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.     

Massera type theorem for almost automorphic solutions of functional differential equations of neutral type

In this work, we study the existence of almost automorphic solutions for functional differential equations of neutral type. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution.     

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.     

C n -almost automorphic solutions for partial neutral functional differential equations

In this work, we study the existence of C ⁿ -almost periodic solutions and C ⁿ -almost automorphic solutions (n?≥?1), for partial neutral functional differential equations. We prove that the existence of a bounded integral solution on ?⁺ implies the existence of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions.     

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.     

Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces

We give sufficient conditions ensuring the existence, uniqueness and global attractiveness of a pseudo compact almost automorphic solution of the following differential equation:

x^′(t)=f(t,x(t))     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

Almost automorphy for abstract neutral differential equations via control theory

In this paper we study the existence of almost automorphic solutions for a class of linear neutral functional differential equations with finite delay and values in a Banach space. We show that the existence of an almost automorphic mild solution is related to the approximate controllability of a distributed control system. We applied our results to establish the existence of an almost automorphic solution for a neutral wave equation with delay.     

Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L¹(R) and the generator A of an L¹-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L¹(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.     

Almost Automorphic Solutions for Partial Functional Differential Equations with Infinite Delay

In this work, we study the existence of almost automorphic solutions for partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. We use the so-called reduction principle developed recently in [3], to show the existence of an almost automorphic solution under minimal condition. More precisely, the existence of an almost automorphic solution is proved when there is at least one bounded solution in the positive real half line. We give an application to the Lotka-Volterra model with diffusion.     

Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators

In this paper, by using the Leray-Schauder alternative, we have investigated the existence of mild solutions to first-order impulsive partial functional integrodifferential equations with nonlocal conditions in an α-norm. We assume that the linear part generates an analytic compact bounded semigroup, and that the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part. An example is also given to illustrate our main results.     

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.     

抽象空间中无穷时滞微分方程概自守解的存在性

研究抽象空间中无穷时滞微分方程概自守解的存在性,证明了在正实轴上存在有界解蕴含存在概自守解,并给出了结论在L otka-V o lterra型方程中的应用.我们的结果推广了经典的关于非齐次线性概周期微分方程概周期解存在性的结论.     

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S ^p -a.a.) functions. We establish a composition theorem about S ^p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S ^p -a.a. functions are discussed. Some theorems extend earlier results.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

Variation of constants formula and reduction principle for a class of partial functional differential equations with infinite delay

In this work, the dynamic behavior of solutions is investigated for a class of partial functional differential equations with infinite delay. We suppose that the undelayed homogeneous part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator. Firstly, a variation of constants formula is obtained in the corresponding α-norm space, which is mainly used to establish a reduction principle of complexity of the considered equation. The reduction principle proves that the dynamics of the considered equation is governed by an ordinary differential equation in finite dimensional space. As an application, we investigate the existence of periodic, almost periodic and almost automorphic solutions for the original equation.     

Pseudo almost automorphic solutions for some partial functional differential equations with infinite delay

In this work, we study the existence of pseudo almost automorphic solution for some partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition. We use the variation of constant formula developed recently in 1 Hino, Y, Murakami, S and Naito, T. 1991. “Functional Differential Equations with Infinite Delay”. In Lecture Notes in Mathematics, Vol. 1473, New York: Springer-Verlag.  [Google Scholar] to get the existence and uniqueness of pseudo almost automorphic solution when the linear equation has an exponential dichotomy. We also give an application of the abstract results to a Lotka–Volterra model with diffusion.     

Bounded mild solutions to fractional integro-differential equations in Banach spaces

We study the existence and uniqueness of bounded solutions for the semilinear fractional differential equation $$D^\alpha u(t)= Au(t)+ \int_{-\infty}^t a(t-s)Au(s)ds+ f \bigl(t,u(t) \bigr), \quad t \in\mathbb{R}, $$ where A is a closed linear operator defined on a Banach space X, α>0, a∈L ¹(?₊) is a scalar-valued kernel and f:?×X→X satisfies some Lipschitz type conditions. Sufficient conditions are established for the existence and uniqueness of an almost periodic, almost automorphic and asymptotically almost periodic solution, among other.     

Bounded mild solutions of perturbed Volterra equations with infinite delay

We study existence and regularity of bounded mild solutions on the real line to perturbed integral equations with infinite delay in the space of almost periodic functions (in the Bohr sense), the space of compact almost automorphic functions, the space of almost automorphic functions and the space of asymptotically almost automorphic functions.