Special Issue on Set Valued Analysis 2021

Preface

Set Valued Analysis plays an important role in the study of statistics, biology, economics, social sciences, optimal control, differential inclusions, image reconstruction and fixed point theory. It provides an efficient tool for the modelling and the analysis of dynamic phenomena which evolve from external actions, either under control or in a state of uncertainty. Research in these fields have been carried out during the past decade, and significant results have been obtained in both the qualitative theory and constructive methods.

Within Set Valued Analysis, Fixed Point Theory has been applied, for example, in physical sciences, computing sciences, and engineering. An overview of Nadler, Hadžić, Itoh, and Miheţ extensions of the Fixed Point Theorems for multivalued functions in probabilistic metric spaces is given in [1]. Further investigations into the considered topic are given together with some hints. In [2], the random controllers to stabilize pseudo Riemann–Liouville fractional equations in MB-spaces and the existence and uniqueness of their solutions are investigated. Moreover, the optimum error of the estimate is computed. In order to accomplish this, a fixed point technique derived from the alternative Fixed Point Theorem to investigate random HUR stability of the Riemann–Liouville fractional equations in MB-spaces is used. As an application, a class of random Wright control function is studied and the existence–uniqueness and Wright stability of these equations in MB-spaces is investigated.

Topics such as viability, adaptive control, and stability of uncertain systems are among the main motivations of the development of Differential Inclusions. In [3], the study of perturbation evolution problems involving time-dependent m-accretive operators is considered. Results about the existence of absolutely continuous solutions and BRVC solutions are presented for a specific class of m-accretive operators with convex weakly compact-valued perturbation. Applications to various domains such as relaxation results, second-order evolution inclusions, fractional-order equations coupled with m-accretive operators, and Skorohod differential inclusions are obtained. In [4], a first-order differential inclusion with periodic boundary conditions and Stieltjes derivative with respect to a left-continuous non-decreasing function is studied. The involved set-valued mapping is assumed to have neither compact or convex values, nor to be upper semicontinuous concerning the second argument everywhere. A condition involving the contingent derivative relative to the non-decreasing function is imposed on the set where the upper semicontinuity and the assumption of having compact convex values fail. The solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side. The existence of different solutions is also derived for dynamic inclusions on time scales with periodic boundary conditions. In [5], semilinear integro-differential inclusions in Banach spaces are examined, under the action of infinite impulses. Existence results of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterative sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one, are established. As an example of application, the controllability of a population dynamics process with distributed delay and impulses is obtained.

Finally, in recent years, particular attention has been paid to the study of Multivalued Functions and of Non-Additive Set Functions and nonlinear integrals because they have various applications in the representation of uncertainty, interval-probability, subjective evaluation, decision making, and in the context of fractal image coding and of max-product type operators involving both real and interval/set valued functions. In [6], the max-product neural network operators of the Kantorovich type based on certain linear combinations of the sigmoidal and ReLU activation functions are considered. Inverse approximation problems for the above family of sub-linear operators are faced, establishing their saturation order for a certain class of functions. Furthermore, a local inverse theorem of approximation is proved while, in [7], different properties of the Riemann–Lebesgue integral of a real function with respect to a non-additive set function were established, providing some limit theorems, such as Lebesgue-type and Fatou, for sequences of Riemann–Lebesgue integrable functions. Then, these results are extended to the case of integrable interval-valued functions.

Moreover, having in mind applications to multicriteria decision support, image processing, and fuzzy-ruler-based classification, four different types of convolutions of aggregation functions (the upper, the lower, the super-, and the sub-convolution) have been examined in [8] in the setting of sub-(super)-decomposition integrals defined on a finite space.

We would like to conclude this preface by thanking all reviewers that helped us in the realization of this Special Issue through their expertise and valuable suggestions. A special thanks goes to the Section Managing Editor of Mathematics, Patty Hu, who promoted us to Guest Editors and supported us during the realization process.