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In this paper, we study a class of second-order abstract linear hyperbolic equations with infinite memory that involve time-dependent unbounded linear operators. We obtain the well-posedness and stability of solutions to those nonautonomous second-order evolution equations under some appropriate assumptions. Our results generalize a number of previously known results in the autonomous case. Some specific examples are given to illustrate our abstract results, such as the nonautonomous Petrovsky type and wave equations. [ABSTRACT FROM AUTHOR]

In this paper we study and obtain the existence of asymptotically almost periodic solutions to some classes of second-order hyperbolic integrodifferential equations of Gurtin–Pipkin type in a separable Hilbert space H. To illustrate our abstract results, the existence of asymptotically almost periodic mild solutions to the well-known Kirchoff plate equation is studied. [ABSTRACT FROM AUTHOR]

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all trajectories of the semigroup T are exponentially stable, but if X is an arbitrary Banach space this result is no longer valid.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 paper we study the well-posedness for some damped elastic systems in a Banach space. To illustrate our abstract results, we study the existence of classical solutions to some elastic systems with viscous and strong dampings. [ABSTRACT FROM AUTHOR]

In this paper we make extensive use of the p -adic Laplace transform and p -adic integration to study and obtain some new identities and congruences for Fubini numbers. Further, we introduce the concept of generalized Fubini numbers and obtain some new identities and congruences for these, which generalize in a natural fashion those identities associated with Fubini numbers. [ABSTRACT FROM AUTHOR]

This paper is aimed at studying the existence of square-mean almost periodic solutions to some singular stochastic differential equations with square-mean almost periodic coefficients. Using the block diagonal form of a linear operator on the suitable space, we establish an existence result under some appropriate assumptions. [ABSTRACT FROM AUTHOR]

In this paper we study and obtain the existence of a positive almost periodic solution to the nonlinear differential equation with delays that describes the so-called hematopoiesis model. The main tool utilized to establish our existence result consists of the fixed-point theorem in a cone. Two examples are given at the end of the paper to illustrate our main result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study and obtain the existence of an almost automorphic solution to a Beverton–Holt dynamic equation with a nonconstant survival rate under some reasonable assumptions. [Copyright &y& Elsevier]

Abstract: In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem. [Copyright &y& Elsevier]

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]

Abstract: In this paper we introduce a new theory for asymptotically almost automorphic sequences on which is a natural generalization of the concept of asymptotically almost periodicity for sequences due to Ky Fan. We next make extensive use of such a newly introduced concept as well as discrete dichotomy techniques to study and establish the existence of globally attracting almost automorphic solutions to some higher-order nonautonomous difference equations. [Copyright &y& Elsevier]

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]

Abstract: In this paper we study and obtain the existence of -pseudo almost automorphic solutions to some higher-order differential equations with operator coefficients whose forcing is -pseudo almost automorphic. We next use these abstract results to study the existence of -pseudo almost automorphic solutions to some second-order differential equations in Hilbert space which, among other things, model the so-called linear beam equation. [Copyright &y& Elsevier]

In this paper we introduce and study a new concept called C(n)-pseudo-almost automorphy, which generalizes in a natural fashion both the notions of C(n)-almost periodicity and that of C(n)-almost automorphy. Properties of such a new concept will be studied. Furthermore, we prove the existence of C(n)-pseudo-almost automorphic solutions to some n-order differential equations. [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]

This paper is concerned with the existence of almost periodic mild solutions to some second-order equations. Using dichotomy tools and the Schauder fixed point theorem, the existence of almost periodic mild solutions to those second-order evolution equations is established. To illustrate our abstract results, the existence of almost periodic solutions to a damped second-order boundary value problem is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we make extensive use of the well-known Schauder fixed point principle and the exponential stability of the evolution family involved to study and obtain the existence of almost periodic mild solutions to some classes of nonautonomous higher-order differential equations with operator coefficients in a separable infinite dimensional complex Hilbert space. To illustrate our main result, we study and obtain the existence of an almost periodic mild solution to a wave equation with fractional damping. [Copyright &y& Elsevier]

