Over the past two decades, means and mean equations have been popular directions of research in the field of functional equations. This paper first introduces the definition and properties of mean as well as the development of the Gauss iteration, invariant equations, and the M-S problem. It then reviews the Bajraktarević mean, the Cauchy mean, and other related means, especially elaborating on the progress of researching corresponding equality and invariance problems. Both the Bajraktarević mean and the Cauchy mean contain two derivative functions, and the corresponding equation problems have been basically solved. However, the invariance of these two symmetric means, due to the presence of four unknown functions, has not been fully solved. Finally, this paper introduces the applications of means in other fields.
In this paper, by using the noncompact measure, we study the exact controllability of mild solutions for semilinear differential equation when the nonlocal term is Lipschitz continuous in general Banach space. Here, the regularity conditions of noncompact measure assumption in previous literature are not used, but instead we adopt the estimate of noncompact measure in different topological spaces, give the proper control function by the property of quotient space, and obtain the exact controllability of the differential equation. Therefore, our results can be regarded as a Generalization of the controllability field.
In this paper, we consider the
then for any