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We obtain the existence of pseudo almost automorphic solutions to the -dimensional heat equation with -pseudo almost automorphic coefficients.

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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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 paper we study and obtain the existence of almost periodic solutions to the following class of semilinear difference equations whereA, Bare singularsquare matrices () andis almost periodic in the first variable uniformly in the second one. Next, we use these results to study the existence of almost periodic to some second-order (and higher-order) singular difference equations. [ABSTRACT FROM PUBLISHER]

In this article, a new q-generalization of the Apostol-Euler polynomials is introduced using the usual q-exponential function. We make use of such a generalization to derive several properties arising from the q-umbral calculus. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new class of functions called S p,q(x)-pseudoalmost periodic (or weighted Stepanov-like pseudo-almost periodic functions with variable exponents), which generalizes the class of weighted Stepanov-like pseudoalmost periodic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost periodic solutions to some first-order differential equations with S p,q(x)-pseudo-almost periodic coefficients will also be studied. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new class of functions called S p,q(x)-pseudoalmost periodic (or weighted Stepanov-like pseudo-almost periodic functions with variable exponents), which generalizes the class of weighted Stepanov-like pseudoalmost periodic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost periodic solutions to some first-order differential equations with S p,q(x)-pseudo-almost periodic coefficients will also be studied. [ABSTRACT FROM AUTHOR]