Solutions To Boundary Value Problems For Fractional Differential Equations

The concept of fractional calculus is from one letter between L'Hospital andLeibniz in 1695. Since then, many famous scholars such as Langrang, Euler,Laplace, Liouvile, Riemann, Caputo and Gru¨nuald showed the di?erent defini-tions and properties of fractional calculus. Especially in recent decades, all sorts offractional dierential equations have resulted from describing physical systems indynamic behavior, biological engineering, dynamic system, control system, signalprocessing and so on. Meanwhile, a lot of attention has been paid to fractionalorder dierential equations (see [5]-[6], [8], [14]-[15], [25], [30]-[32], [35]-[41]).With solving these problems, many important methods and theory such asupper and lower solutions method, fixed point theory, topological degree methodand the theory of cone have been developed gradually. This paper makes use offixed point theorem, cone compression and the approximation method to get theexistence of positive solutions. The dissertation contains two chapters.Chapter 1 investigates the existence, uniqueness and continuous dependence ofsolutions for the following fractional boundary value problemwhere 3 <α≤4, 0 <β≤1, CD0α+ and CD0β+ are the Caputo's fractional derivativesand f : [0, 1]×R×R→R , g : R→R are continuous.In chapter 2, we consider the existence and multiplicity of positive solutions to the fractional singular boundary problemwhereα∈(n-1, n], n≥4 the nonlinearity f(t,y)may be singular at t=0,t=1and y = 0, or may be negative for some values of t and y. The approaches used hereare the approximation method, the fixed point index theory, and a new constructedcone.