Fuzzy Mappings Research Articles

Generalized nonlinear implicit quasi-variational inclusions with fuzzy mappings

In this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings. An existence theorem of solutions is proved without compactness assumptions. A new iterative algorithm is suggested and analysed. The convergence of the iterative sequence generated by the new algorithm is also given. As special cases, some known results are also discussed.

Generalization of preinvex and B-vex fuzzy mappings

A class of fuzzy mappings, called B-preinvex, and strictly B-preinvex fuzzy mappings, is introduced by relaxing the definition of preinvexity of a fuzzy mapping. Similarly, based on a notion of differentiability of fuzzy mappings different from the usual one, the class of pseudo-B-vex, B-invex, and pseudo-B-invex fuzzy mappings is defined as a generalization of pseudo-convex, invex, and pseudo-invex fuzzy mappings. We prove that a strictly B-vex, or B-preinvex fuzzy mapping has at most one global minimum point, and that the class of B-vex fuzzy mappings forms a subset of the class of quasi-convex fuzzy mappings. B-vex (resp. B-preinvex) fuzzy mappings satisfy most of the basic properties of convex (resp. preinvex) fuzzy mappings. In addition characterizations and sufficient optimality conditions are obtained for B-vex and B-preinvex fuzzy mappings.

Common fixed point theorems for a countable family of fuzzy mappings

In this paper we prove fixed point theorems for countable families of fuzzy mappings satisfying contractive-type conditions and a rational inequality in left K-sequentially complete quasi-pseudo-metric spaces. These results generalize the corresponding ones obtained by others authors.

The w-derivatives of fuzzy mappings in Banach spaces

This paper introduces w-derivatives of set-valued mappings and fuzzy mappings, gives some w-derivability properties of such mappings, also shows the relations between the w-derivatives and the weak integrals.

Fuzzy mapping of momentum fluxes in complex shear flows with limited data

This study is divided into three parts centred around modelling the complex turbulent fluxes across strong shear layers, such as exist between the channel and floodplain flow in an over bank flood flow. The three stages utilize Adaptive Neuro-Fuzzy Inference Systems (ANFIS) to make fuzzy mappings between the fluxes and different data types. The de-fuzzification stage commonly used in Fuzzy Inference Systems is adapted to avoid the generation of crisp outputs, a process which tends to hide the underlying uncertainty implicit in the fuzzy relationship. Each stage of the study utilizes conditioning data that makes the fuzzy mappings more tenuously linked with what would normally be considered physically based relationships. The need to make such mappings in distributed models of complex systems, such as flood models, stems from the sparsity of available distributed information (e.g. roughness) with which to condition the models. If patterns in distributed observables which clearly affect, or are affected by, the river hydraulics can be linked to the local fluxes, then the conditioning of the model would improve. Mappings such as these often suffer from scaling effects, an issue addressed here through training the fuzzy rules on the basis of both laboratory and field collected data.

Differentiability and convexity of fuzzy mappings

Goetschel and Voxman [1] have introduced the notion of a derivative for fuzzy mappings of one variable in a manner different from the usual one. In this paper, we define a differentiable fuzzy mapping of several variables in ways that parallel the definition, proposed by Goetschel and Voxman [1], for a fuzzy mapping of one variable, and then study some basic differentiability properties of fuzzy mappings from the standpoint of convex analysis.

Invex and generalized convex fuzzy mappings

In this paper we introduce the concepts of pseudo-convexity, invexity and pseudo-invexity for fuzzy mappings of one variable based on the notion of differentiability proposed by Goetschel and Voxman [4], and investigate the relationship between convex fuzzy mappings, preinvex fuzzy mappings and these classes of fuzzy mappings. We shall prove that pseudo-convex fuzzy mappings and invex fuzzy mappings are pseudo-invex, and that a differentiable convex (resp. preinvex) fuzzy mapping is pseudo-convex (resp. invex). In addition sufficient optimality conditions are obtained for pseudo-convex, invex, and pseudo-invex fuzzy mappings.

Iterative algorithm for finding approximate solutions to completely generalized strongly quasivariational inequalities for fuzzy mappings

In this paper, we construct a new iterative algorithm, which includes many known algorithms as special cases to solve variational inequalities and quasivariational inequalities for fuzzy mappings. Further, we prove the convergence of the iterative sequences generated by this algorithm. Our results are the extension and improvements of the earlier and recent results in this field.

