Positive Solutions For A BVP Of Nonlinear Fractional Differential Equation With Integral Boundary Conditions

Fractional derivatives aregeneralizations for derivative of integral order.There are several kinds of fractional derivatives,such as Riemann-Liouville fractional derivative,Marchaud fractional derivative,Caputo fractional derivative,Griinwald-Letnikov fractional derivative,etc.In the last few decades,fractional-order mod-els have been found to be more adequate than integer-order models for some real world problems.Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.This is the main advantage of fractional differential equations in comparison with classical integer-order models.In consequence,the subject of fractional differential equations is gaining much importance and attention.This paper is concerned with the following boundary value problem of nonlinear fractional differential equation where 2<q?3,0<??1,?>0,?,?,? are nonnegative constants satisfying 0<?:=(?+?)?+??/?(2-?)<?[?+??(q)?(q-?)],f:[0,1]×[0,+?]?[0,+?]and hi(i=1,2):[0,1]?[0,+?)are continuous.CD0+q denotes the standard Caputo fractional derivative.In chapter 1,we mainly introduce the background and basic contents(definitions,theorems,lemma,etc)of nonlinear fractional differential equations,and these basic works are the theoretical basis for our subsequent research work.In chapter 2,first,Green's function of the corresponding linear boundary value problem is constructed.Next,some useful properties of the Green's function are obtained.Finally,existence and multiplicity of positive solutions for the above problem are established by imposing some suitable conditions on f and hi(i=1,2)and applying Guo-Krasnoselskii fixed point theorem and Leggett-williams fixed point theorem.In chapter 3,by using monotone iterative method and some inequalities asso-ciated with the Green's function,we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solu-tions to the above problem.It is worth mentioning that these iterative sequences start off with zero function or linear function,which is useful and feasible for com-putational purpose.An example is also included to illustrate the main results of this paper.