A relation-theoretic set-valued version of Prešić-Ćirić theorem and applications

A general concept of majority rule

Hopf's theorem has been recently extended to compact genus zero surfaces with constant mean curvature H in a product space , where is a surface with constant Gaussian curvature k ≠ 0 [Abresch, Rosenberg, Acta Math. 193: 141–174, 2004]. It also has been observed that, rather than H = const., it suffices to assume that the differential dH of H is appropriately bounded [Alencar, do Carmo, Tribuzy, Analysis Geometry 15: 283–298, 2007]. Here, we consider the case of simply-connected open surfaces with boundary in such that dH is appropriately bounded and certain conditions on the boundary are satisfied, and show that such surfaces can all be described.

Communicating between information systems

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper.

In this study, we consider a non-linear matrix equation of the formwhere Q is a Hermitian positive definite matrix, A * i stands for the conjugate transpose of an n n matrix A i and F j are order-preserving continuous mappings from the set of all Hermitian matrices to the set of all positive definite matrices such that F (O) = O.We discuss sufficient conditions that ensure the existence of a unique positive definite solution of the given matrix equation.For this, we derive some fixed point results for Suzuki-implicit type mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order).We provide adequate examples to validate the fixed-point results and the importance of related work, and the convergence analysis of non-linear matrix equations.

Preference fusion when the number of alternatives exceeds two: indirect scoring procedures

The aim of this paper is to introduce a concept of quasiconvexity for set-valued maps in a general framework, by only considering an abstract convexity structure in the domain and an arbitrary binary relation in the codomain. It is shown that this concept can be characterized in terms of usual quasiconvexity of certain real-valued functions. In particular, we focus on cone-quasiconvex set-valued maps with values in a partially ordered vector space.

The Elementary Theory of Well-Odering—A Metamathematical Study—

We consider methods for aggregating preferences based on discrete optimization. The preferences are represented by arbitrary binary relations (possibly weighted) or matrices of paired comparisons (possibly incomplete). The case of incomplete preferences remains practically unexplored so far. We examine properties of several known methods and propose one new method. Some results are established that characterize solutions of the related optimization problems. Necessary conditions of a new axiom called Self-Consistent Monotonicity are proved. The generalized row sum method is shown to satisfy Self-Consistent Monotonicity. The results suggest that there are general limitations of the discrete optimization approach to preference aggregation.

We extend the notion of generalized weakly contraction mappings due to Choudhury et al. (2011) to generalized α-β-weakly contraction mappings. We show with examples that our new class of mappings is a real generalization of several known classes of mappings. We also establish fixed point results for such mappings in metric spaces. Applying our new results, we obtain fixed point results on ordinary metric spaces, metric spaces endowed with an arbitrary binary relation, and metric spaces endowed with graph.

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of j -adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estevez and Herves will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.

This paper presents a probabilistic version of generalized rough set models. It generalizes the standard algebraic and probabilistic rough set models in two aspects. An arbitrary binary relation is used instead of an equivalence relation. A probability function on the universe is used instead of computing probabilities from the cardinality of sets. Fundamental issues related to probabilistic rough set models are examined.

Set-valued versions of Ky Fan's inequality with application to variational inclusion theory

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.