Fixed point theorems for fuzzy mappings in quasi-metric spaces

We obtain fixed point theorems for fuzzy mappings in Smyth-complete and left K-complete quasi-metric spaces, respectively. Well-known theorems are special cases of our results.

A perturbed algorithm of variational inclusions for fuzzy mappings

In this paper, we introduce a class of variational inclusions for fuzzy mappings and show its equivalence with a class of fixed point problems. Using this equivalence, we develop the perturbed iterative algorithms for this class of variational inclusions. Further, we prove the existence of solutions for the class of variational inclusions and discuss the convergence criteria for the perturbed algorithms.

M-convex fuzzy mappings and fuzzy integral mean

We introduce the notions of m-convex fuzzy mapping and fuzzy integral mean. We study their properties and we give some applications.

Generalized quasi-variational inequalities for fuzzy mappings

In this paper, we use the projection technique to prove that generalized quasi-variational inequality for fuzzy mappings are equivalent to the implicit Wiener–Hopf equations for fuzzy mappings. This equivalence is used to analyze a number of iterative algorithms for solving variational inequalities for fuzzy mappings. We also study the convergence of these iterative algorithms.

Generalization of shadows and fixed point theorems for fuzzy sets

In this paper, we define a fuzzy set concerning shadows without assuming the existence of the metric projection and extend a well-known result in a Hilbert space to that in a normed linear space. We shall observe that the fuzzy set presented here is in essence a generalization of shadows. Further, we study fuzzy mappings in the framework of fixed point theory. We first prove two theorems extending Nadler's fixed point theorem. Lots of fixed point theorems for fuzzy mappings have been proved, however, the method used here is completely different from those in the previous results. Next, we prove a theorem by applying Fan's fixed point theorem and conversely, we show that this is a generalization of this theorem.

Investigating a relevance of fuzzy mappings

The study introduces a concept of relevance of fuzzy mappings regarded as fundamental constructs of granular computing and rule-based systems, in particular. The notion of relevance of the fuzzy mappings is instrumental in the quantification of the quality of such mappings prior to their detailed construction. For the purposes of such quantification, we introduce shadowed sets and discuss as an algorithmic framework to be instrumental in expressing and quantifying the property of relevance of the fuzzy mappings. It is revealed that shadowed sets provide an interesting three-valued quantification of this property (such as acceptable mapping, marginal mapping, and a lack of mapping). The paper includes a number of detailed calculations concerning two commonly exploited classes of triangular and Gaussian fuzzy sets. Numerical studies are discussed as well.

Existence of fixed points of fuzzy mappings via theory of fuzzy differential equations

Recently, the study of fuzzy differential equations has gained importance due to its application. In this paper we prove the existence of a unique solution for fuzzy initial value problems. Further, employing stability theory and contraction mapping principle, we obtain fixed points of fuzzy nonlinear mappings.

Completely generalized nonlinear variational inclusions for fuzzy mappings

In this paper, we introduce and study a new class of completely generalized nonlinear variational inclusions for fuzzy mappings and construct some new iterative algorithms. We prove the existence of solutions for this kind of completely generalized nonlinear variational inclusions and the convergence of iterative sequences generated by the algorithms.

Nonlinear equations for fuzzy mappings in probabilistic normed spaces

In this paper, we introduce the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the existence of solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic normed spaces.

Fixed point theorems for fuzzy mappings

In this paper, we obtain the common fixed points of fuzzy mappings defined on a metric linear space.

Completely generalized strongly quasivariational inequalities for fuzzy mappings

In this paper, we prove an existence theorem of solutions and develop the algorithms of approximating solutions for a completely generalized strongly variational inequality problem for fuzzy mappings and a completely generalized strongly complementarity problem for fuzzy mappings by using the projection method.

Variational inequalities for fuzzy mappings (III)

In this paper, we establish the equivalence between the variational inequalities for fuzzy mappings and the resolvent equations for fuzzy mappings. We use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings. We also study the convergence criteria of these iterative algorithms. The auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution. The results proved in this paper represent an improvement of previously known results.