Positive Solutions for a Coupled Nonlinear Fractional Differential System with Coefficients that Change Signs

This paper investigates the existence of positive solutions of the nonlinear fractional differentialsystem{Dsu=λa(t)f(v),0< t <1,Dpv=μb(t)g(u),0< t <1,where 0< s,p <1,Ds,Dpare the standard Riemann-Liouville fractional derivatives,λ,μ >0are parameters. The peculiarity of this coupled equations is the coefficient functionsa(t) andb(t)change signs, unlike the works in the literature keeping the signs ofa(t),b(t) unchanged. On thebasis of a nonlinear alternative of Leray-Schauder type and Krasnoselskii′s in a cone, sufficientconditions ona(t),b(t) guarantee the existence of positive solution of the coupled equations areobtained. The results are illustrated with an example