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 results of ψ-Hilfer integro-differential equations with fractional order in Banach space

2020 ◽  
Vol 19 (1) ◽  
pp. 171-192
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Banach Fixed Point Theorem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Mönch Fixed Point Theorem

AbstractIn this article we present the existence and uniqueness results for fractional integro-differential equations with ψ-Hilfer fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Mönch fixed point theorem and the Banach fixed point theorem. Furthermore, we discuss Eα -Ulam-Hyers stability of the presented problem. Also, we use the generalized Gronwall inequality with singularity to establish continuous dependence and uniqueness of the δ-approximate solution.

scholarly journals Controllability Analysis for Impulsive Integro-differential Equation via AB Fractional Derivative

2021 ◽  
Author(s):  
KALIMUTHU KALIRAJ ◽  
E. Thilakraj ◽  
Ravichandran C ◽  
Kottakkaran Nisar
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Integro Differential Equation

In this work, we analyse the controllability for certain classes of impulsive integro - differential equations(IIDE) of fractional order via Atangana Baleanu derivative involving finite delay with initial and nonlocal conditions using Banach fixed point theorem.

scholarly journals Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type

2021 ◽  
Vol 26 (5) ◽  
pp. 914-927
Author(s):  
Sergey Smirnov
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Point Theorem ◽  
Boundary Value ◽  
Nonlocal Conditions ◽  
Integral Condition ◽  
Banach Fixed Point Theorem ◽  
Third Order

The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.

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 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND Ψ–HILFER FRACTIONAL DERIVATIVE

2019 ◽  
Vol 24 (4) ◽  
pp. 564-584 ◽  
Author(s):  
Mohammed S. Abdo ◽  
Satish K. Panchal ◽  
Hussien Shafei Hussien
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Uniqueness Of Solutions ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem

Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.

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.

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

2016 ◽  
Vol 3 (2) ◽  
pp. 1-6
Author(s):  
Ravichandran C ◽  
Valliammal N
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Control Systems ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Impulsive Control ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
Impulsive Control Systems

The paper is concerned with the controllability of impulsive functional integrodifferential equations with nonlocal conditions. Using the measure of noncompactness and Monch fixed point theorem, we establish some sufficient conditions for controllability and also our theorems extend some analogous results of (impulsive) control systems.

Fuzzy Solutions for Neutral Functional Differential Equations with Nonlocal Conditions

2004 ◽  
Vol 11 (1) ◽  
pp. 35-42
Author(s):  
Amaria Arara ◽  
Mouffak Benchohra
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Functional Differential Equations ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Neutral Functional Differential Equations ◽  
Functional Differential ◽  
Fuzzy Solutions

Abstract The Banach fixed point theorem is used to investigate the existence of fuzzy solutions for first and second order neutral functional differential equations with nonlocal conditions.

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.

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