The Existence Of Positive Solutions For Fractional Differential Equation And Equations On The Boundary Value Problems

In recent years, the fractional order calculus and the theory of fractionaldifferential equation (equations) in continuous development and improvement, theyhave got very extensive application in many fields. The studies of fractional ordercalculus and fractional order differential equation (equations) has very importanttheoretical significance and practical application value, especially the abstracted fromthe practical problems of fractional differential equation (equations) have become thefocus in the many mathematics workers.In the master’s degree paper, we mainly consider the multi-point boundary valueproblems of fractional differential equations solution and the existence of positivesolutions of the coupling of fractional order differential equations. The positivesolutions we considers is point to the two component is positive. This and we usuallycalled positive solutions of(one of the component is positive, another is negative) havea significant difference. In order to make sure the solutions of the two components arepositive which we got, we use the fixed point theorem of corresponding continuousmaps on product cone to solve it. Tightening up the fixed point index on product coneof the continuous mapping of the calculation, we must establish the product of thefixed formula and we get the product fixed point theorem on product cones. Asapplications, we consider existence of positive solutions of the super-linear fractionalorder differential equations on boundary values.To this end, in the second chapter, we give the deformation of Leray-Schaudertheorem and basic concept and basic properties of topological degree, others by usingthe topological degree of knowledge we established the topological degree formula onthe product of cone of all kinds of mapping in the product space, especially thecontinuous maps on product cone.Further more, we give the fixed point theorem onproduct of cone.As application, in the third chapter, we consider the existence of solution onmulti-point boundary value problems of fractional differential equations. In chapterfour and five, we considered the Super-linear time fractional differential equations ofintegral boundary value problem and the existence of positive solutions of boundaryvalue problems. Through selecting the appropriate open set on the product cone, andverifying it acquire the certain boundary conditions on the boundary of the open set ofthe corresponding continuous maps, we mainly use the related fixed point theorem what we give in the second chapter. Finally, we got the existence of positivesolutions of the two types of fractional differential equations.