The Studies Of The Existence And Uniqueness Of Solutions About Several Types Of Differential Equations And Integral Equations

The integral equations and differential equations have extensive applications in the many fields of economics and military science and so on.And many practical problems in chemical process and economic systems and so on can be transformed into integral equations or periodic boundary value problems of delay differential equations to study.In this thesis,we have done more thorough researches on the existence and uniqueness of solutions of several classes of differential equations and integral equations.The main chapters of this thesis are arranged as follows:In chapter one,we briefly introduce the development background and research significance of integral equations and differential equations.In chapter two,we mainly study the solutions of the Volterra integral equation and the initial value problem of the fourth-order ordinary differential equation.We first prove the existence and uniqueness of the solutions of the equations by using Banach contraction mapping principle.Then we obtain the numerical solutions for a concrete integral equation via numerical integration algorithm.In chapter three,we mainly study the harmonic solutions of periodic boundary value problems of the second-order generalized delay Liénard equation,delay Rayleigh equation,neutral delay Duffing equation and neutral delay Liénard equation.By using the Mawhin continuity theorem in topological degree theory,we obtain some existence and uniqueness results of harmonic solutions for above equations.Then we give concrete applications of our theoretical results.In chapter four,we mainly study the periodic solutions of the first-order delay differential equations with single delay quantity or several delay quantities.Firstly,by transforming the first-order delay differential equation with single delay quantity into a system of ordinary differential equations to discuss,we obtain some existence results of simple four periodic solutions for these problems under different conditions.Secondly,we obtain some existence results of periodic solutions for the first-order delay differential equations with several delay quantities by using the fixed point theorem for completely continuous operators.Finally,a concrete application of our results is given.