On Fréchet and Gateaux derivatives for interval and fuzzy-valued functions in the setting of gH-differentiability (original) (raw)

GATEAUX and FRÉCHET DERIVATIVE IN INTUITIONISTIC FUZZY NORMED LINEAR SPACES

New Mathematics and Natural Computation, 2012

In this paper, we introduce intuitionistic fuzzy derivative, intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative and some of their properties are studied. The relations between intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative are studied.

Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

Nonlinear Analysis: Theory, Methods & Applications, 2009

In the present paper we introduce and study a generalization of the Hukuhara di¤erence and also generalizations of the Hukuhara di¤erentiability to the case of interval valued functions. We consider several possible de…nitions for the derivative of an interval valued function and we study connections between them and their properties. Using these concepts we study interval di¤erential equations. Local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval di¤erential equation by ODE systems and by di¤erential algebraic equations. We also show some connection with di¤erential inclusions. The thoretical results are turned into practical algorithms to solve interval di¤erential equations.

Single-Level Differentiability for Interval-valued Functions

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

This study uses the theory of single-level difference for interval-valued functions to propose the concept of single-level differentiability, illustrate its calculations, and investigate how its single-level derivative (SL-derivative) relates to other mathematical derivatives.

New Results in the Calculus of Fuzzy-Valued Functions Using Mid-Point Representations

Information Processing and Management of Uncertainty in Knowledge-Based Systems

We present new results in the calculus for fuzzy-valued functions of a single real variable. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in fuzzy calculus. Concepts related to convergence and limits, continuity, level-wise gH-differentiability of first and second orders have nice and useful midpoint expressions. Using mid-point representation of fuzzy-valued functions, partial orders and properties of monotonicity and convexity are discussed and analysed in detail. Periodicity is easy to represent and identify. Graphical examples and pictures accompany the presentation.

Analysis and Computation of Fuzzy Differential Equations via Interval Differential Equations with a Generalized Hukuhara-type Differentiability

One of the most efficient ways to model the propagation of epistemic uncertainties (in dynamical environments/systems) encountered in applied sciences, engineering and even social sciences is to employ Fuzzy Differential Equations (FDEs). The FDEs are special type of Interval Differential Equations (IDEs). The IDEs are differential equations used to handle interval uncertainty that appears in many mathematical or computer models. The concept of generalized Hukuhara (gH) differentiability shall be applied in analyzing such equations. We further apply a highly efficient computational method to approximate the solution of some modeled FDEs. The results obtained clearly showed that the method adopted in the research is efficient and computationally reliable.

On the space of Type-2 interval with limit, continuity and differentiability of Type-2 interval-valued functions

arXiv: Functional Analysis, 2019

This paper deals with the new concept of interval whose both the bounds themselves are also intervals. We name this new type of interval as Type-2 interval. Here we have introduced Type-2 interval-valued function and its properties. To serve this purpose, we have defined a distance on the set of all Type-2 intervals, named as extended Moore distance for Type-2 intervals which is a metric on the set of all Type-2 intervals. Then we have shown that the space of Type-2 interval is a complete metric space with respect to extended Moore distance. Then we have introduced the concept of limit-continuity for Type-2 interval-valued function of single variable and also, we have derived some elementary properties of this concept. Subsequently, we have presented the idea of generalised Hukuhara difference on the set of Type-2 intervals. Finally, using this difference, we have defined gH-differentiability of Type-2 interval-valued function and discussed some of its properties.

Generalized Differentiability of Continuous Functions

2020

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

Generalized fuzzy differentiability with LU-parametric representation

Fuzzy Sets and Systems, 2014

In the present paper, we use a new generalization of the Hukuhara di¤erence and derivative for fuzzy-valued functions, and we study several properties of the new concepts in the setting of the LU-parametric representation of fuzzy numbers, assessed both from theoretical and computational points of view.