Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable—Part II: Extremal Points, Convexity, Periodicity

Luciano Stefanini,Laerte Sorini,Benedetta Amicizia

doi:10.3390/axioms8040114

Abstract

We continue the presentation of new results in the calculus for interval-valued functions of a single real variable. We start here with the results presented in part I of this paper, namely, a general setting of partial orders in the space of compact intervals (in midpoint-radius representation) and basic results on convergence and limits, continuity, gH-differentiability, and monotonicity. We define different types of (local) minimal and maximal points and develop the basic theory for their characterization. We then consider some interesting connections with applied geometry of curves and the convexity of interval-valued functions is introduced and analyzed in detail. Further, the periodicity of interval-valued functions is described and analyzed. Several examples and pictures accompany the presentation.

Highlights

Interval arithmetic and interval analysis initiated with [1,2,3], in relation with the algebraic aspects of how arithmetical operations can be defined for intervals and how to solve problems like algebraic, differential, integral equations or others when intervals are involved
In the first part of this work we have introduced a general framework for ordering intervals, using a comparison index based on gH-difference; the suggested approach includes most of the partial orders proposed in the literature and preserves important desired properties, including that the space of real intervals is a lattice with the bounded property
In this paper, we define concepts related to extremal points, using the same general partial orders described in part I, and we analyze the important concepts of concavity and convexity; we establish the connections between extremality, convexity, and concavity with first-order and second-order gH-derivatives

Summary

Introduction

Interval arithmetic and interval analysis initiated with [1,2,3], in relation with the algebraic aspects of how arithmetical operations can be defined for intervals and how to solve problems like algebraic, differential, integral equations or others when intervals are involved. In the more general setting of fuzzy valued functions (see, e.g., [5,6]), were developed in [7,8,9] Contributions in this area are contained, among others, in the papers on gH-differentiability (see, e.g., [10,11,12,13,14,15]; see [16,17]) and in papers on interval and fuzzy optimization and decision making (e.g, [11,18,19,20,21,22,23,24,25,26]). The multidimensional convex case is analyzed in [27]

Results

Discussion

Conclusion