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Let T be a strongly continuous semigroup acting on a complex Banach space X and let A be its infinitesimal generator. It is well-known [29, 33] that the uniform spectral bound s0(A) of the semigroup T is negative provided that all solutions to the Cauchy problems u(t) = Au(t) + eiμtx;t ≥ 0; u(0) = 0; are bounded (uniformly with respect to the parameter μ ∊ R). In particular, if X is a Hilbert space, then this yields all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid. Let X denote the space of all continuous and 1-periodic functions f : B → X whose sequence of Fourier-Bohr coefficients (cm(f))m∊Z belongs to ℓ¹(Z,X). Endowed with the norm ||f||1 := k(cm(f))m∊Zk1 it becomes a non-reexive Banach space [15]. A subspace AT of X (related to the pair (T;X)) is introduced in the third section of this paper. We prove that the semigroup T is uniformly exponentially stable provided that in addition to the above-mentioned boundedness condition, AT = X. An example of a strongly continuous semigroup (which is not uniformly continuous) and fulfills the second assumption above is also provided. Moreover an extension of the above result from semigroups to 1-periodic and strongly continuous evolution families acting in a Banach space is given. We also prove that the evolution semigroup T associated with T on X does not verify the spectral determined growth condition. More precisely, an example of such a semigroup with uniform spectral bound negative and uniformly growth bound non-negative is given. Finally we prove that the assumption AT = X is not needed in the discrete case. [ABSTRACT FROM AUTHOR]

In this article we introduce and study a new class of functions called Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes in a natural way classical Stepanov-like pseudo-almost automorphic spaces. Basic properties of these new spaces are investigated. The existence of pseudo-almost automorphic solutions to some first-order diffeerential equations with Sp,q(x)-pseudo-almost automorphic coefficients will also be studied. [ABSTRACT FROM AUTHOR]

In this article, first we introduce and study the concept of Sγp-pseudo almost automorphy (or generalized Stepanov-like pseudo almost automorphy), which is more general than the notion of Stepanov-like pseudo almost automorphy due to Diagana. We next study the existence of solutions to some classes of nonautonomous differential equations of Sobolev type in Sγp-pseudo almost automorphic spaces. To illustrate our abstract result, we will study the existence and uniqueness of a pseudo almost automorphic solution to the heat equation with a negative time-dependent diffusion coefficient. [ABSTRACT FROM AUTHOR]

In this paper we use the well-known Schauder fixed point principle to obtain the existence of square-mean almost periodic solutions to some classes of nonautonomous second order stochastic differential equations on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper we obtain the existence of quadratic-mean almost periodic solutions to some classes of partial hyperbolic stochastic differential equations. The main result of this paper generalizes in a natural fashion some recent results by authors. As an application, we consider the existence of quadratic-mean almost periodic solutions to the stochastic heat equation with divergence terms. [ABSTRACT FROM AUTHOR]

In this paper we obtain the existence of almost automorphic solutions for some neutral first-order functional differential equations with Spalmost automorphic coefficients. [ABSTRACT FROM AUTHOR]

This paper concerns the square-mean almost periodic solutions to a class of nonautonomous stochastic differential equations on a separable real Hilbert space. Using the so-called 'Acquistapace-Terreni' conditions, we establish the existence and uniqueness of a square-mean almost periodic mild solution to those nonautonomous stochastic differential equations. [ABSTRACT FROM AUTHOR]

This paper studies some sufficient conditions for the existence and uniqueness of a pseudo almost periodic mild solution to abstract hyperbolic differential equations with infinite delay. Applications include the existence and uniqueness of a pseudo almost periodic mild solution to the heat equation. [ABSTRACT FROM AUTHOR]