Existence And Multiplicity Of The Solutions For Several Kinds Of Fractional Differential And Difference Equations

In this doctoral dissertation, by using the nonlinear analysis methods and tech-niques, we study the existence and multiplicity of the solutions for several class of fractional differential equations and discrete fractional boundary value problems. This dissertation consists of four chapter.In Chapter1, we sketch the historical background, status, the up-to-date progress for the discussed problems, our main work, preliminary facts on fractional calculus and the main tools used in the dissertation.In Chapter2, we study the existence of the solutions on the finite interval for four types of fractional differential systems. First, we deal with a class of differential equa-tions of fractional order with variable coefficients, which plays an important role in the description and modelling of control systems. Second, by Mawhin’s continuation theorem, we investigate a class of sequential fractional differential equation involving the Caputo fractional derivative. Third, by the same tool, based on a new concept of a piecewise continuous solution, we established sufficient conditions of the existence of the solutions for the fractional multi-point boundary value problem with impulse effects at resonance. Last, by the recent Leggett-Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima, we present a new result on the existence of positive solutions for a class of differential inclusion of fractional order with boundary conditions at resonance. And our results improve and generalize the existing results.In Chapter3, we study the existence of the solutions on the unbounded interval for two types of fractional differential systems. First, by constructing special Ba-nach spaces and establishing an appropriate compactness criterion, we present some existence results of the solutions about the boundary value problem at resonance. Second, we give some new results on the existence of positive solutions for the fraction-al two-point boundary value problems at resonance by the recent Leggett-Williams norm-type theorem due to O’Regan and Zima.In Chapter4, we study two classes of discrete fractional boundary value problems. By using the fixed point index theorey, the existence of multiple positive solutions are obtained, and the uniqueness of the solution is proved by a new theorem on an ordered metric space established by M. Jleli et al. Second, we point out the differences between a class of fractional difference equation and that of integer order. We show that under the same boundary conditions, the problem of fractional order is nonresonant, while the integer order one is resonant. Then uniqueness of the solution is proved by a new theorem on a metric space established by H. K. Nashine, and the existence of multiple positive solutions are obtained by a new fixed point theorem.117references...