Limit properties of positive solutions of fractional boundary value problems

We investigate the sequence of fractional boundary value problemscDαnu=∑k=1mak(t)cDμk,nu+f(t,u,u′,cDβnu),u′(0)=0,u(1)=Φ(u)-Λ(u′),where limn→∞αn=2,limn→∞βn=1, limn→∞μk,n=1,ak∈C[0,1] (k=1,2,…,m), f∈C([0,1]×D),D⊂R3, and Φ,Λ:C[0,1]→R are linear functionals. cD is the Caputo fractional derivative. It is proved, by the Leray-Schauder degree theory, that for each n∈N the problem has a positive solution un, and that there exists a subsequence {un′} of {un} converging to a positive solution of the differential boundary value problemu″=u′∑k=1mak(t)+f(t,u,u′,u′),u′(0)=0,u(1)=Φ(u)-Λ(u′).