Hukuhara differentiability of interval-valued functions and interval differential equations on time scales

Lupulescu, Vasile

doi:10.1016/j.ins.2013.06.004

Cited by 82 publications

(29 citation statements)

References 45 publications

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“…We now give Riemann's definition of integrability, which is equivalent to the Riemann ∆-integral given in [24,Definition 13]. The following two theorems can be easily verified and so the proofs are omitted.…”

Section: The Interval Darboux and Riemann Delta Integralsmentioning

confidence: 99%

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Some inequalities for interval-valued functions on time scales

Zhao

Liu

et al. 2018

Soft Comput

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We introduce the interval Darboux delta integral (shortly, the ID ∆-integral) and the interval Riemann delta integral (shortly, the IR ∆-integral) for interval-valued functions on time scales. Fundamental properties of ID and IR ∆-integrals and examples are given. Finally, we prove Jensen's, Hölder's and Minkowski's inequalities for the IR ∆-integral. Also, some examples are given to illustrate our theorems.

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Section: The Interval Darboux and Riemann Delta Integralsmentioning

confidence: 99%

“…In 2013, Lupulescu introduced the Riemann ∆-integral for interval-valued functions on time scales and presented some of its basic properties [24]. Nonetheless, to our best knowledge, there is no systematic theory of integration for interval-valued functions on time scales.…”

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confidence: 99%

Some inequalities for interval-valued functions on time scales

Zhao

Liu

et al. 2018

Soft Comput

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“…Markov has introduced nonstandard substraction between two closed intervals( known as Markov difference, ⊖ M ), which has explored calculus of functions with bounded parameters and its application in several areas in recent times (see [1], [2], [9], [15], [16] etc.). Another nonstandard difference, known as gH-difference, in the set of intervals is introduced in [18] and further developed in [4], [7], [17] etc. .…”

Section: Introductionmentioning

confidence: 99%

On Critical Point for Functions with Bounded Parameters

Roy

Panda

2019

2019 IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT)

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Selection of descent direction at a point plays an important role in numerical optimizationfor minimizing a real valued function. In this article, a descent sequence is generated for the functions with bounded parameters to obtain a critical point. First, sufficient condition for the existence of descent direction is studied for this function and then a set of descent directions at a point is determined using linear expansion. Using these results a descent sequence of intervals is generated and critical point is characterized. This theoretical development is justified with numerical example.keywords Line search technique, Markov difference, Interval ordering, Descent direction, Critical point.

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“…Convex optimization on mixed domains have been investigated by Adivar et al [1]. Recently, Luplescu in [12] develop calculus for interval-valued functions on time scales, using the concept of generalized Hukuhara difference provided by Stefanini and Bede [18].…”

Section: Introductionmentioning

confidence: 99%

Convex optimization of interval valued functions on mixed domains

Younus

Nisar

2019

Filomat

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In this paper, we study a class of convex type interval-valued functions on the domain of the product of closed subsets of real numbers. By considering LW order relation on the class of closed intervals, we proposed some optimal solutions. LW convexity concepts and generalized Hukuhara differentiability (viz. delta and nabla) for interval-valued functions yield the necessary and sufficient conditions for interval programming problem. In addition, we compare our results with the results given in the literature. These results may open a new avenue for modeling and solve a different type of optimization problems that involve both discrete and continuous variables at the same time. Definitions, notations and prerequisitesFor a self-contained presentation of our study, we recall briefly the necessary background material.

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Hukuhara differentiability of interval-valued functions and interval differential equations on time scales

Cited by 82 publications

References 45 publications

Some inequalities for interval-valued functions on time scales

Some inequalities for interval-valued functions on time scales

On Critical Point for Functions with Bounded Parameters

Convex optimization of interval valued functions on mixed domains

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