The Existence Of Solutions For Three Classes Of Fractional Differential Equations

In recent years,with the development of fractional calculus theory and its application,the boundary value problem of fractional differential equations has been widely concerned by mathematician.Among them,the existence of the solutions for fractional differential equations is one of the hot research directions.In addition to the field of mathematics,fractional calculus has a lot of applications in rheology,chemical physics,biology and other fields.In this paper,we study the existence of solutions for three classes of fractional differential equations by using of the fixed point theorem on ordered interval,the method of iteration,the fixed point index theory,the fixed point theorem of Guo-Krasnoselskii,the correlation theory of?-(h,e)concave operator,and give the sufficient conditions for the existence of solutions for differential equations.The full text is divided into four chapters:In the first chapter,we introduce the research background of this article,the main task and some important preliminaries.In the second Chapter,we study the existence positive solutions for fractional q-difference equation#12where parameter ?>0,2<??3,f:(0,?)?[0,?)is a continuous function,Dq? is a Riemann-Liouville fractional q-derivative by using of the fixed point theorem on the ordered interval,the method of iteration and the fixed point index theory.The problem of existence of the solutions for equations is transformed into the problem of operator fixed point,we give the sufficient conditions of the existence of positive solutions depending on parameter A for this fractional q-difference equation by analyzing the properties of Green's function.In the third Chapter,we study the existence of positive solutions for the caputo fractional differential equation#12where parameter ?>0,3<??4,f:[0,?)?(0,?)is a continuous function,C D0+? is a caputo fractional derivative by using of the fixed point theorem of Guo-Krasnoselskii.The problem of existence of the solutions for equations is transformed into the problem of operator fixed point,we give the sufficient conditions of the existence of at least one positive solution when the parameter is in a certain range for this caputo fractional differential equation by analyzing the properties of Green's function.In the fourth Chapter,we study the existence and uniqueness of solution for the Riemann-Liouville fractional differential equation#12where 3<??4,f?C([0,1]×(-?,+?),(-?,+?)),b?0,D0+? is a Riemann-Liouville fractional derivative by using of the correlation theory of ?-(h,e)concave operator,we give the existence and uniqueness of the solution of this Riemann-Liouville fractional differential equation by analyzing the properties of Green's function.