, wegive some sufficient conditions for the existence of a single positive solution for BVPs(1.1). In Section 4, some sufficient criteria for the existence of multiplepositive solutions for BVPs (1.1) are obtained. Finally, some examples are provided toillustrate our main results in Section 5.
For the convenience of the reader, we present here the necessary definitions fromfractional calculus theory. These definitions and properties can be found in theliterature.
provided that the right-hand side is pointwise defined on .
where , provided that the right-hand side is pointwise defined on.
Lemma 2.1 (see )
for some, () anddenotes the integer part of the real number q.
Lemma 2.2 (see )
Let E be a Banach space. Assumethatis a completely continuous operator and thesetis bounded. Then T has a fixedpoint in E.
Lemma 2.3 (Schauder fixed point theorem, see )
If U is a close bounded convex subset of a Banachspace E andis completely continuous,then T has at least one fixed point in U.
Lemma 2.4 (see )
, and, , or
hold. Then A has at least one fixed pointin.
Now we present Green’s function for a system associated with BVPs (1.1).
where , and are defined by (2.2)-(2.4).
where , and are defined by (2.2)-(2.4). The proof iscomplete. □
, , andfor all, where.
- (ii)The functions, andhave the following properties:(2.16)
The proof is completed. □
3 Existence of single positive solutions
In this section, we discuss the existence of positive solutions for BVP (1.1).
Obviously, is a Banach space with the norm . is a positive cone.
In the following, we need the assumptions and some notations as follows:
(B1) , , where , .
(B2) and for all .
(B3) , .
From Lemma 2.4, we can obtain the following lemma.
Then, by Lemma 3.1, the existence of solutions for BVPs (1.1) is translated intothe existence of the fixed point for , where T is given by (3.3). Thus, the fixed pointof the operator T coincides with the solution of problem (1.1).
Lemma 3.2 Assume that (B1)-(B3) hold,thenanddefined by (3.3) are completelycontinuous.
Proof Firstly, we shall show that is completely continuous through three steps.
Step 1. Let , in view of the nonnegativity and continuity of functions, , , , , and , we conclude that is continuous.
which imply that .
which means that is equicontinuous on all the subintervals, . Thus, by means of the Arzela-Ascoli theorem, we have that is completely continuous.
So for every , which implies . Similar to the above arguments, we can easily concludethat is a completely continuous operator. The proof iscomplete. □
Theorem 3.1 Assume that (B1)-(B3) hold,and suppose that the following assumptions hold:
(A1) There exists a constantsuch thatfor eachand all.
(A2) There exists a constantsuch thatfor all, .
(A3) There exists a constantsuch thatfor all, .
If, then problem (1.1) has a unique solutionin.
where . Consequently, T is a contraction mapping.Moreover, from Lemma 3.2, T is completely continuous. Therefore, by theBanach contraction map principle, the operator T has a unique fixed point in which is the unique positive solution of system (1.1).This completes the proof. □
Theorem 3.2 Assume that (B1)-(B3) hold,and suppose that the following assumptions hold:
(A4) There exists a constantsuch thatfor eachand all.
(A5) There exists a constantsuch thatfor all, .
(A6) There exists a constantsuch thatfor all, .
Then BVPs (1.1) have at least one positive solutionin.
Thus, for every , we have , which indicates that the set Ω is bounded. Accordingto Lemma 2.2, T has a fixed point . Therefore, BVPs (1.1) have at least one positivesolution. The proof is complete. □
In the following, we present an existence result when the nonlinearity and the impulsefunctions have sublinear growth.
Theorem 3.3 Assume that (B1)-(B3) hold andsuppose that the following assumptions hold:
(A7) There exist, andsuch thatfor eachand all.
(A8) There exist constantsandsuch thatfor all, .
(A9) There exist constantsandsuch thatfor all, .
(A10) , , where, .
Then BVPs (1.1) have at least one positive solutionin.
which imply that . When , then . When , then . Taking , we have for any solution of (3.4). This shows that the set Ωis bounded. According to Lemma 2.2, T has at least one fixed point in. Therefore, BVPs (1.1) have at least one positive solutionin . The proof is complete. □
Theorem 3.4 Assume that (B1)-(B3) hold.And suppose that one of the following conditions is satisfied:
(H1) (particularly, ).
(H2) There exists a constantsuch thatfor, .
(H3) There exists a constantsuch thatfor, .
Then BVPs (1.1) have at least one positive solution.
Proof Case 1. Considering , there exists such that for all , , where satisfies .
Therefore, . From Lemma 3.2, we have that is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3.
Therefore, . From Lemma 3.2 we have that is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3.
Therefore, . From Lemma 3.2 we have is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3. We complete the proof ofTheorem 3.4. □
4 Existence of multiple positive solutions
In this section, we discuss the multiplicity of positive solutions for BVPs (1.1) by theGuo-Krasnoselskii fixed point theorem.
Theorem 4.1 Assume that (B1)-(B3) hold,and suppose that the following two conditions are satisfied:
(H4) and (particularly, ).
(H5) There exists a constantsuch thatfor, .
Thus, applying Lemma 2.4 to (4.2)-(4.4) yields that T has the fixed point and the fixed point . Thus it follows that problem (1.1) has at least twopositive solutions and . Noticing (4.4), we have and . Therefore (4.1) holds. The proof iscomplete. □
Theorem 4.2 Assume that (B1)-(B3) hold.Further suppose that there exist three positivenumbers () withsuch that one of the following conditions issatisfied:
(H6) , , .
(H7) , , .
Thus, applying Lemma 2.4 to (4.6)-(4.8) yields that T has the fixed point and the fixed point . Thus it follows that BVPs (1.1) have at least twopositive solutions and . Noticing (4.7), we have and . Therefore (4.5) holds. The proof iscomplete. □
Similar to the above proof, we can obtain the general theorem.
Theorem 4.3 Assume that (B1)-(B3) hold.Suppose that there existpositive numbers () withsuch that one of the following conditions issatisfied:
(H8) , , ;
(H9) , , .
5 Illustrative examples
If we let , , , , , , , .
Thus, all the assumptions of Theorem 3.1 are satisfied. Hence, BVPs (5.1) have aunique solution on .
In addition, in this case, let , . It is clear that , , . Thus, BVPs (5.1) have at least one solution on by Theorem 3.2.
Thus it follows that BVPs (5.2) have at least two positive solutions, with by Theorem 4.1.
The authors thank the referees for their valuable comments and suggestions for theimprovement of the manuscript. This work is supported by the National NaturalSciences Foundation of Peoples Republic of China under Grant No. 11161025 and YunnanProvince Natural Scientific Research Fund Project under Grant No. 2011FZ058.
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