Abstract
In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.
Highlights
In recent years, impulsive differential and partial differential equations have become more important in some mathematical models of real phenomena, especially in control, biological and medical domains
Since many systems arising from realistic models heavily depend on histories, there is a real need to discuss partial functional differential systems with infinite delay, where numerous properties of their solutions are studied and detailed bibliographies are given
Chalishajar and Acharya [22] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators
Summary
Impulsive differential and partial differential equations have become more important in some mathematical models of real phenomena, especially in control, biological and medical domains. Hernández et al [18] studied existence of solutuions for impulsive partial neutral functional differential equations for first and second order systems with infinite delay. Arthi and Balachandran [19] proved controllability of the second order impulsive functional differential equations with state dependent delay using fixed point approach and cosine operator theory. Chalishajar and Acharya [22] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Fu et al [27] studied the existence of PC-mild solutions for Cauchy and nonlocal problems of impulsive fractional evolution equations for which the impulses are not instantaneous, by using the theory of operator semigroups, probability density functions, and some suitable fixed point theorems.
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