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Approximate Controllability of a Second Order Neutral Differential Equation with State Dependent Delay

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Abstract

This paper deals with the existence and uniqueness of mild solution and approximate controllability of a second order neutral partial differential equation with state dependent delay. The Hausdorff measure of noncompactness and Darbo Sadovskii theorem is used to prove the existence of mild solution of the system. Some restrictive conditions such as the compactness assumption on the associated cosine or sine family of operators and the Lipschitz conditions on the nonlinear functions are replaced by simple and natural assumptions. The conditions for approximate controllability are investigated for the distributed second order neutral system by assuming the approximate controllability of the corresponding linear system in a Hilbert space.

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Acknowledgments

The authors would like to express sincere gratitude to the reviewer for his valuable suggestions. The first author would like to thank Ministry of Human Resource and Development with Grant No. MHR-02-23-200-429/304 for their funding.

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Authors and Affiliations

  1. Department of Mathematics, IIT Roorkee, Roorkee, 247667, Utarrakhand, India

    Sanjukta Das, Dwijendra N. Pandey & N. Sukavanam

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  1. Sanjukta Das
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  2. Dwijendra N. Pandey
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  3. N. Sukavanam
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Correspondence to Sanjukta Das.

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Das, S., Pandey, D.N. & Sukavanam, N. Approximate Controllability of a Second Order Neutral Differential Equation with State Dependent Delay. Differ Equ Dyn Syst 24, 201–214 (2016). https://doi.org/10.1007/s12591-014-0218-6

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