Existence Of Positive Solutions For Fractional Differential Equations With Integral Boundary Value Conditions

In this thesis, we study on the existence and multi-existence of positivesolutions for Caputo fractional differential equations with integral boundary valuecondions in two cases, where in one case nonlinear term is independent ofderivative of state variable, while in the other the term is dependent upon thederivative. First, we prove the theorem of existence of at least one positive solutionfor a class of fractional differential equations with integral boudary conditionwhich is independent of derivative of unknown function in nonlinear term by usingthe fixed point theorem of cone expansion and compression. Second, we study themultiple existence of positive solutions for a class of fractional differentialequations with integral boudary condition involving derivative of unknownfunction in nonlinear term by using the Leggett-Williams fixed point theorem.In Chapter one, we elaborate the significance of this paper, introduceresearches both at home and abroad on boundary value problems of fractionaldifferential equations, summarize the content and the structure of this paper,analyse the methods by which we solve the problems.In Chapter two, we introduce some fix point theorems which are used in thisthesis, such as the fixed point theorem of cone expansion and compression,Leggett-Williams fixed point theorem and the generalization of theLeggett-Williams fixed point theorem. And we give the concept and nature offractional derivative and integral.In chapter three, we investigate the existence of at least one positive solutionfor a class of Caputo fractional differential equations with integral boudarycondition independing on derivative of unknown function in nonlinear term byusing the fixed point theorem of cone expansion and compression. The sufficientconditions of the existence of one positive solution are obtained. At the end of thischapter, we give an example to illustrate how the results we obtained can be usedin practice.In chapter four, we study the multiple existence of positive solutions for a classof Caputo fractional differential equations with integral boudary condition byusing the Leggett-Williams fixed point theorem. Because the nonlinear termdepend on Caputo fractional derivative, the defination of norm changed, correspondingly, the problem is more difficult than the problem with nonlinearindependence on fractional derivatives. As a result, we find that we can not solvethe prblem with dependence on fractional derivatives both in nonlinear term andboundary value condions by using the traditional Leggett-Williams fixed pointtheorem, we have to use the generalization of the Leggett-Williams fixed pointtheorem, details about the generalization of the Leggett-Williams are introduced inChapter two. In this chapter, we define the norm which is different from before,then, we investigate the multiple existence of positive solutions for fractionaldifferential equations with integral boudary condition by using the generalizationof the Leggett-Williams fixed point theorem on special cones.