Abstract: In a recent paper by Liang et al. (2010) , the original question which is that of the existence of a weighted mean for almost periodic functions was raised. In particular, they showed through an example that there exist weights for which a weighted mean for almost periodic functions may or may not exist. In this note we give some sufficient conditions which do guarantee the existence of a weighted mean for almost periodic functions, which will then coincide with the classical Bohr mean. Moreover, we will show that under those conditions, the corresponding weighted Bohr transform exists. [Copyright &y& Elsevier]

In this paper we make extensive use of the well-known Schauder fixed point principle and dichotomy techniques to obtain the existence of almost automorphic solutions to some classes of nonautonomous (higher-order) differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, under Acquistapace–Terreni conditions, we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations. Applications include the existence of weighted pseudo-almost periodic solutions to a nonautonomous heat equation with gradient coefficients. [Copyright &y& Elsevier]

In this paper, we make extensive use of the well-known Krasnoselskii fixed point theorem to obtain the existence of square-mean almost periodic solutions to some classes of hyperbolic stochastic evolution equations with infinite delay. Next, the existence of square-mean almost periodic solutions to not only the heat equation but also to a boundary value problem with infinite delay arising in control systems are studied. [ABSTRACT FROM AUTHOR]

In this paper we give some sufficient conditions, which do ensure the existence and uniqueness of a compact almost automorphic solution to some hyperbolic differential equations. As an illustration, we study the existence and uniqueness of almost automorphic solutions to a one-dimensional heat equation with small delays. [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]

Abstract: In this paper we introduce the concept of weighted pseudo-almost periodicity in the sense of Stepanov, also called -weighted pseudo-almost periodicity and study its properties. Next, we present a result on the existence of weighted pseudo-almost periodic solutions to the -dimensional heat equation with -weighted pseudo-almost periodic coefficients. [Copyright &y& Elsevier]

Abstract: In this paper we obtain the existence of solutions to some classes of partial fractional differential equations. Applications include the existence of solutions to a fractional heat-like equation. [Copyright &y& Elsevier]

Abstract: The paper establishes the existence and uniqueness of asymptotically almost automorphic mild solution to an abstract partial neutral integro-differential equation with unbounded delay. An example is given to illustrate our results. [Copyright &y& Elsevier]

Abstract: This paper introduces and studies the notion of -pseudo-almost automorphy (or Stepanov-like pseudo-almost automorphy), which generalizes in a natural fashion the concept of pseudo-almost automorphy that was recently introduced in the literature by Liang, Xiao and Zhang. We next obtain the existence of pseudo-almost automorphic solutions to some differential equations with -pseudo-almost automorphic coefficients. [Copyright &y& Elsevier]

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]

Abstract: This paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations. Upon making some suitable assumptions, the existence and uniqueness of a weighted pseudo almost periodic solution is obtained. To illustrate our main result, we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional-differential equations with feedback control. [Copyright &y& Elsevier]

Abstract: This paper studies suitable sufficient conditions to ensure the existence and uniqueness of weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type introduced by T.A. Burton in the literature. The abstract results are then utilized to characterize weighted pseudo-almost periodic solutions to the well-known logistic equation. [Copyright &y& Elsevier]

Abstract: The paper revisits the concept of -pseudo-almost periodicity recently introduced by the author. In particular, we study the existence of pseudo-almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and -pseudo-almost periodic for . Applications include the existence of pseudo-almost periodic solutions to the heat equation with a forcing term, which is -pseudo-almost periodic and jointly continuous. [Copyright &y& Elsevier]

The paper studies the notion of Stepanov almost periodicity (or S²-almost periodicity) for stochastic processes, which is weaker than the notion of quadratic-mean almost periodicity. Next, we make extensive use of the so-called Acquistapace and Terreni conditions to prove the existence and uniqueness of a Stepanov (quadratic-mean) almost periodic solution to a class of nonautonomous stochastic evolution equations on a separable real Hilbert space. Our abstract results will then be applied to study Stepanov (quadratic-mean) almost periodic solutions to a class of n-dimensional stochastic parabolic partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract: The paper considers the existence and uniqueness of almost automorphic mild solutions to some classes of first-order partial neutral functional-differential equations. Sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the above-mentioned equations are obtained. As an application, a first-order boundary value problem arising in control systems is considered. [Copyright &y& Elsevier]

