The Existence Of Solutions To Several Types Of Nonlinear Fractional Boundary Value Problems

This paper studies the existence of solutions for fractional-order q-difference equation and differential equation.The paper consists of three chapters.The first chapter is an intro-duction,which briefly describes the background significance and research status of fractional q-difference equation and differential equation.The second chapter studies the fractional singular q-difference equation.The main results of this chapter are divided into two parts.In the first part,we investigate the following singular fractional q-difference equation with nonlinear term: where 0<q<1,n-1<??n,n?3,??0,f?C([0,1]x[0,+?),[0,+?)),p may be singular at t=0,1.Dq? is the fractional q-derivative of the Riemann-Liouville type.By applying the fixed point theory in cones,the existence of solution for the fractional q-difference equation(2.1.1)can be obtained.In the second part,considering the possibility of f(t,x)is singular at t=0,1 and x=0,we study the following equation: where 0<q<1,n-1<??n,n?3,??0,f?C((0,1)×(0,+?),[0,+?)),Dq?is the fractional q-derivative of the Riemann-Liouville type,f(t,x)may be singular at t=0,1 and x=0.By applying the cone compression and expansion fixed point theorem,the existence of at least one positive solution for the singular fractional q-difference equation can be gained.In the third chapter,we investigate the existence of positive solution of the fractional boundary value problem of the form where 3<?<4,?>1,D0+?is the Riemann-Liouville fractional derivative,I0+?is the Riemann-Liouville fractional integral,f:[0,1]×[0,?)×[0,?)?[0,?),g:[0,?)?(-?,0]are continuous functions.By using the mixed monotone operator method,we obtain the existence and uniqueness of positive solutions.