Research On Theoretical Analysis And Its Application For Fractional Differential Equations

Fractional order differential equation is differential equation containing arbitrary order derivative. In recent thirty years, fractional calculus has got attention from many researchers which is mainly due to its application as model for physical and engineering.Many researchers point out that the fractional order calculus more precisely describe the property of some materials and process with memory and heredity than integer order calculus. On the other hand, the properties of nonlocal and singular of the fractional differential operator make it difficult in theory research. Thus it is important to study on the fractional differential equation. This thesis mainly studies some properties of fractional order differential equations with a parameter and the existence of solutions for fractional q-difference equation and system with Riemann-Liouville q-derivative and the multiplicity results for fractional impulsive differential inclusion with left and right fractional order derivative. The main topics of this thesis are consisted of the following aspects:1. The existence, uniqueness, multiplicity and nonexistence of solutions for a nonlinear fractional differential equation with parameter and p-Laplacian operator. Based on the structure of the fractional differential system, we transform the problem into its equivalent integral system and introduce a special functional space, apply some knowledge of nonlinear analysis combined with the basic theory of fractional order calculus,and some inequality techniques, some criteria for existence, uniqueness, multiplicity and nonexistence of positive solution of fractional differential equations in terms of different value of parameter are established. The obtained results supplement and perfect the existing literature.2. The problem of existence, uniqueness and the existence of positive solutions for fractional q-difference equations. Mainly discussed a nonlinear fractional q-difference equation with the nonlinear term contains a fractional q-derivative of Rieman-Liouville type. By using some knowledge of fractional q-derivative and fractional q-integral and some properties of p-Laplace operator, through three fixed point theorem, Schauder’s fixed point theorem, Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem, the conditions for the existence, uniqueness and the existence of positive solutions for fractional q-difference equations were obtained.3. The problem of existence of extremal solutions for the fractional differential equations with integral boundary conditions and fractional q-difference system. First, we discussed the existence of extremal solution for a fractional differential equation with integral boundary conditions. By using the properties of Green function of the system and successively iterative method and some inequality techniques, two successively iterative sequences are constructed by two power functions, the solution is the limit of the sequence is proved. The method we use only require the local continuity and local monotonicity of nonlinear term, weaken the existing literature on the nonlinear term.Moreover, we obtained the conditions for the existence of sign-changing solutions and existence of positive solution. Second, we discussed the existence of extremal solution for a fractional q-difference system. We constructed two operators, using successively iterative give out the extremal solution of the system. Different from the first system, the iterative functions are zero function and power function, and the method than the existing literature is simple and easy to calculate.4. The problem of multiplicity results of impulsive fractional differential inclusion with left and right fractional derivative. Mainly by constructing suitable variational space, and constructed some appropriate function, converted the problem to the existence of the critical point of the function. By using methods of variational and non-smooth critical point theory, the conditions for the existence of three solutions are obtained.