The Existence Of Positive Solutions For Boundary Value Problems Of Fractional Differential Equations With Sign-changing Nonlinear Terms

The theory of fractional differential equations is becoming more and more important not only in mathematics but also in physics,chemistry and engineering.So fractional calculus equation has become an important part of differential equation.In particular,many experts and scholars focus on the boundary value problems of fractional differential equations,especially the existence direction of positive solutions.They use Banach contraction mapping principle,Mawhin coincidence degree theory,Leray-Schauder topological degree theory and critical point theory to study the boundary value problems of nonlinear ordinary differential equations in depth.All the above studies are based on the conclusion that the nonlinear term is non-negative.The boundary value problem of differential equation with sign-changing non-linear term is also one of the important research problems.However,up to now,there are few studies on the existence of positive solutions for boundary value problems of fractional differential equations with sign-changing nonlinearity and singularities.In this paper,we will study the following boundary value problems.1)The existence of positive solutions for boundary value problems of fractional differential equations with sign-changing nonlinear terms;2)The existence of positive solutions for boundary value problems of a system of fractional differential equations with sign-changing nonlinear terms.To solve the these problems,first,we use the Green function for a given boundary value problem and convert the differential equation into an equivalent integral equation.Then,under the condition that the nonlinear term f(t,x)or g(t,x)satisfies Caratheodory conditions(that is to say,the non-linear term f(t,x)or g(t,x)is measurable function when the variable x was chosen arbitrarily,and the non-linear term f(t,x)or g(t,x)is continuous function of x when the variable t was fixed).By constructing a suitable Banach space,we use the cone-stretching and compression fixed point theorem and Leray-Schauder nonlinear selection to obtain the sufficient conditions for the existence of positive solutions of boundary value problems.The nonlinear term f(t,x)or g(t,x)of this paper can be singular at any point t in the interval of [0,1].In addition to this,the range of the eigenvalue ?,? which makes the solution of the boundary value problem exist was changed.