The Existence Of The Fixed Point Theorem Of Operators And The Solutions Of Fractional Differential Equations

With the development of nonlinear fractional differential equations in recent decades and im-proving, it has been widely applied in different fields, such as physics, chemistry, economics, engineering, biological sciences and so on. Thus, they have captured a wide range of attention of many researchers and obtained large quantities of results.In this paper, we consider a class of fixed point theorems for -?-convex operator, ?-concave operator, mixed monotone operators and use the mixed monotone method and some fixed point theory to obtain the existence results of positive solutions for a class of p-Laplacian fractional differential equations boundary value problems in a Banach space.This article we shall divide into three sections:In the first chapter, we discuss the existence and uniqueness of positive solutions for a class of operator equations as follow A{x,x)+B(x,x)= x, (1.1) Ax+Bx= x, (1.2) Ax-Bx= x. (1.3)Via using fixed point theorems, the existence and uniqueness results for operator equations (1.1), (1.2) and (1.3) are established. Finally, we give an example to illustrate the correctness and effectiveness of our main theorems. To the best of author's knowledge, the appropriate method which figures out the problem of the existence and uniqueness results of positive solution for equations (1.2) and (1.3) can not been founded in the existing literature. Therefore, our main results not only extend and complement paper [33] but also obtain some new fixed point theorems for -?-convex operator, ?-concave operator, mixed monotone operators on ordered Banach spaces.In the second chapter, we consider the existence of positive solutions for p-Laplacian fractional differential equations boundary value problems on unbounded domains in a Banach space as belowvia using the Darbo's fixed point theorem and noncompactness measure method, we obtain the existence results of positive solutions for boundary value problem (2.1). Compare with paper [18], the variable t in problems (2.1) is more widely and from the equation form, equation (2.1) is more general then paper [21]. Thus, our main results extend paper [18] and [21], so, it has more widely application.In the third chapter, we investigate the following singular p-Laplacian fractional differential system involving the Riemann-Stieltjes integral boundary conditionCombining mixed monotone method with the Guo-Krasnosel'skii fixed point theorem, the existence and uniqueness results of positive solution for problem (3.1) is established. Compar-ing with paper [1] and [4], the integer-order problem is extended into fractional-order problem. Moreover, under some particular conditions, the equation (3.1) is equivalent to the main equation in paper [4], for instance, the parameters and x(t),y(t) in equation (3.1) satisfy ?1=?1= 1, ?2=?2=2, and where p? 2, c1? 0 and c2 are constants. Hence, from the form of equations, the system (3.1) is more general.