Interval-valued Functions Research Articles

Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond–Weir-type weak and strong duality theorems, and illustrate our results with an example.

Fréchet and Gateaux gH-differentiability for interval valued functions of multiple variables

In this work we extend concepts of differentiability for Interval-Valued functions (IV-functions) of multiple variables, based on generalized Hukuhara gH-difference. In this context, we introduce a new concept of gH-linearity and characterize the class of gH-linear IV-functions, which are fundamental for a general approach to Fréchet-type and Gateaux-type gH-differentiability of the first order. We moreover consider vector IV-functions and outline the definition of the gH-Jacobian; by representing intervals and IV-functions in midpoint-radius notation, we establish properties and relations between pointwise, Fréchet and Gateaux gH-differentiability. Finally, higher-order differentiability and gH-Hessian matrix are considered. These intuitive concepts are mathematically and computationally easy to work with.

NEW FRACTIONAL INTEGRAL INEQUALITIES FOR EXPONENTIALLY (h,m)-PREINVEX INTERVAL-VALUED FUNCTIONS

We introduce the concept of exponentially [Formula: see text]-preinvex interval-valued functions (EP-([Formula: see text])-IVFs) by using the pseudo-order relation, and derive some interesting properties. Based on such a definition, we present some new Hermite–Hadamard and Hermite–Hadamard–Fejér-type inequalities for [Formula: see text]-Riemann–Liouville fractional integrals. These inequalities not only generalize some existing research findings in the literature, but also provide a new direction for the further study of interval integral estimation. Additionally, graphical representations and numerical estimates of some examples are given to illustrate the validity of the main results.

Characterization results of generalized differentiabilities of fuzzy functions

This paper presents new properties of the so-called generalized derivatives of fuzzy and interval-valued functions, which consist of the gH, gH⁎, and g-derivatives of these functions. Several recent properties concerning the generalized derivatives of interval-valued functions are extended to the fuzzy case, for which general characterization results are provided. Lastly, in order to illustrate the presented results, a study on the Malthusian growth and decay models considering fuzzy populations is made.

Conditions for chainability of inverse limits on [0,1] with interval-valued functions

For an inverse sequence on [0,1] with interval-valued functions, we establish necessary conditions on the bonding functions for chainability of the inverse limit space. We also characterize chainability of the inverse limit in this setting in terms of properties of the bonding functions fi and the induced functions Fn:[0,1]→G′(f1,…,fn−1). The properties, in both cases, are related to how triods arise in the partial graphs associated with the inverse sequence when each graph G(fi) is chainable.

On conditional monotonicities of interval-valued functions

On conditional monotonicities of interval-valued functions

Fractional calculus of interval-valued functions on time scales

This paper aims to develop fractional interval-valued calculus on time scales that unify the continuous and discrete cases. The definitions of fractional interval-valued calculus, which encompass the integral, Riemann–Liouville (R-L) derivative, and Caputo derivative of nabla type, are established within the framework of time scales. Moreover, some fundamental properties and nabla Laplace transform of these new operators are discussed. Based on these findings, explicit solutions of several nabla fractional interval-valued differential equations on time scales are presented.

On Semi-Infinite Optimization Problems with Vanishing Constraints Involving Interval-Valued Functions

In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. We establish these theorems by creating duality models, which establish a relationship between SIVOPVC and its corresponding dual models, assuming generalized convexity conditions. Some examples are also given to illustrate the results.

Interval-based KKT framework for support vector machines and beyond

Our article proves inequalities for interval optimization and shows that feasible and descent directions do not intersect in constrained cases. Mainly, we establish some new interval inequalities for interval-valued functions by defining LC-partial order. We use LC-partial order to study Karush–Kuhn–Tucker (KKT) conditions and expands Gordan's theorems for interval linear inequality systems. By applying Gordan's theorem, we can determine the best outcomes for interval optimization problems (IOPs) that have constraints, such as Fritz John and KKT conditions. The optimality conditions are observed with inclusion relations rather than equality. We can use the KKT condition for binary classification with interval data and support vector machines(SVMs). We present some examples to illustrate our results.

New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.

