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:
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
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 )
Lemma 2.1 (see )
Lemma 2.2 (see )
Assume thatwith a fractional derivative of order, then, where, (). Herestands for the standard Riemann-Liouville fractional integral of orderanddenotes the Riemann-Liouville fractional derivative as Definition 2.1.
has the unique solution
Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation
By (2.5) and Lemma 2.1, we have
So, from (2.6), we have
The proof is completed. □
This completes the proof. □
Now let us consider the following modified problem of BVP (1.1)
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 2.7 (see )
3 Main results
whereMis defined by (2.8). Then BVP (1.1) has at least one nontrivial solution.
On the other hand, from (3.2), we know
thus we have, by hypothesis (3.1),
Consequently, from (3.3), 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. □
Proof The proof is similar to that of Theorem 3.2, so it is omitted. □
and (3.2) holds. Then BVP (1.1) has a unique nontrivial solution.
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). □
From the proof of Theorem 3.1, we only need to prove
(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. □
Then BVP (1.1) has at least one nontrivial solution.
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
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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