Existence And Uniqueness Of Solutions For Fractional Differential Equations

In recent decades, fractional order differential equation is more and moreattention, It is not only in science, engineering and mathematics, and other fieldsimportant applications, For example, has been successfully applied to the viscoelasticmaterial, signal processing, control, biological and other fields, And in other scientificresearch areas, also played a positive role of fractional calculus, Such as: optical andthermal system, rheology and materials and mechanical system, signal processing andsystem identification, control and robotics and other applications. Fractional ordercalculus theory has also been more and more attention of scholars both at home andabroad, especially from the practical problems of abstracting the fractional orderdifferential equations have become the focus in the many mathematics workers.Fractional order calculus is theory of arbitrary order differential and integral, it isunified with the integer order differential and integral calculus, is a generalization ofthe fractional calculus. So the existence and uniqueness of fractional order differentialequations not only has theoretical significance, but also has important practicalapplication value.Paper on the basis of the theory of fractional order differential equations onseveral kinds of fractional order differential equation is studied and existence anduniqueness of the solution.First of all, using the green’s function and Guo-Krasnoselskii fixed pointtheorem, the fractional order differential equations is discussed the existence ofpositive solutions of integral boundary value problem, the problem has at least onepositive solution of the sufficient conditions.Secondly, using the Banach contraction mapping theory, and discusses thefractional order differential equation of the existence and uniqueness of periodicsolutions of a, the fractional order differential equation discriminant basis have uniquesolution.Finally, study the half fractional order linear differential equations with pulse ismoderately counter-cyclical existence and uniqueness of the solution, using the banach fixed point theorem, Schaefer, fixed point theorem, Krasnoselkill fixed point theorem,is given and proved pulse fractional order semilinear differential equation of theexistence and uniqueness of periodic solutions of a condition, and an example is givento verify the above conditions.