Research Article | Open Access
Multiple Positive Solutions for Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions and a Parameter
By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.
In recent years, the interest in the study of fractional differential equations has been growing rapidly since it has many applications in biology, mechanics, electrochemistry, and dynamical processes in self-similar structures, etc.; see [1–3], for instance. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. Fractional-order models have proved to be more accurate than integer order models and have more applications. In consequence, many meaningful results in these fields have been obtained. For details, see [4–14] and references therein.
In addition, several new types of fractional derivatives are also investigated in some excellent papers. In , Yang studied a class of fractional derivatives of constant and variable orders for the first time. In , the authors introduced a new fractional derivative without singular kernel, and this was an extension of the Riemann-Liouville fractional derivative with singular kernel and had some important applications in the modeling of the fractional-order heat flow.
Meanwhile, there are a lot of papers (see [3, 17–28], for instance) dealing with the existence of positive solutions of nonlinear fractional differential equations by use of fixed-point theorems, monotone iterative technique, and upper and lower solution method. For example, in , by using the method of upper and lower solutions and Schauder fixed theorem, Vong investigated the positive solutions for the following nonlocal fractional BVP:where , and is a function of bounded variation. may be singular at
In , The authors studied a class of higher order fractional BVP. In view of monotone iterative method, they got an existence and uniqueness result of positive solution.
Goodrich  dealt with a problem similar to the system in  but with local conditions. By deriving properties of the Green’s function and by means of Guo-Krasnoselskii’s fixed-point theorem, the author established some existence results of at least one positive solution provided that satisfies some growth conditions.
However, compared with the methods of fixed-point theorems, Leray-Schauder theory, and monotone iterative technique, there is little work investigating positive solution of fractional integral boundary value problem by use of the differentiable operator method. On the other hand, the results about such system with a parameter are relatively scarce.
Motivated by all the above-mentioned works, we aim to establish some existence criteria of multiple positive solutions for the following nonlinear fractional BVP with integral boundary conditions and a parameter: where , , , , is the Caputo fractional derivative, and is continuous. It is worth mentioning that there is no paper studying the multiple positive solutions for the fractional differential systems by using the differentiable operator method. Besides, to the best of our knowledge, no contribution exists investigating the existence of positive solutions for BVP (2). It should be noted that is more general than those in [23, 25, 26], in which equals one.
Contributions of this article are as follows. Firstly, we derive the corresponding Green’s function for fractional BVP (2) and argue its important properties which will play an important role in our proof. Also, this is the key to establish an appropriate cone. Secondly, by using the differential operator method, spectrum theory, and the properties of Green’s function, we firstly investigate some existence results of multiple positive solutions for the considered fractional system. The methods used in the present paper are different from all the above-mentioned works.
This paper is organized in the following way. In Section 2, we give some necessary preliminaries and lemmas. Section 3 establishes sufficient conditions for the existence of multiple positive solutions to system (2). In Section 4, some examples are given to highlight the value of the new contributions.
2. Preliminaries and Some Lemmas
Definition 1. The Riemann-Liouville fractional integral of order of a continuous function is given byprovided that the right side integral is pointwise defined on .
Definition 2. The Caputo fractional derivative of order of a continuous function is given byprovided that the right side integral is pointwise defined on
Lemma 3. Let Then for some
Let be endowed with the norm for each Clearly, is a Banach space.
Lemma 4. Suppose function . Then the unique solution of the following systemwhere , , , , is given bywhere is the Green’s function of system (6) given by
Proof. From Lemma 3, it follows that where .
So, one has From , we get So we have,By direct integration to (13), it is easy to getThrough the conditions , (12), and (14), one hasThen, we obtainTherefore, we can get the following solution of (6) by (13),In the following, we divide the proof into two cases.
Case 1 Case 2 The conclusion of this lemma follows
Next, we give some important properties about Green’s function belonging to system (6).
Lemma 5. The Green’s function of system (6) satisfies the following inequalities:
Proof. We denote and take .
For , one hasandThis means that is concave on . So we get
For , we have and These three facts can also indicate
For , we have , andSo, we can get that
For , we obtain , andHence, we also have
Consequently, one gets the validity of and . The establishment of and is obvious.
Lemma 6 (see ). Let be a Banach space, be a cone in , and be a bounded open subset in . Suppose that is a completely continuous operator. Then the following results hold:
(i) if there exists such that , for any , then ;
(ii) if , , for any , then
Lemma 7 (see ). Let be a cone in a Banach space , be completely continuous, and . Suppose that is differentiable at along and 1 is not an eigenvalue of corresponding to a positive eigenvector. Moreover, if has no positive eigenvectors corresponding to an eigenvalue greater than one, then there exists such that , for , where
Lemma 8 (see ). Let be a cone in a Banach space and be completely continuous. Suppose that is differentiable at along and 1 is not an eigenvalue of corresponding to a positive eigenvector. Moreover, if has no positive eigenvectors corresponding to an eigenvalue greater than one, then there exists such that , for , where
Define a cone by where Let
Define integral operators and by
Lemma 9 (see ). is completely continuous and the spectral radius
By a similar process in the proof of Lemma 9, we have the following lemma.
Lemma 10. Let , and then operator maps cone into and is completely continuous.
3. Main Results
For convenience in the following discussion, we present the following notations and assumptions:(H1), and , (H2)(H3)There exists , such that uniformly holds with respect to on , and .
Adopting regular approaches similar to that in , we give the following two lemmas.
Lemma 11. Suppose that (H1) and (H2) hold. Thus is differentiable at along , and In addition, has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.
Proof. Obviously, by . From mean value theorem, it follows thatfor some . Notice , and we can obtain that for any , there exists a such that andSo for any , noticing (31) and Lemma 5, we getwhich means thatNow, we are ready to show that has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one. Otherwise, there exist and such that Then Notice , and it followsBy direct integration to (35) on , we obtainNotice , from which we infer thatwhich contradicts (H2). The proof is thus completed.
Lemma 12. Suppose that (H3) holds. Thus is differentiable at along and In addition, has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.
Proof. From (H3), it follows thatwhere , and is a constant corresponding to
Thus, for any , it has which implies that when Therefore,
Suppose that has positive eigenvectors corresponding to an eigenvalue greater than or equal to one. Then we can choose a function and a constant satisfying , and thusBy means of , one hasIntegrating (43) with respect to on givesFrom the condition , it follows thatNotice (H3); this is a contradict. Hence, Lemma 12 holds.
The following theorems are main results in this paper.
Theorem 13. Suppose that (H1), (H2), and hold. Then BVP (2) has at least one positive solution.
Proof. From Lemmas 9, 11, and 7, it follows thatwhere is a suitable constant.
LetStandard arguments can show that is completely continuous and Let , and The proof of follows by arguing as in , so we omit the proof. Hence, there exist and such that ; that is According to Krein-Rutman theorem, we can obtain some with such thatThen, for all , one has Thus, , which means
Moreover, using , we can choose a constant such thatChoose Thus for any , it followsIf we can choose a function and a constant such thatthen we know
DenoteObviously, and Noticing (48), (49), (51), and (53), we obtain This is in contradiction with the definition of Then we have Using Lemma 6 yields that This together with (46) gives Therefore, system (2) has at least one positive solution.
Theorem 14. Suppose that (H3) and hold. Then BVP (2) has at least one positive solution.
Proof. Since , we know that there exists a positive constant such thatBy a similar way in Theorem 13, there exists a function such thatNext, we are ready to showIn fact, if not, then there exist and such thatwhich indicates
Denote Obviously, and , for any This together with (58), (59), and (61) gives