| In later years, all sorts of nonlinear problems have resulted from mathemat-ics, physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinear func-tional analysis has been bing one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree method and the variational method. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and tech-nology. And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in1912. J. Leray and J. Schauder had extend the conception to completely con-tinuous field of Banach space in1934, afterward E. Rothc, M. A. Krasnosel’skii, P. H. Rabinowitz, H. Amann, K. Deimling had carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, say Zhang Gongqing. Guo Dajun. Chen Wenyuan, Sun Jingxian etc., had proud works in various fields of nonlinear functional analysis.However, with the development of sciences, researchers gradually realize the orders of established differential equations of mathematical modes are not always integral. Or rather, if the orders of these differential equations are fractional num-bers, these equations can be viewed as mathematics models more accurate than those orders of differential equations are integral. Meanwhile, since plenty of the theories of nonlinear functional analysis can be applied to fractional differential equations, researchers, recently, have gradually paid much attention to these equa-tions. In the research of the integral order differential equations, it is all known that the eigenvalue is important for solving the existence of the solutions of the equations; and the existence of eigenvalue of the fractional differential equations provides a theoretical basis for the fractional diffusion-wave equations, hence it is worthy study of the eigenvalue of fractional differential equations.This paper can divide into two parts; firstly, we discuss the eigenvalue of R-L fractional differential equations with the approximation method; then we will give a triple solutions theorem of second order impulsive differential equation via a critical theorem. The contens are arranged as followed: In charpter I. we will give the definition of R-L fractional calculus and its propositions.In charpter II, we will study the existence of R-L fractional differential equa-tion, as following: Where [a] is the latest integer which is bigger than a. We will divide into two cases to discuss the problem, which are1<a<2and n-1<a<n,(n>2), at the same time, we also give a comparation to the problem of eigenvalue of integer order differential equations, it shows that the eigenvalue of fractional differential will be more extensive. In charpter III, we will study the existence of several solutions of integral order differential equation. At fist, we will introduce two knids of derivative of nonlinear functional analysis and their connections; then we list some lemmas and a critical theorem, which will be used to study the following impulsive differential equations: where0=t0<t1<t2<…<tm<tm+1=1,(Δu’)|t=tj=u’(tj+)-u’(tj-) and a.β,γ, σ>0;f∈C(R,R), Ij∈C(R,R), j=1,2,…, m, and we point that: should be positive real numbers... |
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