This paper is concerned with the existence of almost automorphic mild solutions to equations form \[ \dot u(t)= Au(t)+f(t),\tag *{$(*)$}\] where $A$ generates a holomorphic semigroup and $f$ an function. Since functions may not be uniformly continuous, we introduce notion uniform spectrum By modifying method sums commuting operators used in previous works for case bounded continuous solutions, obtain sufficient conditions $(*)$ terms imaginary $f$.

We consider the existence of invariant manifolds to evolution equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis equals upper A u right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> </mml:mrow>...

This paper is a continuation of previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept spectral dichotomy has been introduced. new notion proved to be useful since it allows apply well known theory linear operators study dynamic properties nonautonomous difference equations. In present we extend our result on equivalence and exponential class differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate set positive integers for...

The Massera Theorem for almost periodic solutions of linear ordinary differential equations the form (*)x′=A(t)x+f(t), where f is periodic, stated and proved. Furthermore, it extended to abstract functional (**)x′=Ax+F(t)xt+f(t), A generator a compact semigroup, F periodic. main techniques used in proofs involve new variation constants formula phase space decomposition theorem solutions.

Abstract Both polarized and unpolarized Raman scattering studies of seven tourmalines from the Lucyen mines in Vietnam are presented. These tourmalines, according to their chemical compositions, can be classified into four groups: G1, liddicoatite; G2, elbaite; G3, uvite; G4, feruvite. The spectra were recorded two spectral ranges, i.e. 150–1600 cm −1 3000–4000 . In lower range, which covers metal ion‐oxygen bond vibrations, all observed A 1 E modes identified. higher we investigated OH...

We consider the almost automorphy of bounded mild solutions to equations form \begin{equation*} (*)\quad \qquad dx/dt = A(t)x + f(t) \quad \end{equation*} with (generally unbounded) $\tau$-periodic $A(\cdot )$ and automorphic $f(\cdot in a Banach space $\mathbb {X}$. Under assumption that {X}$ does not contain $c_0$, part spectrum monodromy operator associated evolutionary process generated by on unit circle is countable. prove every solution $(*)$ real line automorphic.

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form $\dot {x}=A(t)x+f(t) \ (*)$, with $f$ having precompact range, which is then applied find new criteria

This paper is concerned with the existence and stability of solutions a class semilinear nonautonomous evolution equations. A procedure discussed which associates to each equation so‐called semigroup (possibly nonlinear) operators. Sufficient conditions for periodic oscillations are given in terms accretiveness corresponding infinitesimal generator. Furthermore, through integral manifolds abstract evolutionary processes we obtain reduction principle questions mild solutions. The results...