scholarly journals Analysis on ψ-Hilfer Fractional Impulsive Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1895
Author(s):  
Kulandhaivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Dimplekumar N. Chalishajar ◽  
Duraisamy Senthil Raja ◽  
Ponnusamy Sundararajan
Keyword(s):  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Strong Relationship ◽  
Dimensional System ◽  
Structural Investigations ◽  
Sectorial Operators ◽  
Higher Dimensional ◽  
Almost Sectorial Operators ◽  
Special Cases

In this manuscript, we establish the existence of results of fractional impulsive differential equations involving ψ-Hilfer fractional derivative and almost sectorial operators using Schauder fixed-point theorem. We discuss two cases, if the associated semigroup is compact and noncompact, respectively. We consider here the higher-dimensional system of integral equations. We present herewith new theoretical results, structural investigations, and new models and approaches. Some special cases of the results are discussed as well. Due to the nature of measurement of noncompactness theory, there exists a strong relationship between the sectorial operator and symmetry. When working on either of the concepts, it can be applied to the other one as well. Finally, a case study is presented to demonstrate the major theory.

scholarly journals Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators

2021 ◽  
Vol 5 (1) ◽  
pp. 22
Author(s):  
Kulandhaivel Karthikeyan ◽  
Amar Debbouche ◽  
Delfim F. M. Torres
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sectorial Operators ◽  
Fundamental Properties ◽  
Almost Sectorial Operators ◽  
Fixed Point Technique

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

scholarly journals Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 313
Author(s):  
Kulandhaivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Conditions ◽  
Hilfer Fractional Derivative ◽  
Sectorial Operators ◽  
Almost Sectorial Operators ◽  
Suitable Definition ◽  
Definition Of

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered.

Higher dimensional systems of differential equations obtainable by iterative use of complex methods

2015 ◽  
Vol 38 ◽  
pp. 1560077 ◽  
Author(s):  
Asghar Qadir ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Iterative Procedure ◽  
Dimensional System ◽  
Symmetry Generator ◽  
Higher Dimensional ◽  
Symmetry Properties ◽  
Systems Of Odes ◽  
Systems Of Pdes ◽  
Two Dimensional System ◽  
Complex Methods

A procedure had been developed to solve systems of two ordinary and partial differential equations (ODEs and PDEs) that could be obtained from scalar complex ODEs by splitting into their real and imaginary parts. The procedure was extended to four dimensional systems obtainable by splitting complex systems of two ODEs into their real and imaginary parts. As it stood, this procedure could be extended to any even dimension but not to odd dimensional systems. In this paper, the complex splitting is used iteratively to obtain three and four dimensional systems of ODEs and four dimensional systems of PDEs for four functions of two and four variables that correspond to a scalar base equation. We also provide characterization criteria for such systems to correspond to the base equation and a clear procedure to construct the base equation. The new systems of four ODEs are distinct from the class obtained by the single split of a two dimensional system. The previous complex methods split each infinitesimal symmetry generator into a pair of operators such that the entire set of operators do not form a Lie algebra. The iterative procedure sheds some light on the emergence of these "Lie-like" operators. In this procedure the higher dimensional system may not have any or the required symmetry for being directly solvable by symmetry and other methods although the base equation can have sufficient symmetry properties. Illustrative examples are provided.

ADVANCED IMPULSIVE DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

2010 ◽  
Vol 15 (2) ◽  
pp. 175-187 ◽  
Author(s):  
Hüseyin Bereketoglu ◽  
Gizem Seyhan ◽  
Aşar Ogun
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Existence And Uniqueness ◽  
Impulsive Differential Equations ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
First Order ◽  
Special Cases

We prove the existence and uniqueness of the solutions of a class of first order nonhomogeneous advanced impulsive differential equations with piecewise constant arguments. We also study the conditions of periodicity, oscillation, nonoscillation and global asymptotic stability for some special cases.

Almost sectorial operators on Ψ‐Hilfer derivative fractional impulsive integro‐differential equations

2021 ◽  
Author(s):  
Kulandhivel Karthikeyan ◽  
Panjaiyan Karthikeyan ◽  
Haci Mehmet Baskonus ◽  
Kuppusamy Venkatachalam ◽  
Yu‐Ming Chu
Keyword(s):  
Differential Equations ◽  
Sectorial Operators ◽  
Almost Sectorial Operators
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