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Semigroups of operators and abstract dynamic equations on time scales

Author

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  • Hamza, Alaa E.
  • Oraby, Karima M.

Abstract

In this paper we develop the theory of strongly continuous semigroups (C0-semigroups) of bounded linear operators from a Banach space X into itself. Many properties of a C0-semigroup {T(t):t∈T} and its generator A are established. Here T⊆R≥0 is a time scale endowed with an additive semigroup structure. We also establish necessary and sufficient conditions for the dynamic initial value problem {xΔ(t)=Ax(t),t∈Tx(0)=x0∈D(A),0∈Tto have a unique solution, where D(A) is the domain of A. Finally, we unify the continuous Hille–Yosida–Phillips Theorem and the discrete Gibson Theorem.

Suggested Citation

  • Hamza, Alaa E. & Oraby, Karima M., 2015. "Semigroups of operators and abstract dynamic equations on time scales," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 334-348.
    • Handle: RePEc:eee:apmaco:v:270:y:2015:i:c:p:334-348
    DOI: 10.1016/j.amc.2015.07.110
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    Cited by:

    1. Wang, Lingyu & Huang, Tingwen & Xiao, Qiang, 2018. "Global exponential synchronization of nonautonomous recurrent neural networks with time delays on time scales," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 263-275.

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