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论文范文
1. Introduction and Preliminaries Fractional calculus is old as the Newtonian calculus [1–3]. The name fractional was given to express the integration and differentiation up to arbitrary order. Traditionally, there are two approaches to define the fractional derivative. The first approach, Riemann-Liouville approach, is to iterate the integral with respect to certain weight function and replace the iterated integral by single integral through Leibniz-Cauchy formula and then replace the factorial function by the Gamma function. In this approach, the arbitrary order Riemann-Liouville results from the integrating measure and the Hadamard fractional integral results from the integrating measure . The second approach, Grünwald-Letinkov approach, is to iterate the limit definition of the derivative to get a quantity with certain binomial coefficient and then fractionalize by using the Gamma function instead of the factorial in the binomial coefficient. In case of the Riemann-Liouville and Caputo fractional derivatives, a singular kernel of the form is generated for to reflect the nonlocality and the memory in the fractional operator. Through history, hundreds of researchers did their best to develop the theory of fractional calculus and generalize it, either by obtaining more general fractional derivatives with different kernels or by defining the fractional operator on different time scales such as the discrete fractional difference operators (see [4–7] and the references therein) and -fractional operators (see [8] and the references therein). In 2014 [9], Khalil et al. introduced the so-called conformable fractional derivative by modifying the limit definition of the derivative by inserting the multiple inside the definition. The word fractional there was used to express the derivative of arbitrary order although no memory effect exists inside the corresponding integral inverse operator. This conformable (fractional) derivative seems to be kind of local derivative without memory. An interesting application of the conformable fractional derivative in Physics was discussed in [10], where it has been used to formulate an Action Principle for particles under frictional forces. Despite the many nice properties the conformable derivative has, it has the drawbacks that when tends to zero we do not obtain the original function and the conformable integrals inverse operators are free of memory and do not have a semigroup property. It is most likely to call them conformable derivatives or local derivatives of arbitrary order. In connection with this, at the end of reference [11], the author asked whether it is possible to fractionalize the conformable (fractional) derivative by using conformable (fractional) integrals of order or by iterating the conformable derivative. The first part, Riemann-Liouville approach, was answered in [12, 13], where the author iterated the (conformable) integral with weight , to define generalized fractional integrals and derivatives that unify Riemann-Liouville fractional integrals () and derivatives together with Hadamard fractional integrals and derivatives. Actually, the limiting case of that generalization is when leads to Hadamard type. However, the Grünwald-Letinkov approach for conformable derivatives is still open. The conformable time-scale fractional calculus of order is introduced in [14] and has been used to develop the fractional differentiation and fractional integration. After then, many authors got interested in this type of derivatives for their many nice behaviors [10, 15–18]. Motivated by the need of some new fractional derivatives with nice properties and that can be applied to more real world modeling, some authors introduced very recently new kinds of fractional derivatives whose kernel is nonsingular. For the fractional derivatives with exponential kernels we refer to [19]. For fractional derivatives of nonsingular Mittag-Leffler functions we refer to [20–22]. Motivated, as mentioned above, with the need of new fractional derivatives with nice properties we study in this article the eigenvalue problems of Sturm-Liouville into conformable (fractional) calculus. Recently, there are several analytical studies devoted to fractional Sturm-Liouville eigenvalue problems; see [23–27]. In these studies some of the well-known results of the Sturm-Liouville problems are extended to the fractional ones with left- and right-sided fractional derivatives of Riemann-Liouville and Caputo and Riesz derivatives. These results include orthogonality and completeness of eigenfunctions and countability of the real eigenvalues. Another class of fractional eigenvalue problem with Caputo fractional derivative has been studied in [28] using maximum principles and method of upper and lower solutions. 
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