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Singular positone and semipositone boundary value problems of nonlinear fractional differential equations. (English) Zbl 1185.34008

Summary: We present some new existence results for the singular positone and semipositone nonlinear fractional boundary value problem

𝐃 0 α u(t)=μa(t)f(t,u(t)),0<t<1,
u(0)=u(1)=u ' (0)=u ' (1)=0,

where μ>0, a, and f are continuous, α(3,4] is a real number, and 𝐃 0+ α is Riemann-Liouville fractional derivative. The nonlinearity may be singular in its dependent variable. Two examples are given to illustrate the main results.

MSC:
34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE