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.

Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

2020 ◽  
Vol 37 (4) ◽  
pp. 1089-1113
Author(s):  
Yi-rong Jiang ◽  
Qiong-fen Zhang ◽  
Qi-qing Song
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Control Problem ◽  
Mild Solution ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Solution Set ◽  
Impulsive Evolution Equations

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.

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.

Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Fang Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Mild Solution ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations

AbstractIn this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.

scholarly journals Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Lina Ma ◽  
Haibo Gu ◽  
Yiru Chen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Sufficient Conditions

In this paper, we consider a kind of neutral measure evolution equations with nonlocal conditions. By using semigroup theory and fixed point theorem, we can obtain sufficient conditions for the controllability results of such equations. Finally, an example is given to verify the reliability of the results.

scholarly journals EXISTENCE OF SOLUTIONS FOR IMPULSIVE NEUTRAL FUNCTIONAL INTEGRO DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

2017 ◽  
Vol 4 (3) ◽  
pp. 1-6
Author(s):  
Valliammal N ◽  
Ravichandran C
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Mild Solution ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Nonlocal Conditions ◽  
Fractional Power ◽  
Integrodifferential Equations ◽  
Fractional Power Of Operators

In this paper, by using fractional power of operators and Sadovskii’s fixed point theorem, we study the existence of mild solution for a certain class of impulsive neutral functional integrodifferential equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent resultson this issue.

scholarly journals Second order evolution equations with nonlocal conditions

2017 ◽  
Vol 50 (1) ◽  
pp. 309-319 ◽  
Author(s):  
Mouffak Benchohra ◽  
Juan J. Nieto ◽  
Noreddine Rezoug
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Hausdorff Measure ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Fréchet Spaces

Abstract In this paper, we shall establish sufficient conditions for the existence of solutions for second order semilinear functional evolutions equation with nonlocal conditions in Fréchet spaces. Our approach is based on the concepts of Hausdorff measure, noncompactness and Tikhonoff’s fixed point theorem. We give an example for illustration.

scholarly journals New discussion on nonlocal controllability for fractional evolution system of order $1 < r < 2$

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. Mohan Raja ◽  
Velusamy Vijayakumar ◽  
Anurag Shukla ◽  
Kottakkaran Sooppy Nisar ◽  
Shahram Rezapour
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Evolution System ◽  
Cosine Families ◽  
Mönch Fixed Point Theorem ◽  
Fixed Point Techniques

AbstractIn this manuscript, we deal with the nonlocal controllability results for the fractional evolution system of $1< r<2$ 1 < r < 2 in a Banach space. The main results of this article are tested by using fractional calculations, the measure of noncompactness, cosine families, Mainardi’s Wright-type function, and fixed point techniques. First, we investigate the controllability results of a mild solution for the fractional evolution system with nonlocal conditions using the Mönch fixed point theorem. Furthermore, we develop the nonlocal controllability results for fractional integrodifferential evolution system by applying the Banach fixed point theorem. Finally, an application is presented for drawing the theory of the main results.

scholarly journals ON THE CONTROLLABILITY OF IMPULSIVE FUNCTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

2019 ◽  
Vol 6 (1) ◽  
pp. 17-22
Author(s):  
Valliammal N ◽  
Ravichandran C
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Mönch Fixed Point Theorem

In this paper, we established the some sufficient conditions for controllability of impulsive functional integrodifferential equations with nonlocal conditions by using the measure of noncompactness and Monch fixed point theorem.

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.

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
JinRong Wang ◽  
Yong Zhou ◽  
Michal Fečkan
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Growth Condition ◽  
Fractional Order ◽  
Point Theorem ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Time Interval ◽  
Nonlocal Cauchy Problem

AbstractIn this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.

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