Boundaries of semilinear spaces and semialgebras

Worth, R. E.

doi:10.1090/s0002-9947-1970-0273405-5

Cited by 15 publications

(3 citation statements)

References 14 publications

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“…We provide the following definition for a semi-linear space [can see in Galanis ( 2009 ); Phu et al. ( 2019 ); Worth ( 1970 )] with scalar field F , which is the real number field …”

Section: Preliminariesmentioning

confidence: 99%

“…(Galanis 2009 ; Phu et al. 2019 ; Worth 1970 ) A semi-linear space is a set B together two operations, addition and scalar multiplication with nonnegative reals, are defined: The first operation: addition, denoted by , such that to every pair there correspond a element , The second operation: scalar multiplication of with an element , denoted by such that satisfy the following properties for every and : with called a neutral element of B ; where is addition of the field and with and are identity element and neutral element of scalar multiplication, respectively. …”

Section: Preliminariesmentioning

confidence: 99%

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Limit properties in the metric semi-linear space of picture fuzzy numbers

et al. 2022

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The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338–353, 1965). The picture fuzzy number (PFN) is an ordered value triple, including a membership degree, a neutral-membership degree, a non-membership degree, of a PFS. The PFN is a useful tool to study the problems that have uncertain information in real life. In this paper, the main aim is to develop basic foundations that can become tools for future research related to PFN and picture fuzzy calculus. We first establish a semi-linear space for PFNs by providing two new definitions of two basic operations, addition and scalar multiplication, such that the set of PFNs together with these two operations can form a semi-linear space. Moreover, we also provide some important properties and concepts such as metrics, order relations between two PFNs, geometric difference, multiplication of two PFNs. Next, we introduce picture fuzzy functions with a real domain that is also known as picture fuzzy functions with time-varying values, called geometric picture fuzzy function (GPFFs). In this framework, we give definitions about the limit of GPFFs and sequences of PFN. The important limit properties are also presented in detail. Finally, we prove that the metric semi-linear space of PFNs is complete, which is an important property in the classical mathematical analysis.

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“…We provide the following definition for a semi-linear space [can see in Galanis ( 2009 ); Phu et al. ( 2019 ); Worth ( 1970 )] with scalar field F , which is the real number field …”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Limit properties in the metric semi-linear space of picture fuzzy numbers

et al. 2022

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“…In this paper, we assume that the H-difference always exists and for i = 1, 2, define E i c as a space of fuzzy numbers ∈ E with H-continuous. According to [29,31,33], fuzzy number spaces E and E 1 c are semilinear spaces having the cancellation property. i.e., for any ζ, θ, φ ∈ S, where S is a semilinear space, if ζ + θ = φ + θ, then ζ = φ.…”

Section: Preliminariesmentioning

confidence: 99%

Generalized Hukuhara Weak Solutions for a Class of Coupled Systems of Fuzzy Fractional Order Partial Differential Equations without Lipschitz Conditions

Zhang

Lan

2022

Mathematics

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As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research.

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An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty

Long

Dong

2018

J. Fixed Point Theory Appl.

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Boundaries of semilinear spaces and semialgebras

Cited by 15 publications

References 14 publications

Limit properties in the metric semi-linear space of picture fuzzy numbers

Limit properties in the metric semi-linear space of picture fuzzy numbers

Generalized Hukuhara Weak Solutions for a Class of Coupled Systems of Fuzzy Fractional Order Partial Differential Equations without Lipschitz Conditions

An extension of Krasnoselskii’s fixed point theorem and its application to nonlocal problems for implicit fractional differential systems with uncertainty

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