Integrable resolvent operators for integrodifferential equations in Hilbert space

1984 ◽  
pp. 269-278
Author(s):  
R. L. Wheeler
Keyword(s):  
Hilbert Space ◽  
Integrodifferential Equations ◽  
Resolvent Operators

scholarly journals Controllability for neutral stochastic functional integrodifferential equations with infinite delay

2016 ◽  
Vol 1 (2) ◽  
pp. 493-506 ◽  
Author(s):  
Tomás Caraballo ◽  
Mamadou Abdoul Diop ◽  
Aziz Mane
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Separable Hilbert Space ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
Real Separable Hilbert Space ◽  
Stochastic Functional

AbstractIn this work, we study the controllability for a class of nonlinear neutral stochastic functional integrodifferential equations with infinite delay in a real separable Hilbert space. Sufficient conditions for the controllability are established by using Nussbaum fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

2016 ◽  
Vol 16 (06) ◽  
pp. 1650014 ◽  
Author(s):  
Mamadou Abdoul Diop ◽  
Tomás Caraballo ◽  
Mahamat Mahamat Zene
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Resolvent Operators ◽  
Infinite Delays ◽  
Fixed Point Principle ◽  
Exponentially Stable

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [13], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in [Formula: see text]th-moment. An example is provided to illustrate the abstract results.

scholarly journals A characterisation and two examples of Riesz operators

1968 ◽  
Vol 9 (2) ◽  
pp. 106-110 ◽  
Author(s):  
T. A. Gillespie ◽  
T. T. West
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Spectral Projection ◽  
Resolvent Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

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