Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

Abstract

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problemDtαu(t)=Au(t)+f(t),  t>0, where0<α<1.WhenAis the generator of aC0-semigroup(T(t))t≥0on a Banach spaceX, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary conditionu(0)=u(1)admits a unique mild solution for eachf∈C([0,1];X)if and only if the operatorI-Sα(1)is invertible. Here, we use the representationSα(t)x=∫0∞‍Φα(s)T(stα)x ds,  t>0in whichΦαis a Wright type function. For the first order case, that is,α=1, the corresponding result was proved by Prüss in 1984. In caseXis a Banach lattice and the semigroup(T(t))t≥0is positive, we obtain existence of solutions of the semilinear problemDtαu(t)=Au(t)+f(t,u(t)),t>0,0<α<1.