Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

M. Manjula; K. Kaliraj; Thongchai Botmart; Kottakkaran Sooppy Nisar; C. Ravichandran; M. Manjula; K. Kaliraj; Thongchai Botmart; Kottakkaran Sooppy Nisar; C. Ravichandran

doi:10.3934/math.2023229

AIMS Mathematics

2023, Volume 8, Issue 2: 4645-4665. doi: 10.3934/math.2023229

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Research article

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

1.
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, Tamil Nadu, India
2.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
3.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
4.
Department of Mathematics, Kongunadu Arts and Science College, Coimbatore 641029, Tamil Nadu, India

Received: 25 September 2022 Revised: 07 November 2022 Accepted: 11 November 2022 Published: 06 December 2022
MSC : 34B10, 34K40, 34K45, 47H10

This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.
- fractional calculus,
- impulsive,
- fixed point techniques,
- sobolev type
Citation: M. Manjula, K. Kaliraj, Thongchai Botmart, Kottakkaran Sooppy Nisar, C. Ravichandran. Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses[J]. AIMS Mathematics, 2023, 8(2): 4645-4665. doi: 10.3934/math.2023229

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Abstract

This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.

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