Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions
© Jia et al.; licensee Springer 2012
Received: 7 April 2012
Accepted: 15 June 2012
Published: 3 July 2012
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.
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.
where x takes values in a reflexive Banach space E and with . denotes the k th 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).
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.
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 )
where is the unique positive integer satisfying and .
Lemma 2.1 (see )
- (1)If , , then
- (2)If , , then
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.
The proof is completed. □
This completes the proof. □
Lemma 2.5 Let, . Then (2.9) can be transformed into (1.1). Moreover, ifis a solution of the problem (2.9), then the functionis a solution of the problem (1.1).
and also . It follows from that . Using , , (2.9) is transformed into (1.1).
Consequently, is a solution of the problem (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 2.6is 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 )
Let X be a real Banach space, Ω be a bounded open subset of X, 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 .
where M is defined by (2.8). Then BVP (1.1) has at least one nontrivial solution.
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. □
whereare nonnegative constants. Then BVP (1.1) has at least one nontrivial solution.
Proof By Lemma 2.6, we know is a completely continuous operator.
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. □
whereare 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. □
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.
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). □
- (1)There exists a constant such that(3.8)
- (2)There exists a constant such that(3.9)
- (3)There exists a constant such that(3.10)
- (4)() satisfy(3.11)
- (1)If (3.8) holds, let , and by using Hlder 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. □
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. □
Thus the condition (3.2) in Theorem 3.1 is satisfied, and from Theorem 3.1, BVP (4.1) has a nontrivial solution. □
Thus Theorem 3.4 guarantees a nontrivial solution for BVP (4.2). □
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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