Research On The Fixed Point Theory Of Mixed Monotone Operators And The Solutions Of Several Differential Equations

This article mainly discusses the following three aspects.The existence and unique-ness of fixed points for some mixed monotone e-concave-convex operators or monotone e-concave operators with perturbation,the existence and the uniqueness of positive solu-tion for singular nonlinear fractional differential equations,and the existence of positive solution of the high order impulsive differential equation on infinite interval.This thesis is divided into four chapters.The chapter 1 describes the importance of the nonlinear operator theory and the fractional differential equation theory.Based on this reason,the study of the fixed point of operator and fractional order differential equations are meaningful.In the chapter 2,we consider the existence and uniqueness of fixed points for the following operators equations by monotone iterative techniques and properties of cone:A(x,x)+ B(x,x)=x,(2.1.1)Ax + Bx = x.(2.1.2)In(2.1.1),A:Ce×Ce ?Ce is a mixed monotone and e-concave-convex operator,B is a sub-homogeneous and mixed monotone operator.In(2.1.2),A:Ce?Ce is an increasing operator and e-concave operator,B is an increasing sub-homogeneous operator.This chapter considers the existence and uniqueness of fixed points for some operators with perturbation.Compared with Zhao and Du published in the Journal of Mathematical Analysis and Applications in 2007,and Zhao published in the Nonlinear Analysis in 2010,our operator equations are more general.If we let B = ?,(2.1.1)or(2.1.2)reduce to operator equations of Zhao and Du or Zhao respectively.Under our assumptions,we also obtain the existence and uniqueness of solutions for operator equations x = A(x,x)and x = Ax.Compared with one of the main theorem of Wu published in the Journal of Mathematical Analysis and Applications in 2008,we need not transform t?(t,x,y)to t[1+?(x,y,t)].It means that we need not 0<t[1 +?(x,y,t)]<1,we need only?(x,y,t)>0.Moreover,this chapter also obtains the solution of the nonlinear eigenvalue equation ?x = A(x,x)and ?x = Ax,and discusses its dependency to the parameter.We don't need compactness or continuity of operators in this chapter.In the chapter 3,we consider the existence and uniqueness of solutions for the fol-lowing singular nonlinear fractional differential equation by applying the mixed monotone operator fixed point theorem where n-1<??n,n>3,1?????n-2,p,q?C((0,1),[0,?)),p(t)and q(t)allowed to be singular at t = 0 or t = 1,f:(0,1)×(0,?)×(0,?)?[0,?)is continuous and f(t,u,v)may be singular at t=0,1 and u = v = 0,g:(0,1)×[0,?)×[0,?)?[0,?)is continuous and g(l,u,v)may be singular at t = 0.1,k:[0,1)?[0,?)is continuous function.The problem(3.1.1)of this chapter has more general form.We consider the singularity in this chapter.We allow p,q are singular at t = 0,1,f may be singular at t = 0,1 and x = y = 0,g may be singular at t = 0.It means that f(t,x,y)is singular both for time and space variables,and g(l,x,y)is singular for time variables.The nonlinear terms not only consider the derivative term,but also consider the operator term,where the operator can be linear or nonlinear.Especially,when p(t)= q(t)= 1,Hu =0,we know that the problem in Zhang and Tian published in the Advances in Difference Equations in 2017 is a particular case of the problem(3.1.1).When k(u)= 0,?=0,p(t)= q(t)= 1,and Hu(t)= u(t),our problem(3.1.1)reduced to the problem in Jleli and Samet published in the Nonlinear Analysis:Modelling and Control in 2015.In the chapter 4,we study the existence of positive solution of the high order impulsive differential equation on infinite interval as follows by applying the Schauder fixed point theorem and Altman fixed point theorem:where u0?R,?,??(n-1,n],n>2,D0+? is the standard Riemann-Liouville fractional derivative,0=t0<t1<t2<…<tm<?,?u(tk)=u(tk+)-u(tk-),u(tk-)=u(tk),u(tk+)=lim u(tk+h)and u(tk+)=lim u(tk-h)represent the right and left limits of u(t)at t=tk,D0+?-1u(?)=lim u(t).f?C((0+?)×R×R×R,R),Ik?C(R,R).In this chapter,the operator term not only involved with fractional order deriva-tive,but also contained fractional integral.Compared with Liu published in the Applied Mathematics and Computation in 2016,Liu and Ahmad published in the The Scientific World Journal in 2014,and Zhao and Ge published in the Applications of Mathematics in 2011,our nonlinear term are more generally.There are many articles that contained derivative term in the nonlinear term,but the articles that contained fractional integral and derivative are fewer.We study the problem on the infinite interval in this chapter.To the best of our knowledge,there are few articles involving the impulsive fractional order differential equations on the infinite interval.Compared with the problem on limited in-terval,the problem on the infinite interval is different when construct the cone.Moreover,our problem is higher order impulsive fractional equations.