Solvability Of Fractional Differential Equations And Stability Of Fractional Inertial Neural Networks

Because of its historical dependence and global correlation,fractional calculus has become an ideal tool to describe the memory and heredity of things.Compared with the integral calculus,the fractional calculus has a better fit with the experimental results in some aspects,so it is widely used in signal processing,hydrodynamics,mathematical biology,electrochemistry and so on.The study of fractional differential equations and the solution of fractional models from the above disciplines can enrich the research results in the field of calculus,expand the research field of differential equations,and have important theoretical significance and application value.Fractional calculus seems to be a simple generalization of integral calculus,but the definition of fractional integral involves defective integral with parameters,and many conclusions and properties of fractional integral calculus cannot be established in fractional order,even if they are not reasonable.Therefore,it is of great significance to systematically study fractional calculus and its equations.In this paper,the existence,uniqueness and stability of the solution are discussed by establishing corresponding fractional Lyapunov inequality,fractional Lyapunov function,fractional comparison theorem and extremal mapping fixed point theorem.The main work of this paper is summarized as follows:1.Lyapunov inequality plays an important role in the existence of nontrivial solutions for integer order differential equations and low order(order less than 1)fractional differential equations.In this paper,the corresponding Lyapunov inequalities are established for linear differential equations with higher order(order between 2 and 3)fractional derivatives,and the uniqueness of solutions and Hyer-Ulam stability for a class of linear differential equations with higher fractional derivatives are obtained.2.For classical integer differential equations,there is an integer comparison theorem.For fractional differential equations,some scholars have established fractional comparison theorems.In this paper,a new comparison theorem containing both integer and fractional terms is established,and by using the upper and lower solution method,fixed point theorem.The existence of solutions for a class of nonlinear two-term fractional boundary value problems is obtained.3.Based on the characteristics of regenerative cone,fixed point theorems of set-valued increase,decrease operators and mixed monotone operators are established.As an application,the solvability of fractional integrals and the existence of solutions for fractional coupled systems are discussed.4.The stability of solutions of differential equations describing fractional stochastic delay inertial neural networks is studied.The fractional stochastic inertial neural network is reduced by appropriate variable substitution,construct the random delay inertial fractional Lyapunov function of neural network,and by using Ito formula,combined with the LMI technique,obtained the sufficient conditions for the robust stability of the finite time randomly,the corresponding design method of state feedback controller is given,and the stochastic stability time function to estimate the upper bound,the effectiveness of the proposed method is verified by numerical simulation.