Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instan
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DOI: 10.1007/s13226-020-0413-9
APPROXIMATE CONTROLLABILITY OF NON-AUTONOMOUS SOBOLEV TYPE INTEGRO-DIFFERENTIAL EQUATIONS HAVING NONLOCAL AND NON-INSTANTANEOUS IMPULSIVE CONDITIONS Arshi Meraj and Dwijendra N. Pandey Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, Uttarakhand, India e-mails: [email protected]; [email protected] (Received 15 October 2018; accepted 25 February 2019) The aim of this article is to study approximate controllability of a class of non-autonomous Sobolev type integro-differential equations having non-instantaneous impulses with nonlocal initial condition. The results will be proved with the help of evolution system and Krasnoselskii fixed point theorem. An example is presented to show how our abstract results can be applied. Key words : Approximate controllability; Krasnoselskii fixed point theorem; evolution system; non-instantaneous impulsive condition; Sobolev type integro-differential equations. 2010 Mathematics Subject Classification : 34A37, 34G20, 93B05.
1. I NTRODUCTION The Sobolev type differential equations appear in several fields such as thermodynamics [8], fluid flow via fissured rocks [4] and mechanics of soil [29]. Brill [5] first established the existence of solution for a semilinear Sobolev differential equation in a Banach space. Lightbourne et al. [21] studied a partial differential equation of Sobolev type. In recent years many researchers paid attention to study the differential equations with instantaneous impulses, which have been used to described abrupt changes such as shocks, harvesting and natural disasters. Particularly, the theory of instantaneous impulsive equations have wide applications in control, mechanics, electrical engineering, biological and medical fields. It seems that models with instantaneous impulses could not explain certain dynamics of evolution process in pharmacotherapy. For example, one considers the hemodynamic equilibrium of a person,
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ARSHI MERAJ AND DWIJENDRA N. PANDEY
the introduction of the drugs in bloodstream and the consequent absorbtion for the body are gradual and continuous process, we can interpret the above situations as an impulsive action which starts abruptly and stays active on a finite time interval. Hern´ andez and O’Regan [16] and Pierri et al. [26], initially studied Cauchy problems for first order evolution equations with non-instataneous impulses. Kumar et al. [20] established the existence and uniqueness of mild solutions for non-instantaneous impulsive fractional differential equations. Chen et al. [9] investigated the existence of mild solutions for first order semi-linear evolution equations with non-instantaneous impulses using noncompact semigroup. Kumar et al. [18] derived a set of sufficient conditions for the existence and uniqueness of mild solutions to fractional integro-differential equations with non-instantaneous impulses. The work on nonlocal initial value problem was first studied by Byszewski. In [6], Byszewski established the results about the existence
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