This paper investigates the existence and uniqueness of nontrivial solutions to a class of fractional nonlocal multi-point boundary value problems of higher order fractional differential equation, this kind of problems arise from viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system. Some sufficient conditions for the existence and uniqueness of nontrivial solutions are established under certain suitable growth conditions, our proof is based on Leray-Schauder nonlinear alternative and Schauder fixed point theorem.
MSC: 34B15, 34B25.
Keywords:fractional differential equation; nontrivial solution; Green function; Leray-Schauder nonlinear alternative
The purpose of this paper is to establish the existence and uniqueness of nontrivial solutions to the following higher fractional differential equation:
where , , , , for , and , , , , , , is the standard Riemann-Liouville derivative, and is continuous.
Differential equations of fractional order occur more frequently in different research areas such as engineering, physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic and signal processing of wireless communication system, etc. [1-6].
For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas et al. , Miller and Ross , Podlubny , the papers [10-24] and the references therein.
Recently, Salem  has investigated the existence of Pseudo solutions for the nonlinear m-point boundary value problem of a fractional type. In particular, he considered the following boundary value problem:
where x takes values in a reflexive Banach space E and with . denotes the kth Pseudo-derivative of x and denotes the Pseudo fractional differential operator of order α. By means of the fixed point theorem attributed to O’Regan, a criterion was established for the existence of at least one Pseudo solution for the problem (1.2).
More recently, Zhang  has considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of unknown functions:
where is the standard Riemann-Liouville fractional derivative of order αq may be singular at and f may be singular at . By using the fixed point theorem of a mixed monotone operator, a unique existence result of positive solution to the problem (1.3) was established. And then, Goodrich  was concerned with a partial extension of the problem (1.3) by extending boundary conditions
The author derived the Green’s function for the problem (1.4) and showed that it satisfies certain properties. Then, by using cone theoretic techniques, a general existence theorem for (1.4) was obtained when satisfies some growth conditions.
In recent work , Rehman and Khan have investigated the multi-point boundary value problems for fractional differential equations of the form
where with . By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for BVP (1.5) provided that the nonlinear function is continuous and satisfies certain growth conditions. However, Rehman and Khan only considered the case and the case of the nonlinear term f was not considered comprehensively.
Notice that the results dealing with the existence and uniqueness of solution for multi-point boundary value problems of fractional order differential equations are relatively scarce when the nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. Thus, the aim of this paper is to establish the existence and uniqueness of nontrivial solutions for the higher nonlocal fractional differential equations (1.1) where nonlinear term f and the boundary conditions all involve fractional derivatives of unknown functions. In our study, the proof is based on the reduced order method as in  and the main tool is the Leray-Schauder nonlinear alternative and the Schauder fixed point theorem.
2 Basic definitions and preliminaries
Definition 2.1 A function x is said to be a solution of BVP (1.1) if and satisfies BVP (1.1). In addition, x is said to be a nontrivial solution if for and x is solution of BVP (1.1).
For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later. These and other related results and their proofs can be found, for example, in [6-9].
Definition 2.2 (see )
Let with . Suppose that then the αth Riemann-Liouville fractional integral is defined by
whenever the right-hand side is defined. Similarly, with with , we define the αth Riemann-Liouville fractional derivative to be
where is the unique positive integer satisfying and .
Remark 2.1 If with order , then
Lemma 2.1 (see )
(1) If , , then
(2) If , , then
Lemma 2.2 (see )
Assume that with a fractional derivative of order , then , where , ( ). Here stands for the standard Riemann-Liouville fractional integral of order and denotes the Riemann-Liouville fractional derivative as Definition 2.1.
Lemma 2.3If , and , then the boundary value problem
has the unique solution
where is the Green function of BVP (2.1), and
Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation
Note that and (2.4), we have . Consequently, a general solution of (2.3) is
By (2.5) and Lemma 2.1, we have
So, from (2.6), we have
By , combining with (2.7), we obtain
So, substituting into (2.5), the unique solution of the problem (2.1) is
The proof is completed. □
Lemma 2.4 , for , where
Proof Obviously, for , we have , . Thus
This completes the proof. □
Now let us consider the following modified problem of BVP (1.1)
Lemma 2.5Let , . Then (2.9) can be transformed into (1.1). Moreover, if is a solution of the problem (2.9), then the function is a solution of the problem (1.1).