NEW FRACTIONAL INTEGRAL INEQUALITIES FOR LR-ℏ-PREINVEX INTERVAL-VALUED FUNCTIONS

Based on the pseudo-order relation, we introduce the concept of left and right [Formula: see text]-preinvex interval-valued functions (LR-[Formula: see text]-PIVFs). Further, we establish the Hermite–Hadamard and Hermite–Hadamard–Fejér-type estimates for LR-[Formula: see text]-PIVFs using generalized fractional integrals. Finally, an example of interval-valued fractional integrals is provided to illustrate the validity of the results derived herein. Our results not only extend some existing inequalities for Hadamard, Riemann–Liouville, and Katugampola fractional integrals, but also provide new insights for future research on generalized convexity and IVFs, among others.

Generalized Hukuhara weak subdifferential and its application on identifying optimality conditions for nonsmooth interval-valued functions

Generalized Hukuhara weak subdifferential and its application on identifying optimality conditions for nonsmooth interval-valued functions

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

<abstract><p>Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.</p></abstract>

HERMITE–HADAMARD TYPE INEQUALITIES FOR ℏ-CONVEX FUNCTION VIA FUZZY INTERVAL-VALUED FRACTIONAL q-INTEGRAL

Fractional [Formula: see text]-calculus is considered to be the fractional analogs of [Formula: see text]-calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) [Formula: see text]-integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving [Formula: see text]-convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF [Formula: see text]-integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided.

Novel fractional inequalities measured by Prabhakar fuzzy fractional operators pertaining to fuzzy convexities and preinvexities

<abstract><p>In this article, we implemented the idea of a fuzzy interval-valued function with the well-known generalized fuzzy fractional operators, associated with different types of convexities and preinvexities. We developed the Prabhakar fuzzy fractional operators using the fuzzy interval-valued function. We presented the novel extensions of Hermite-Hadamard fuzzy-type and trapezoidal fuzzy-type inequalities, based on the $ h $-Godunova-Levin convex and $ h $-Godunova preinvex fuzzy interval-valued functions.</p></abstract>

Generalized Hukuhara Hadamard derivative of interval-valued functions and its applications to interval optimization

Generalized Hukuhara Hadamard derivative of interval-valued functions and its applications to interval optimization

A New Fuzzy Joint Choquet Integral Method Under Interval-Valued Function

A new fuzzy group decision-making method considering multi-attributes correlation under interval-valued function is presented, which mainly includes (1) acquiring the group fuzzy preference matrix and (2) handling the interactions between multiple evaluation attributes. To do that, firstly, the fuzzy joint Choquet integral based on an interval-valued function is proposed, which not only reflects the interaction between multiple attributes in a complex and uncertain environment, but also retains the initial preference of the decision maker. Secondly, a Shapley value with fuzzy measure is applied to assign each decision maker's weight, and the fuzzy group preference matrix is acquired by fusing the fuzzy preference matrices of all decision makers. Finally, a nursing home selection case is depicted to explain the effectiveness of the proposed technique. The corresponding sensitivity analysis is operated, which clarifies the reliability and flexibility of the proposed technique.

New quantum integral inequalities for left and right log-ℏ-convex interval-valued functions

Abstract We introduce the concept of quantum integration for interval-valued functions and establish new q-Hermite–Hadamard and q-Hermite–Hadamard–Fejér inequalities for left and right log ⁢ - ⁢ h {\mathrm{log}\text{-}h} -convex interval-valued functions. Our results generalize the known ones in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. We illustrate our results with examples.

Second-order optimality conditions for interval-valued functions

This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of second-order optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the first-order conditions. We will define new concepts such as second-order gH-derivative for interval-valued functions, 2-critical points, and 2-KKT-critical points. We obtain and present new types of interval-valued functions, such as 2-pseudoinvex, characterized by the property that all their second-order stationary points are global minima. We extend the optimality criteria to the semi-infinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with interval-valued functions.

Ostrowski and Čebyšev type inequalities for interval-valued functions and applications.

As an essential part of classical analysis, Ostrowski and Čebyšev type inequalities have recently attracted considerable attention. Due to its universality, the non-additive integral inequality takes several forms, including Sugeno integrals, Choquet integrals, and pseudo-integrals. Set-valued analysis, a well-known generalization of classical analysis, is frequently employed in studying mathematical economics, control theory, etc. Inspired by pioneering work on interval-valued inequalities, this paper establishes specific Ostrowski and Čebyšev type inequalities for interval-valued functions. Moreover, the error estimation to quadrature rules is presented as some applications for illustrating our results. In addition, illustrative examples are offered to demonstrate the applicability of the mathematical methods presented.