- Open Access
Spectral problems for fractional differential equations from nonlocal continuum mechanics
© Li and Qi; licensee Springer. 2014
- Received: 19 November 2013
- Accepted: 26 February 2014
- Published: 13 March 2014
This paper studies the spectral problem of a class of fractional differential equations from nonlocal continuum mechanics. By applying the spectral theory of compact self-adjoint operators in Hilbert spaces, we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system. Furthermore, we obtain the lower bound of the eigenvalues.
MSC:26A33, 34L15, 34B10, 47E05.
- fractional differential equation
- eigenvalue problem
where q is a real-valued function, and are the left and right Riemann-Liouville fractional derivatives of order α, respectively, whose definitions are given later, μ is a real constant and λ is the spectral parameter.
Recently, fractional differential equations have drawn much attention. It is caused both by the development of fractional calculus itself and by the applications in various fields of science and engineering such as control, electrochemistry, electromagnetic, porous media, and viscoelasticity. For details, see [1–5].
Fractional differential equations with both left and right fractional derivatives are also applicable to many fields, such as the extremal problems of fractional Euler-Lagrange equations [6, 7] and the optimal control theory for functionals involving fractional derivatives . For the study of FDEs with left and right fractional derivatives, we mention the papers [9–12].
They demonstrated that the eigenvalues of the above two fractional eigenvalue problems are real, and the eigenfunctions corresponding to different eigenvalues are orthogonal.
where is the axial displacement of the bar at x, is the longitudinal force per unit volume, E is Young’s modulus, and are the left and right Riemann-Liouville fractional derivatives of order α (), respectively, and is a material constant. In a recent work, Qi and Chen  studied analytically the eigenvalue problem of order associated to the equation (1.1). They also proposed that the spectral problem associated to (1.1) with is more interesting. The reason is that (1.1) with can overcome some mechanical inconsistencies arising when the order of fractional derivative is in the range . Therefore, this paper will focus on the spectral problem which was proposed in .
In this study, by using the spectral theory of self-adjoint compact operators in Hilbert spaces, we prove that the spectrum of the spectral problem associated to (1.1) with consists of only countable real eigenvalues with finite multiplicity and the orthogonal completeness of the corresponding eigenfunction system in the Hilbert spaces. Furthermore, we obtain the lower bound of the eigenvalues.
The rest of the paper is organized as follows. In Section 2, we give some preliminary knowledge for fractional derivatives and the spectral theory of Sturm-Liouville problems that will be needed to develop this work. In Section 3, we obtain the main results of this paper.
2.1 Fractional derivatives
We will use the following properties of fractional integrals and derivatives which can be found in .
Definition 1 (cf. [, p.69])
respectively, where Γ denotes the Euler gamma function.
Definition 2 (cf. [, p.70])
Proposition 2.1 (cf. [, Lemma 2.2, p.73])
2.2 Eigenvalue problems for second order differential equations
These results can be obtained by Prüfer transformation [, Theorem 8.18].
the domain is dense in ;
is Hermitian, or for ;
, where is the adjoint of .
We also call the function G the Green function associated to the operator .
Throughout this paper we always assume that and .
The next lemma reveals the rationality of .
Lemma 3.1 If and , then T is a linear operator from to if and only if .
Let , then , by calculation, and hence , which implies that T is a linear operator from to .
By the arbitrariness of y, we can choose such that and . Then by (3.5), we have , which implies that . □
and hence T is symmetric. □
We will use the following result to prove the self-adjointness of T.
Lemma 3.3 (cf. [, Theorem 5.19, p.107])
If for some , the range of is the entire space , then T is self-adjoint.
In order to apply Lemma 3.3, we first prepare three lemmas.
and , are the solutions of such that , , on .
Remark 1 The details of the computation of (3.11) are presented in the appendix.
Lemma 3.5 If , then T is self-adjoint.
A combination of (3.13) and (3.14) gives (3.12).
Step 2. We prove that if , then there exists such that for , .
Step 3. We prove that T is self-adjoint for .
Let be defined as in (3.19). It follows from Lemma 3.4 that y is a solution of , if and only if , where K and F are defined as in (3.8) and (3.9), respectively. From the definition of , we conclude that is continuous on . Hence the operator K is compact and (3.8) is a Fredholm integral equation of the first kind. Note that for by the definition of F. If is a solution of , then (3.18) implies , and hence has only zero solution in . It follows from the Fredholm Alternative Theorem that the equation has a unique solution, which implies that for has a unique solution. By the arbitrariness of f, we know that , and hence T is self-adjoint by Lemma 3.3. □
Lemma 3.6 Let be defined as in (3.19). If , then the resolvent is well defined, self-adjoint, compact and positive, i.e. for .
Proof The existence and self-adjointness of follow from the discussion in Lemma 3.5. The positivity follows from (3.18). It remains to prove that is compact. Let be a bounded sequence in , say, for . Set , or . Then (3.18) gives , , and hence . Since K is compact in , possesses a convergent subsequence, denoted by again for simplicity. Since G is also compact, we can find a convergent subsequence of in . Therefore, (3.8) means that is convergent in . Thus, is compact. □
The following theorem is the main result of this paper.
and the set of all corresponding eigenfunctions , , forms a complete, orthogonal system of . Moreover, the first eigenvalue satisfies .
which implies for . □
where is defined as in (3.9).
The authors wish to thank the anonymous referees for providing valuable comments and suggestions which improved this paper. This research was partially supported by the National NSF of China (Grants 11271229, 11201263), the NSF of Shandong Province (Grants ZR2012AM002, ZR2012AQ004) and Independent Innovation Foundation of Shandong University (IIFSDU), China.
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