Proof Substituting into (1.1), by Lemmas 2.1 and 2.2, we can obtain that
and also . It follows from that . Using , , (2.9) is transformed into (1.1).
Now, let be a solution for the problem (2.9). Then, from Lemma 2.1, (2.9) and (2.10), one has
which implies that . Thus from (2.10), for , we have
Moreover, it follows from the monotonicity and property of that
Consequently, is a solution of the problem (1.1). □
Now let us define an operator by
Clearly, the fixed point of the operator T is a solution of BVP (2.9); and consequently is also a solution of BVP (1.1) from Lemma 2.5.
Lemma 2.6 is a completely continuous operator.
Proof Noticing that is continuous, by using the Ascoli-Arzela theorem and standard arguments, the result can easily be shown. □
Lemma 2.7 (see )
LetXbe a real Banach space, Ω be a bounded open subset ofX, where , is a completely continuous operator. Then, either there exists , such that , or there exists a fixed point .
3 Main results
For the convenience of expression in rest of the paper, we let .
Theorem 3.1Suppose for any . Moreover, there exist nonnegative functions such that
whereMis defined by (2.8). Then BVP (1.1) has at least one nontrivial solution.
Proof Since , there exists such that
By condition (3.1), we have , a.e. , thus
On the other hand, from (3.2), we know
Now let , suppose , such that . Then
Moreover, for ,
thus we have, by hypothesis (3.1),
Consequently, from (3.3), we have
This contradicts . By Lemma 2.7, T has a fixed point , since ; so then, by Lemma 2.5, BVP (1.1) has a nontrivial solution . This completes the proof. □
Theorem 3.2Suppose for any . Moreover, there exist nonnegative functions such that
where are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.
Proof By Lemma 2.6, we know is a completely continuous operator.
and define a ball . For every , we have
On the other hand, it follows from (3.4) that
In view of (3.5), we have the following estimate:
Therefore, . Thus we have . Hence the Schauder fixed point theorem implies the existence of a solution in for BVP (2.9). Since , then by Lemma 2.5, BVP (1.1) has a nontrivial solution . This completes the proof. □
Theorem 3.3Suppose for any . Moreover, there exist nonnegative functions such that
where are nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.
Proof The proof is similar to that of Theorem 3.2, so it is omitted. □
Theorem 3.4Suppose for any . Moreover, there exist nonnegative functions such that
and (3.2) holds. Then BVP (1.1) has a unique nontrivial solution.
Proof In fact, if , then we have
From Theorem 3.1, we know BVP (1.1) has a nontrivial solution.
But in this case, we prefer to concentrate on the uniqueness of a nontrivial solution for BVP (1.1). Let T be given in (2.11), we shall show that T is a contraction. In fact, by (3.7), a similar method to Theorem 3.1, we have
Then (3.2) implies that T is indeed a contraction. Finally, we use the Banach fixed point theorem to deduce the existence of a unique nontrivial solution to BVP (1.1). □
Corollary 3.1Suppose for any , and (3.1) holds. Then BVP (1.1) has at least one nontrivial solution if one of the following conditions holds
(1) There exists a constant such that
(2) There exists a constant such that
(3) There exists a constant such that
(4) ( ) satisfy
From the proof of Theorem 3.1, we only need to prove
(1) If (3.8) holds, let , and by using H lder inequality,
(2) In this case, it follows from (3.9) that
(3) In this case, it follows from (3.10) that
(4) If (3.11) is satisfied, we have
This completes the proof of Corollary 3.1. □
Corollary 3.2Suppose for any . Moreover,
Then BVP (1.1) has at least one nontrivial solution.
Proof Take such that
by (3.12), there exists a large enough constant such that for any , , one has
Then for any , we have
Then it follows from Theorem 3.1 that BVP (1.1) has at least one nontrivial solution. □
Example 4.1 Consider the boundary value problem
Proof Let , , , , and set
Thus we have
Thus the condition (3.2) in Theorem 3.1 is satisfied, and from Theorem 3.1, BVP (4.1) has a nontrivial solution. □
Example 4.2 Consider the boundary value problem
Thus Theorem 3.4 guarantees a nontrivial solution for BVP (4.2). □
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. Each of the authors contributed to every part of this study equally and read and approved the final version of the manuscript.
The authors thank the referee for helpful comments and suggestions which led to an improvement of the paper. The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2010091), the National Science Foundation for Post-doctoral Scientists of China (Grant No. 2012M510956).
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