On the semigroup approach to the interval-valued differential evolution equations
- PDF / 450,567 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 79 Downloads / 163 Views
On the semigroup approach to the interval-valued differential evolution equations Nguyen Thi Kim Son1,2 · Hoang Viet Long3 Received: 29 May 2019 / Accepted: 8 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we introduce the semigroups of semilinear mappings on the space of nonempty compact intervals of R. Some important properties of strongly continuous semigroups of interval-valued mappings are investigated. Depending on the different types of generalized Hukuhara differences, the interval-valued infinitesimal generators and resolvent operators are defined and the Hille–Yosida like representation of resolvent operators is given. As an application, the unique existence of mild solutions for Cauchy problems of interval-valued evolution equations is given. Keyword Semigroups of interval-valued mappings · Infinitesimal generators · Resolvent operators · Semilinear metric spaces · Generalized Hukuhara differentiability · Interval-valued evolution equations Mathematics Subject Classification 28B10 · 58C06 · 54C60
1 Introduction We recall the space KC of all compact subintervals of R. The Minkowski addition and scalar multiplication are defined for all C, D ∈ KC and λ ∈ R by C + D = {c + d : c ∈ C, d ∈ D} , λC = {λc : c ∈ C} . KC is closed under the operations of addition and scalar multiplication. However, the structure is that of a cone rather than a vector space because, in general, K + (−1)K = {0} , K ∈ KC .
B
Hoang Viet Long [email protected] Nguyen Thi Kim Son [email protected]
1
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Information Technology, People’s Police University of Technology and Logistics, Bac Ninh, Vietnam
123
N. T. K. Son, H. V. Long
Here we denote {0} = [0, 0] by the zero element with respect to addition. Thus adding −1 times a set does not constitute a natural operation of subtraction. Instead, we define the Hukuhara difference of K , C ∈ KC by K C, provided there exists a set D ∈ KC satisfying K = C + D. Clearly, the Hukuhara difference K C does not exist for all K , C ∈ KC , for example [0, 0] [0, 1] does not exist because of no D ∈ KC satisfying [0, 1] + D = [0, 0]. Thus, this space is not a group with respect to addition. Moreover, in general the scalar multiplication is not distributive with respect to usual addition. It follows that (KC , +, .) is not a linear space over R and, consequently, (KC , ||.||) cannot be a normed space, where ||K || = d(K , {0}), K ∈ KC , defined from Hausdorff metric d(K , B) = max sup inf |k − b|, sup inf |k − b| , K , B ∈ KC . k∈K b∈B
b∈B k∈K
Therefore KC is not a Banach space. Unfortunately, for models with abstract systems, the constructive approach which allows to find explicitly solutions is based on interior estimations in the framework of Banach spaces or Hilbert space
Data Loading...
