The existence theory of solution in Sobolev space for fractional differential equations

Abstract

We study the solution theory in Sobolev space for fractional differential equations of Caputo type: CDxαu(x)=f(x,u),0<α<1, with given initial value. Under proper conditions, we establish the existence theorem of solution in space Wp1(0,b). We emphasize that our existence theorem has no any continuity requirement on f(x,u) with respect to x∈[0,b]. The uniqueness condition is also given. Several examples are provided to support our theoretical analysis.