In this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

In this paper we obtain results on the existence and uniqueness of a solution to a fractional nonlinear Cauchy problem containing the Riemann-Liouville derivative, in a fractional counterpart of the set of ℝn-valued absolutely continuous functions. We also derive a Cauchy formula for the solution to the linear problem of such a type.

Using Riemann-Liouville derivatives, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of some imbeddings. An application to boundary value problems is given as well.

In the paper, we derive results concerning a continuous dependence of solutions on the right-hand side for a semilinear operator equation Lu={g(u), by assuming that L : D(L)/H Ä H (H&a Hilbert space) is self-adjont, with a closed range, and g: H Ä R is continuous convex on H and Ga^teaux differentiable on D(L). Using these results, we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order u +au= D u F(t, u, |), t # [0, ?] a.e., with the Dirichlet boundary conditions u(0)= u(?)=0, where a 1, and | is a functional parameter. Academic Press

No abstract