Study On The Existence And Stability Of Solutions For Two Classes Of Fractional Order Systems

Fractional systems are widely used in physics,chemistry,aerodynamics,biophysics and other disciplines.In practice,it is very important to study the existence of positive solutions of fractional order systems and discuss the application of fractional order equa-tions in infectious disease models.In this paper,we will study the existence of positive solutions for a class of discrete fractional-order systems with semi-positive and non-linear differential boundary values and the stability of a class of fractional-order HIV epidem-ic models with time-delay and non-linear infection rates.The main content is given as follows:In the first part,we study the existence of positive solutions for a class of discrete fractional order systems with semi-positive nonlinear differential boundary value.The existence of positive solutions for the system satisfying certain boundary conditions is obtained by constructing suitable cones,using directionality absence,homotopy invariance and fixed point theorem.In the second part,we study the HIV epidemic model with time delay and non-linear infection rate.Firstly,we analyze the stability of the integer order HIV reaction-diffusion epidemic model with time delay and non-linear infection rate satisfying Neumann bound-ary conditions.By using the theory of solution of reaction-diffusion equation,linearization method and Hurwitz theorem,the existence and stability of the solution of the model are analyzed.Sufficient conditions for the existence and local stability of disease-free equilib-rium point and infection-disease equilibrium point are obtained.Secondly,the stability of fractional-order HIV epidemic model with time-delay and non-linear infection rate is s-tudied.By using the related theory of fractional-order equation,linearization method and Lyapunov function,the related properties and stability of the solution of the fractional-order model arc analyzed.Sufficient conditions for the local stability and global stability of the disease-free equilibrium point are obtained,and the local stability of the infection disease equilibrium point is obtained.