Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

Almost automorphic solutions to some classes of partial evolution equations

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.     

Almost automorphic solutions to some damped second-order differential equations

In this paper we make extensive use of dichotomy tools and the well-known Schauder fixed point principle to study and obtain the existence of almost automorphic solutions to some nonautonomous damped second-order differential equations. To illustrate our abstract result, we study the existence of almost automorphic solutions to the so-called plate-like equation.     

Some properties of pseudo-almost automorphic functions and applications to abstract differential equations

This paper is concerned with some properties of pseudo-almost automorphic functions, which are more general and complicated than pseudo-almost periodic functions. Using these properties, we establish an existence and uniqueness theorem for pseudo-almost automorphic mild solutions to semilinear differential equations in a Banach space.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Weighted pseudo-almost automorphic solutions for some partial functional differential equations

In this paper, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some nonhomogeneous partial functional differential equations. We use the variation of constants formula developed in Ezzinbi and N’Guérékata (2007) [11] and the spectral decomposition of the phase space to show the main result of this work. To illustrate our main result, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some diffusion equations with delay.     

非自治中立型无穷时滞泛函微分方程的加权伪概自守解

首先引入h型Stepanov 加权伪概自守函数和∞型Stepanov加权伪概自守函数的概念, 接着建立了其函数空间的完备性以及相应组合定理, 最后证明了一类非自治无穷时滞偏中立型泛函微分方程在S^p-加权伪概自守系数下加权伪概自守解的存在唯一性.     

Measure theory and pseudo almost automorphic functions: New developments and applications

In this work, we establish a new concept of weighted pseudo almost automorphic functions using the measure theory. We present new results on weighted ergodic functions like completeness and composition theorems. The theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions. For illustration, we provide some applications for evolution equations which include reaction-diffusion systems and partial functional differential equations.     

Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.     

Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications

In this paper, we introduce a new concept of bi-almost automorphic functions and obtain new existence and uniqueness theorems for pseudo-almost automorphic mild solutions to several nonautonomous differential equations. Moreover, two examples are given to illustrate the general theorems.     

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

The aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.     

k-Hypermonogenic automorphic forms

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the k-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of k-hypermonogenic automorphic forms.     

Weyl almost automorphic solutions for a class of Clifford-valued dynamic equations with delays on time scales

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford-valued shunt inhibitory cellular neural networks with time-varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results.     

Almost automorphic solutions for some stochastic differential equations

The concept of distributional almost automorphy for stochastic processes is introduced. The existence and uniqueness of distributionally almost automorphic solutions to some linear and nonlinear stochastic differential equations are established for cases where the coefficients satisfy certain conditions.     

Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy~ and the results also hold for some nonlinear equations with the form x＇（t） = f（t,x（t）） ＋ λg（t,x（t））, where f,g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.     

Modified Bers operators and the duality of multiplicative holomorphic automorphic forms

The integral Bers operator, related to a reflection in some quasicircle, plays an important role in the theory of single-valued automorphic forms [1, 2]. Study was started in [3] of the normed spaces of measurable and multiplicative holomorphic automorphic forms for a Fuchsian group. In the present article we introduce some basic multiplicative modifications of the Bers operator and the corresponding bilinear pairing in connection with duality in the spaces of multiplicative holomorphic automorphic forms. Under study we obtain a universal norm estimate and establish selfadjointness for all operators.     

Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain

In this work we study the existence and uniqueness of compact almost automorphic solutions to a first-order differential equation with a linear part dominated by a Hille-Yosida type operator with non dense domain.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).