Resolvent Operators for Some Classes of Integro-Differential Equations

2016 ◽  
pp. 535-558
Author(s):  
I. N. Parasidis ◽  
E. Providas
Keyword(s):  
Differential Equations ◽  
Resolvent Operators

scholarly journals Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zhenbin Fan ◽  
Gisèle Mophou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Mild Solutions ◽  
Resolvent Operators ◽  
Nonlocal Problems ◽  
Lipschitz Continuous ◽  
Resolvent Method ◽  
Fractional Differential ◽  
Analytic Resolvent ◽  
Cauchy Operator

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

scholarly journals Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

2020 ◽  
Vol 18 (1) ◽  
pp. 1097-1112
Author(s):  
Louk-Man Issaka ◽  
Mamadou Abdoul Diop ◽  
Hasna Hmoyed
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Analysis ◽  
Mild Solutions ◽  
Hurst Parameter ◽  
Resolvent Operators ◽  
Final Point ◽  
Analysis Theory ◽  
Non Local

Abstract This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

scholarly journals Approximate controllability of semi-linear stochastic integro-differential equations with infinite delay

2020 ◽  
Vol 37 (4) ◽  
pp. 1133-1167
Author(s):  
Hai Huang ◽  
Xianlong Fu
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Approximate Controllability ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Fundamental Solutions ◽  
Resolvent Operators ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Approximately Controllable

Abstract In this work, by constructing fundamental solutions and using the theory of resolvent operators and fractional powers of operators, we study the approximate controllability of a class of semi-linear stochastic integro-differential equations with infinite delay in $L_p$ space ($2<p<\infty $). Sufficient conditions for approximate controllability of the discussed equations are obtained under the assumption that the associated deterministic linear system is approximately controllable. An example is provided to illustrate the obtained results.

scholarly journals A study of controllability of impulsive neutral evolution integro-differential equations with state dependent delay in Banach space

2016 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
A. Anguraj ◽  
Kulandhivel Karthikeyan ◽  
Malar Ganeshan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Phase Space ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Neutral Evolution ◽  
Resolvent Operators ◽  
State Dependent ◽  
State Dependent Delay

In this paper, we study the problem of controllability of impulsive neutral evolution integrodifferential equations with state dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii's fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.

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