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Almost automorphic solutions of certain abstract differential equations

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Summary

Almost automorphic (=a.a.) functions with values in Banach spaces have been introduced. Eight theorems have been proved usinga.a. functions. The following are, perhaps the main representative results:

  1. 1)

    If f(t) isa.a. from R into X, a uniformly convex space, then\(F(t) = \int\limits_{t_0 }^t {f(s)ds} \) isa.a. if and only if it is bounded on (− ∞, ∞).

  2. 2)

    X is any B-space. Tt a strongly continuous semigroup such that\(\mathop {\lim }\limits_{t \to \infty } T_t x = 0\) for every x in X. f(t) isa.a. and\(u(t) = T_{t - t_0 } u(t_0 ) + \int\limits_{t_0 }^t {T_{t - 1} f(s)ds} \). If the trajectory of u(t) is relatively compact then u(t) isa.a.

  3. 3)

    X is any B-space. G(t) a one parameter strongly continuous group such that\(\mathop {\sup }\limits_{ - \infty< t< \infty } \) ∥G(t)∥ < ∞. If S={f(t): t rational} isa.a. under G(t) and f(t) isa.a. then G(t)f(t): R→X isa.a. A consequence of this result is the following: If X is uniformly convex and A is the infinitesimal operator associated with G(t), then a bounded solution u of u′=Au + f isa.a.

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Entrata in Redazione il 10 agosto 1972.

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Zaki, M. Almost automorphic solutions of certain abstract differential equations. Annali di Matematica 101, 91–114 (1974) doi:10.1007/BF02417100

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Keywords

  • Differential Equation
  • Banach Space
  • Representative Result
  • Convex Space
  • Continuous Semigroup