Abstract: The paper considers some new classes of functions called weighted pseudo-almost periodic functions, which implement in a natural fashion the classical pseudo-almost periodic functions due to Zhang. Properties of these weighted pseudo-almost periodic functions are discussed, including a composition result for weighted pseudo-almost periodic functions. The results obtained are subsequently utilized to study the existence and uniqueness of a weighted pseudo-almost periodic solution to the heat equation with Dirichlet conditions. [Copyright &y& Elsevier]

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]

The paper studies a Beverton-Holt difference equation, in which both the recruitment function and the survival rate vary randomly. It is then shown that there is a unique invariant density, which is asymptotically stable. Moreover, a basic theory for random mean almost periodic sequence on [image omitted] is given. Then, some sufficient conditions for the existence of a mean almost periodic solution to the stochastic Beverton-Holt equation are given. [ABSTRACT FROM AUTHOR]

This paper introduces and examines a new class of functions called Stepanov-like pseudo almost periodic functions (or Sp-pseudo almost periodic functions), which generalizes in a natural fashion the classical notion of pseudo almost periodicity. We then make extensive use of these new functions to study the existence and uniqueness of a pseudo almost periodic solution to the semilinear equation u′(t)=Au(t) + F(t,u(t)), where A : (D)(A) ⊂ X ↦ X is the infinistesimal generator of an exponentially stable C0-semigroup on a Banach space X and F: ℝ × X ↦ X is Sp-pseudo almost periodic for p > 1 and jointly continuous. [ABSTRACT FROM AUTHOR]

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence and uniqueness of an almost automorphic solution to the semilinear equation [image omitted] where [image omitted] is the infinitesimal generator of an exponentially stable C0-semigroup on a Banach space [image omitted] and [image omitted] is Sp-almost automorphic for p > 1 and jointly continuous. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give some sufficient conditions, which ensure the existence of pseudo almost periodic solutions to some classes of non-autonomous first-order partial neutral functional-differential with unbounded delay. [Copyright &y& Elsevier]

We establish the basic theory of almost periodic sequences on +. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models. [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]

Abstract: We give in this work some sufficient conditions for the existence and uniqueness of almost automorphic (mild) solutions to some classes of partial evolution equations. Then we use our abstract results to discuss the existence and uniqueness of almost automorphic solutions to some partial differential equations. [Copyright &y& Elsevier]

Abstract: The paper considers the existence and uniqueness of pseudo almost periodic (mild) solutions to some classes of first-order partial neutral functional–differential equations. Upon making some suitable assumptions, existence and uniqueness results are obtained. Applications include both a partial integro-differential equation arising in control systems and the scalar reaction–diffusion equation with delay. [Copyright &y& Elsevier]

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those classes of hyperbolic evolution equations are given. As an application, we consider the existence of pseudo almost periodic solutions to the heat equations with delay. [ABSTRACT FROM AUTHOR]

Abstract: Some sufficient conditions for the existence and uniqueness of pseudo-almost periodic (mild) solutions to some classes of partial evolution equations are given. We then make use of our abstract results to discuss the existence of pseudo-almost periodic solutions to some partial differential equations. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and uniqueness of pseudo-almost periodic solutions to the semilinear differential equations of the form where are densely defined closed linear operators on a Hilbert space , and is a jointly continuous function. Using both the so-called method of the invariant subspaces for unbounded linear operators and the classical Banach fixed-point principle, the existence of a pseudo-almost periodic solution to (*) is obtained under some suitable assumptions. As applications, we examine the existence and uniqueness of pseudo-almost periodic solutions to some second-order hyperbolic equations. [Copyright &y& Elsevier]