scholarly journals A study on controllability of impulsive fractional evolution equations via resolvent operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Resolvent Operator ◽  
Evolution Equations ◽  
Validity Study ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Alpha Beta

AbstractIn this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

scholarly journals Controllability of Impulsive ψ-Caputo Fractional Evolution Equations with Nonlocal Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1358
Author(s):  
Longfei Lin ◽  
Yansheng Liu ◽  
Daliang Zhao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Mild Solution ◽  
Point Theorem ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Exact Controllability ◽  
Fractional Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Controllability Result

This paper is mainly concerned with the exact controllability for a class of impulsive ψ-Caputo fractional evolution equations with nonlocal conditions. First, by generalized Laplace transforms, a mild solution for considered problems is introduced. Next, by the Mönch fixed point theorem, the exact controllability result for the considered systems is obtained under some suitable assumptions. Finally, an example is given to support the validity of the main results.

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators

2020 ◽  
Vol 23 (1) ◽  
pp. 268-291 ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Resolvent Operator ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Control Function ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Theoretical Results

AbstractIn this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical 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.

scholarly journals Existence of Solutions for Fractional Evolution Equations with Infinite Delay and Almost Sectorial Operator

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shanshan Li ◽  
Shuqin Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Sectorial Operator ◽  
Analytic Semigroups ◽  
Measures Of Noncompactness ◽  
Fractional Evolution Equations

This paper discusses a class of semilinear fractional evolution equations with infinite delay and almost sectorial operator on infinite interval in Banach space. By using the properties of analytic semigroups and Schauder’s fixed-point theorem, this paper obtains the existence of mild solutions of the fractional evolution equation. Moreover, this paper also discusses the existence of mild solution when the analytic semigroup lacks compactness by Kuratowski measures of noncompactness and Darbo–Sadovskii fixed-point theorem.

scholarly journals Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

2019 ◽  
Vol 24 (4) ◽  
Author(s):  
Mamadou Abdoul Diop ◽  
Amour Toffodji Gbaguidi Amoussou ◽  
Carlos Ogouyandjou ◽  
Rathinasamy Sakthivel
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Resolvent Operator ◽  
Functional Equations ◽  
Banach Fixed Point Theorem ◽  
Resolvent Operators ◽  
Poisson Jumps ◽  
Rosenblatt Process ◽  
Potential Benefits ◽  
Equations With Delay

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results.

scholarly journals Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Haide Gou ◽  
Baolin Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Fractional Evolution Equations

We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

scholarly journals On a fixed point theorem of Greguš

1986 ◽  
Vol 9 (1) ◽  
pp. 23-28 ◽  
Author(s):  
Brian Fisher ◽  
Salvatore Sessa
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Point Theorem ◽  
Convex Subset ◽  
Unique Common Fixed Point

We consider two selfmapsTandIof a closed convex subsetCof a Banach spaceXwhich are weakly commuting inX, i.e.‖TIx−ITx‖≤‖Ix−Tx‖   for   any   x   in   X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}for allx,yinC, where0<a<1. It is proved that ifIis linear and non-expansive inCand such thatICcontainsTC, thenTandIhave a unique common fixed point inC.

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