Banach algebras defined by fractional Mikhlin-type conditions are continuously contained in Besov spaces, in such a way that the difference between the corresponding degrees of derivation can be made arbitrarily small. In this note a proof of this inclusion is given which is based on the Hadamard fractional operator and its adjoint integration operator on the positive half-line. §

By virtue of Γ−convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional p−Laplacian operator, in the singular limit as the nonlocal operator converges to the p−Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.

We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a time-fractional memory term.

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conducti...

Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical a...

Let D denote the Riemann-Liouville fractional differential operator of order α. Let 1 < α < 2 and 0 < β < α. Define the operator L by L = D − aD where a ∈ R. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem Lu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) = 0.