We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem. The crucial ingredients of our proofs are the understanding of the blow up profiles and a Liouville theorem.

MSC 2010: 26A33, 33E12, 33C60, 35R11In this paper we derive an analytic solution for the fractional Helmholtz equation in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Leffler function. In all three cases the solutions are represented also in terms of Fox's H-function.

This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

We are concerned with the problem of differentiability of the derivatives of order $m+1$ of solutions to the “nonlinear basic systems” of the type $${(-1)}^m\sum _{\left|\alpha \right|=m}{D}^{\alpha }{A}^{\alpha }\left(D^{\left(m\right)}u\right)+\frac{\partial u}{\partial t}=0\text{in}Q.$$ We are able to show that $$D^{\alpha }u\in L^2(-a,0,H^{\vartheta }(B\left(\sigma \right),{ℝ}^N)),|\alpha |=m+1,$$ for $\vartheta \in (0,1/2)$ and this result suggests that more regularity is not expectable.

In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal...