Editorial

2 pages, 155 KiB

Open AccessEditorial

Special Issue on Set Valued Analysis 2021

by Anca Croitoru, Radko Mesiar, Anna Rita Sambucini and Bianca Satco

Mathematics 2022, 10(15), 2703; https://doi.org/10.3390/math10152703 - 30 Jul 2022

Cited by 2 | Viewed by 746

Set Valued Analysis plays an important role in the study of statistics, biology, economics, social sciences, optimal control, differential inclusions, image reconstruction and fixed point theory [...] Full article

(This article belongs to the Special Issue Set-Valued Analysis)

Research

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21 pages, 346 KiB

Open AccessArticle

Some New Extensions of Multivalued Contractions in a b-metric Space and Its Applications

by Reny George and Hemanth Kumar Pathak

Mathematics 2021, 9(1), 12; https://doi.org/10.3390/math9010012 - 23 Dec 2020

Cited by 4 | Viewed by 1238

Abstract

The

H^{β}

-Hausdorff–Pompeiu b-metric for

β \in [0, 1]

is introduced as a new variant of the Hausdorff–Pompeiu b-metric H. Various types of multi-valued

H^{β}

-contractions are introduced and fixed point theorems are proved for such contractions [...] Read more.

The

H^{β}

-Hausdorff–Pompeiu b-metric for

β \in [0, 1]

is introduced as a new variant of the Hausdorff–Pompeiu b-metric H. Various types of multi-valued

H^{β}

-contractions are introduced and fixed point theorems are proved for such contractions in a b-metric space. The multi-valued Nadler contraction, Czervik contraction, q-quasi contraction, Hardy Rogers contraction, weak quasi contraction and Ciric contraction existing in literature are all one or the other type of multi-valued

H^{β}

-contraction but the converse is not necessarily true. Proper examples are given in support of our claim. As applications of our results, we have proved the existence of a unique multi-valued fractal of an iterated multifunction system defined on a b-metric space and an existence theorem of Filippov type for an integral inclusion problem by introducing a generalized norm on the space of selections of the multifunction. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

17 pages, 1407 KiB

Open AccessFeature PaperArticle

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

by Danilo Costarelli, Anca Croitoru, Alina Gavriluţ, Alina Iosif and Anna Rita Sambucini

Mathematics 2020, 8(12), 2250; https://doi.org/10.3390/math8122250 - 20 Dec 2020

Cited by 9 | Viewed by 2293

Abstract

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the

R L

integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the [...] Read more.

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the

R L

integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

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15 pages, 288 KiB

Open AccessFeature PaperArticle

On Regulated Solutions of Impulsive Differential Equations with Variable Times

by Diana Caponetti, Mieczysław Cichoń and Valeria Marraffa

Mathematics 2020, 8(12), 2164; https://doi.org/10.3390/math8122164 - 04 Dec 2020

Cited by 1 | Viewed by 1542

Abstract

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to [...] Read more.

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

23 pages, 321 KiB

Open AccessArticle

Applications of Stieltjes Derivatives to Periodic Boundary Value Inclusions

by Bianca Satco and George Smyrlis

Mathematics 2020, 8(12), 2142; https://doi.org/10.3390/math8122142 - 01 Dec 2020

Cited by 7 | Viewed by 1314

Abstract

In the present paper, we are interested in studying first-order Stieltjes differential inclusions with periodic boundary conditions. Relying on recent results obtained by the authors in the single-valued case, the existence of regulated solutions is obtained via the multivalued Bohnenblust–Karlin fixed-point theorem and [...] Read more.

In the present paper, we are interested in studying first-order Stieltjes differential inclusions with periodic boundary conditions. Relying on recent results obtained by the authors in the single-valued case, the existence of regulated solutions is obtained via the multivalued Bohnenblust–Karlin fixed-point theorem and a result concerning the dependence on the data of the solution set is provided. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

22 pages, 329 KiB

Open AccessArticle

Inequalities in Triangular Norm-Based ∗-fuzzy ( L + ) p Spaces

by Abbas Ghaffari, Reza Saadati and Radko Mesiar

Mathematics 2020, 8(11), 1984; https://doi.org/10.3390/math8111984 - 06 Nov 2020

Cited by 2 | Viewed by 1454

Abstract

In this article, we introduce the ∗-fuzzy

{(L^{+})}^{p}

spaces for

1 \leq p < \infty

on triangular norm-based ∗-fuzzy measure spaces and show that they are complete ∗-fuzzy normed space and investigate some properties in these space. Next, we [...] Read more.

In this article, we introduce the ∗-fuzzy

{(L^{+})}^{p}

spaces for

1 \leq p < \infty

on triangular norm-based ∗-fuzzy measure spaces and show that they are complete ∗-fuzzy normed space and investigate some properties in these space. Next, we prove Chebyshev’s inequality and Hölder’s inequality in ∗-fuzzy

{(L^{+})}^{p}

spaces. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

30 pages, 402 KiB

Open AccessArticle

Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

by Charles Castaing, Christiane Godet-Thobie and Le Xuan Truong

Mathematics 2020, 8(9), 1395; https://doi.org/10.3390/math8091395 - 20 Aug 2020

Cited by 13 | Viewed by 1812

Abstract

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order [...] Read more.

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

14 pages, 329 KiB

Open AccessArticle

Kuelbs–Steadman Spaces for Banach Space-Valued Measures

by Antonio Boccuto, Bipan Hazarika and Hemanta Kalita

Mathematics 2020, 8(6), 1005; https://doi.org/10.3390/math8061005 - 19 Jun 2020

Cited by 5 | Viewed by 1680

Abstract

We introduce Kuelbs–Steadman-type spaces (

K S^{p}

spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of

L^{q}

-type spaces into

K S^{p}

spaces, considering both the [...] Read more.

We introduce Kuelbs–Steadman-type spaces (

K S^{p}

spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of

L^{q}

-type spaces into

K S^{p}

spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

13 pages, 355 KiB

Open AccessArticle

Decompositions of Weakly Compact Valued Integrable Multifunctions

by Luisa Di Piazza and Kazimierz Musiał

Mathematics 2020, 8(6), 863; https://doi.org/10.3390/math8060863 - 26 May 2020

Cited by 8 | Viewed by 1516

Abstract

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition [...] Read more.

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic. Full article

(This article belongs to the Special Issue Set-Valued Analysis)

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