Solvability Of Some Kinds Of Boundary Value Problems For Fractional Differential Equations With P-Laplacian Operators

In recent years,with the application of fractional differential equation more and more widely,many scholars begin to pay attention to fractional differential equations,and have done a lot of research on fractional boundary value problems.On this basis,this paper studies the solvability of boundary value problems for some fractional differ-ential equations with p-Laplacian operators by means of variational method,upper and lower solutions method,monotone iterative method,coincidence degree method and fixed point theorem,and obtains some results on the existence of solutions.The results obtained extend and enrich some known work to some extent.The full text is divided into six chaptersChapter 1 briefly introduces the background,significance of the research and the research status and the main work of this paper,as well as some related definitions and properties of fractional calculusIn Chapter 2,the solvability of two kinds of Sturm-Liouville boundary value prob-lems of fractional p-Laplacian equation is studied by using the critical point theory Firstly,by using the Nehari manifold method,the existence theorem of the ground s-tate solution is given when the nonlinear term satisfies the condition of weaker than over p times Ambrosetti-Rabinowtiz type condition.To our knowledge,the existence of the ground state solution to this problem has not been studied.Secondly,when the nonlinear termf=f1+f2,f1 satisfies the condition of weaker than over p times Ambrosetti-Rabinowtiz type condition,f2 is the sub-linear growth at infinity,the exis-tence theorem of two non-trivial weak solutions is obtained by using the critical point theory.The previous work on the research of this kind of problem mostly used the Ambrosetti-Rabinowtiz condition,so the results of this chapter improve and enrich the previous related resultsIn Chapter 3,we study the multiplicity of solutions for Sturm-Liouville boundary value problems and the coupled systems of fractional p-Laplacian equation with impul-sive effects under the variational framework.When the nonlinear term satisfies a new condition and the impulsive function satisfies the sub-linear condition,we prove that the above problems possess at least three weak solutions by using the three critical point theorem,and possess infinite solutions for Sturm-Liouville boundary value problems by using the genus properties.There is no similar study on Sturm-Liouville boundary value problems of fractional p-Laplacian equation with impulsive effects.Compared with the existing related work,the equations and boundary value conditions are gener-alized to a more general form and the existing related conditions are weakened.In Chapter 4,the existence and uniqueness of the maximal(minimal)solution for boundary value problems of fractional p-Laplacian equation are studied by upper and lower solutions method.Firstly,the existence and uniqueness of the maximal(mini-mal)solution for a class of fractional p-Laplacian differential equation under different boundary conditions are studied.By establishing a new comparison principle and using upper and lower solutions and monotone iterative method,the existence and uniqueness of the maximal(minimal)solutions of the above problems are obtained.Secondly,we study the existence of the maximal(minimal)solution for a class of fractional bound-ary value problems with p-Laplacian operators under the condition of reverse upper and lower solutions.By establishing several new comparison principles under reverse upper and lower solutions and using monotone iterative method,the existence of the maxi-mal(minimal)solution for this kind of problem is obtained.For the positive upper and lower solutions,the existing problems with linear differential operators are extended to the boundary value problems with quasi-linear differential operators.For the reverse upper and lower solutions,due to the reverse order of the upper and lower solutions,it is very difficult to establish a comparison theorem based on the reverse upper and lower solutions,so the existence of the maximum(minimal)solution of the fractional boundary value problems with p-Laplacian operators has not been studied so far by the method of the inverse upper and lower solutions.In Chapter 5,the solvability of boundary value problems of fractional p(t)-Laplacian equation with variable index at resonance is studied.Firstly,the existence of solutions for a class of fractional periodic boundary value problems with p(t)-Laplacian operator at resonance is studied.Because the variable exponential operator p(t)-Laplacian is non-linear,Mawhin continuation theorem can not be used directly.Therefore,a new continuation theorem is established in this chapter.On this basis,the existence results of solutions for periodic boundary value problems at resonance are obtained.Secondly,in the case of high-dimensional kernel space,the existence of so-lutions for a class of fractional integral boundary value problems with p(t)-Laplacian operator at resonance is studied.By appropriate transformation,the non-linear p(t)-Laplacian operator equation is transformed into a linear differential operator equation,and then the existence of solutions for integral boundary value problems at resonance is proved by the continuation theorem.Compared with the existing related work,the problem studied is more general,and the corresponding kernel space has higher dimen-sionChapter 6 summarizes the main work of this paper and looks forward to the future research.