Some remarks on almost automorphic solutions of some abstract differential equations. (English) Zbl 1040.34068

Diagana, T.; N’Guerekata, G. M.

Far East J. Math. Sci. (FJMS) 8, No. 3, 313-322 (2003).

Summary: We are concerned with both almost automorphic and almost periodic solutions of the differential equation given by \[ u'(t)= Au(t)+ f(t). \] The cases where \(A\) is of the simplest type, nilpotent, maximal monotone, or normal, respectively, are considered. In particular, we establish a connection between almost automorphic functions, provided by both the semigroup \(S(t)\) generated by \(A\) and the semigroup \(S_\lambda(t)\) generated by Yosida’s approximation \(A_\lambda\) of \(A\).

MSC:

34G10	Linear differential equations in abstract spaces
47D06	One-parameter semigroups and linear evolution equations

Keywords:

almost automorphic function; almost periodic function; \(C_0\)-semigroups; nilpotent operators; operators of the simplest type; Banach spaces