The Application And Existence Of Solutions Of Some Fractional Nonlinear Differential Equations

Fractional differential equations have various applications in the fields of physics, chemistry, aerodynamics, control theory, signal, and image processing, biophysics, electrodynamics of complex medium, and so on. This dissertation mainly investigates the problems for solutions of nonlinear fractional differential(integro-differential) equation(s) and nonlinear fractional impulsive differential equations. It consists five chapters.In chapter one, we mainly introduce research background and meaning, current development situations of this study, and the main conclusions and motive of this thesis.In chapter two, firstly, by applying an iterative technique, sufficient conditions are obtained for the existence of the unique solution of the nonlinear neutral fractional integro-differential equation involving two Riemann-Liouville derivatives of different fractional orders. Secondly, applying the monotone iterative method combined with the upper and lower solutions, we investigate the existence of solutions for a coupled system of nonlinear neutral fractional differential equations, which involves Riemann-Liouville derivatives of different fractional orders. Our results generalize and improve some former corresponding results.In chapter three, firstly, by using the lower and upper solutions method and fixed point theorem on cone, we consider the existence and uniqueness of positive solution of the integral boundary value problem for nonlinear differential equation involving Riemann-Liouville fractional derivative. Secondly, by employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary. Our obtained results improve and extend many recent results.In chapter four, by employing the monotone iterative method, we not only establishes the existence of the minimal and maximal positive solutions for multipoint fractional boundary value problem on an unbounded domain, but also develops two computable explicit monotone iterative sequences for approximating the two positive solutions. Our results improve and extend some recent results.In chapter five, firstly, by applying Schauder fixed point theorem and Banach contraction mapping principle, we investigate a new impulsive multi-orders nonlinear fractional differential equation. The existence and uniqueness results are obtained for the a nonlinear problem with fractional integral boundary conditions. Our results generalize and improve some former corresponding results.