The Existence Of Positive Solutions For Some Semipositone Fractional Differential Equations

Fractional differential equations attract the attention of many scholars because of its wide application in physics, chemistry and other scientific fields. In recent years, there are many papers which deal with the existence and multiplicity of solution of nonlinear fractional differential equations (see [1-34] and the references therein).In this thesis, we study the existence of positive solutions for a few of semipositone fractional differential equations with boundary conditions. By means of Banach contraction mapping principle, Guo-Krasnoselskii’s fixed point theorem in a cone, fixed point index and the Banach fixed point theorem, the existence of positive solutions is obtained. This thesis contains four chapters:In chapter1, we present here the necessary definitions, lemmas and theorems from fractional calculus theory and then give a number of results about fixed point index theory and fixed point theorem for the existence of solution.In chapter2, we obtain the existence and multiplicity results of positive solutions for the following nonlinear fractional differential equations by the Banach contraction mapping principle and some fixed point theorems where n-1<α≤n,i∈N and i≤n-2, α>2, λ<αΔ Dα+is the standard Riemann-Liouville derivative.In chapter3, we are concerned with the existence of positive solutions for the following singular semipositone system of fractional differential equations with integral boundary value problem by the means of the Green’s function and the fixed point index theorem where n-1<α≤n,i∈N andi≤n-2,α≥2,λk<aΔ, Dα0+is the Riemann-Liouville derivative, R+=[0,+∞.),R-=(-∞,0], fk∈C((0,1)×R+×R+,R),fk(k=1,2)may be singular at t=0and/or t=1and may take negative values.In chapter4,we consider the existence of positive solutions of the following nonlinear fractional differential equation boundary value problem with changing sign nonlinearity i=1where λ>0,α≥2,n-1<α≤n,i∈N and1≤i≤n-2,,ηj≥0(i=1,2…,m-2),0<ζ1<ζ2<…<ζm-2<1,Δ-(α-1)m-2∑ηjζjα-2>0Dα0+is the Riemann-Liouville derivative,f may be singular at t=0and/or t=1and may take negative values.