Weakly demicompact linear operators and axiomatic measures of weak noncompactness

2019 ◽  
Vol 69 (6) ◽  
pp. 1403-1412 ◽  
Author(s):  
Bilel Krichen ◽  
Donal O’Regan
Keyword(s):  
Fixed Point ◽  
Essential Spectrum ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Linear Operators ◽  
Nonlinear Operators ◽  
Measures Of Weak Noncompactness ◽  
The Relationship

Abstract In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

scholarly journals Relative controllability of a stochastic system using fractional delayed sine and cosine matrices

2021 ◽  
Vol 26 (6) ◽  
pp. 1031-1051
Author(s):  
JinRong Wang ◽  
T. Sathiyaraj ◽  
Donal O’Regan
Keyword(s):  
Fixed Point ◽  
Stochastic System ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Point Theory ◽  
Linear Operators ◽  
Relative Controllability ◽  
Finite Dimensional ◽  
Pure Delay

In this paper, we study the relative controllability of a fractional stochastic system with pure delay in finite  dimensional stochastic spaces. A set of sufficient conditions is obtained for relative exact controllability using fixed point theory, fractional calculus (including fractional delayed linear operators and Grammian matrices) and local assumptions on nonlinear terms. Finally, an example is given to illustrate our theory.

Radius vs diameter

2018 ◽  
pp. 102-115
Author(s):  
Kazimierz Goebel ◽  
Stanisław Prus
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Main Tool ◽  
Normal Structure ◽  
Point Theory ◽  
The Subject ◽  
The Relationship ◽  
Bounded Sets

The subject of the chapter is the relationship between the (Chebyshev) radius and diameter of convex bounded sets. The main tool is the Jung coefficient. Diametral sets and normal structure in connection with the fixed point theory for nonexpansive mappings are presented.

scholarly journals Nonlinear Operators in Fixed Point Theory with Applications to Fractional Differential and Integral Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-2
Author(s):  
Jamshaid Ahmad ◽  
Ahmad Saleh Al-Rawashdeh ◽  
Talat Nazir ◽  
Vahid Parvaneh ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Nonlinear Operators ◽  
Fractional Differential ◽  
Differential And Integral Equations

Fractional Retarded Evolution Equations with Measure of Noncompactness Subjected to Mixed Nonlocal Plus Local Initial Conditions

2018 ◽  
Vol 19 (1) ◽  
pp. 69-81
Author(s):  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Fixed Point Theory ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Point Theory ◽  
Linear Operators ◽  
Nonlinear Operators ◽  
Bounded Linear ◽  
Uniformly Bounded

AbstractWe consider the fractional retarded evolution equations $$^{C}D_{t}^{q}u(t)+Au(t)=f\Big(t,u_t,\int_{0}^{t}w(t,s,u_s)ds\Big),\quad t\in[0,a],$$where $^{C}D_{t}^{q}$, $q\in(0,1]$, is the fractional derivative in the Caputo sense, $-A$ is the infinitesimal generator of a $C_0$-semigroup of uniformly bounded linear operators $T(t)$$(t\geq0)$ on a Banach space $X$ and the nonlinear operators $f$ and $w$ are given functions satisfying some assumptions, subjected to a general mixed nonlocal plus local initial condition of the form $u(t)=g(u)(t)+\phi(t)$, $t\in[-h,0]$. Under more general conditions, the existence of mild solutions and positive mild solutions are obtained by means of fractional calculus and fixed point theory for condensing maps. Moreover, we present an example to illustrate the application of abstract results.

Modeling the dynamics of hepatitis E via the Caputo–Fabrizio derivative

2019 ◽  
Vol 14 (3) ◽  
pp. 311 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Zakia Hammouch ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Numerical Scheme ◽  
Arbitrary Order ◽  
Fixed Point Theory ◽  
Hepatitis E ◽  
Point Theory ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Essential Properties

A virus that causes hepatitis E is known as (HEV) and regarded on of the reason for lever inflammation. In mathematical aspects a very low attention has been paid to HEV dynamics. Therefore, the present work explores the HEV dynamics in fractional derivative. The Caputo–Fabriizo derivative is used to study the dynamics of HEV. First, the essential properties of the model will be presented and then describe the HEV model with CF derivative. Application of fixed point theory is used to obtain the existence and uniqueness results associated to the model. By using Adams–Bashfirth numerical scheme the solution is obtained. Some numerical results and tables for arbitrary order derivative are presented.

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