Park extended the notion of -distance to the quasimetric framework in order to generalize and unify different versions of Ekeland’s variational principle. For example, it is showed that if a sequence of functions from an intuitionistic fuzzy normed space to another intuitionistic fuzzy normed space is pointwise intuitionistic fuzzy convergent on to a function with respect to, then is pointwise -statistical convergent with respect to the intuitionistic fuzzy norm. For instance, given a continuous -norm, say, if and are two -convex type- fuzzy sets, then the generalized union and the generalized intersection are -convex type- fuzzy sets.Ī notion of -pointwise convergence and also of -statistical uniform convergence in an intuitionistic fuzzy normed spaces is introduced in the paper “ - Statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces.” Some properties of these concepts are proved. Some applications of the main results to the hyperstructure convexity invariance of type- fuzzy sets under hyperalgebra operations and to the convexity invariance of fuzzy numbers under basic arithmetic operations are obtained. One of the main results states that if is a real linear topological space and is a -norm, then a function from into a totally ordered set equipped with the order topology is an inverse -convex transformation if and only if the restriction of to every line segment is monotonic.
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In the paper “ Convexity invariance of fuzzy sets under the extension principles,” Zadeh’s (multivariable) extension principle is applied to convexity invariance of fuzzy sets. Also an Arzela-Ascoli theorem is obtained for the space of -valued continuous functions on a locally compact topological space equipped with the compact-open topology. Among other results, a characterization of compact subsets of is presented and it is shown that the space is a separable Baire space. In particular the authors describe the completion of respect to its natural uniformity, that is, the pointwise uniformity.
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The paper “ On the fuzzy number space with the level convergence topology” is devoted to the study of some uniform and topological properties of the space of fuzzy numbers endowed with the level convergence topology. Chains of -hyperconvex subspaces and the approximation of fixed points are also discussed. If is a bounded -hyperconvex -quasimetric space and if is a nonexpansive self-map of, then the fixed point set Fix ( ) of in is nonempty and -hyperconvex. Among other results, they prove the following quasimetric version of the well-known fixed point theorem of R. In the paper q-hyperconvexity in quasipseudometric spaces and fixed point theorems the authors extend several properties of the theory of hyperconvexity in metric spaces to the asymmetric framework. Applications to the problem of obtaining solutions for Lotka-Volterra system with diffusion, for a model of fractional-order chemical autocatalysis with decay, and for a generalized logistic equation are given. 254–272), where the authors show the existence of a global attractor for this type of reaction-diffusion systems. Kapustyan, published in Journal of Mathematical Analysis and Applications (vol. Antecedents of this research are contained in two papers by J.
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More exactly, a weak comparison principle for a reaction-diffusion system in which the nonlinear term satisfies suitable dissipative and growth conditions, ensuring existence of solutions but not uniqueness, is established. The paper “ A weak comparison principle for reaction-diffusion systems” deals with certain differential equations for which uniqueness of the Cauchy problem fails. This special issue addresses work at the interface between function spaces, hyperspaces, fuzzy structures, and asymmetric topology.
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Many of the impactful and highly cited papers in general topology and its applications have been published in the realm of this relationship. The interplay between fuzzy structures and the theory of hyperspaces and the asymmetric topology is a topic of broad appeal, encouraging new research on such topics and in many cases defining emerging research directions and laying the foundation for future innovations. Hyperspaces, asymmetric topology and fuzzy structures are interdisciplinary topics of increasing interest in the development of the general topology, and its applications in other areas of Mathematics and computer science, such as convex analysis and optimization, fractals and dynamical systems, mathematical economics, and image computing. Zadeh proposed in the sixties his theory of fuzzy sets, this subject became a productive field of research: fuzzy structures not only generalize most of the familiar structures but also they allow obtaining interesting applications in engineering.