Positive Solutions For Boundary Value Problems Of Fractional Differential Equations

During the past several decades, nonlinear problems have been paid a huge amount of attention on the scientific fields, such as mathematics. fluid flow, elec-trochemistry, economics, engineering, control theory of dynamic systems and the list goes on. With the development of solving such problems, nonlinear functional analysis has been being one of the most significant research fields in modern math-ematics. It mainly contains topological degree methods, partial ordering methods and variational methods. And these methods lay solid foundations for the research of all sorts of nonlinear problems which are rather complicated. L. E. J. Brouwer had first put up with the conception of topological degree for finite dimensional space in 1912. And J. Leray and J. Schauder had made an extension of the concep-tion to completely continuous field of Banach space in 1934. After that. E. Rothe, M. A. Krasnosel' skii. P. H. Rabinowitz, H. Amann and K. Deimling had carried on embedded research on topological degree and cone theory. Plenty of outstanding mathematicians in China, such as, Gongqing Zhang, Dajun Guo, and Jingxian Sun etc., had made great progress in various fields of nonlinear functional analysis.(See [1-15]).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. M. L'Hospital had first put up with the concept of fractional derivatives in 1695. After that, L. Euler, J. Liouville and J. L. Lagrange also paid their attention to this field. In the past decade,â… . Podlubny, A. A. Kilbas, H. M. Srivastava and J. J. Trujillo had made the theory of this field be more reasonable and systemic than before. (See [16-29])This paper mainly discusses existence of positive solutions, multiplicity for several classes of boundary value problems of nonlinear fractional differential equa-tions by using topological degree theory and some theories established on cones. It is made up of four chapters and the main contents are arranged as follows: In Chapterâ… . we give the definitions and lemmas of fractional integrals, frac-tional derivatives and the fixed point index. Meanwhile, we give several lemmas of fixed point theorems. And these definitions and lemmas play vital roles in the following chapters.In chapterâ…¡, by means of Schauder's fixed point theorem and an extension of Krasnoselskii's fixed point theorem in a cone, we investigate the existence of positive solutions for two-point boundary value problem of the following nonlinear differential equation of fractional order. where 1<α< 2 is a real number. CD0+αis the Caputo fractional derivative and f:[0,1]×[0,+∞)×Râ†'[0,+∞) is continuous.In chapterâ…¢, by using Leray-Schauder's fixed point theorem, we are con-cerned with the existence of positive solutions for three-point boundary value prob-lem of the following singular nonlinear differential equation of fractional order, where CD0+αis the standard Riemann-Liouville fractional calculus andα,β,γ,α,ξ,f satisfy that (H2)f:(0,1)×[0,+∞)×Râ†'[0,+∞) satisfies the Caratheodory conditions with respect to Lp[0, 1](p≥1).In chapterâ…£, by means of the fixed-point index theory, we established the existence of multiply positive solutions for the following m-point boundary value problem of singular nonlinear differential equation of fractional order: where is continuous,f(t,u)is singular at u=0 and (?) is the Caputo fractional derivative.