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In this paper, we prove that the class of Eberlein weakly almost periodic functions in the Stepanov's sense is strictly larger than both Eberlein weak almost periodicity (E‐w.a.p.) and Stepanov's almost periodicity. Some examples are provided. We present a composition result for this class of functions, and using Banach fixed‐point theorem, we prove the existence and uniqueness of E‐w.a.p. solution for a differential equation of neutral type with nondense domain when the forcing terms are only E‐w.a.p. in the Stepanov's sense. We mention that our results can extend and improve previous several works in this direction. An application is given to illustrate the result. [ABSTRACT FROM AUTHOR]

In this article, we prove a number of properties concerning a (new) class of (Stepanov-like) Eberlein-weakly almost periodic (SP-E.w. a. p.) functions with values in a Banach space. We use the results obtained to study the asymptotic behaviour of solutions to the evolution equation:whereAis the generator of an integral resolvent family in a Banach space 𝕏,a ∈ L1(ℝ), andfis a given 𝕏-valued function on ℝ. The objective is to deduce Eberlein-weakly almost periodicity (in Stepanov-like sense) of the solutionufrom corresponding properties on the partf. [ABSTRACT FROM AUTHOR]

By means of the fixed point methods and the properties of the pseudo almost automorphic functions, the existence and uniqueness of pseudo almost automorphic solutions are obtained for differential equations involving reflection of the argument. For the nonscalar, case we use the exponential dichotomy properties. [ABSTRACT FROM AUTHOR]

In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic solutions for some neutral partial functional differential equations. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main result, we propose to study the existence and uniqueness of a weighted pseudo-almost periodic solution for some neutral model arising in physical systems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We study some properties of bounded and almost automorphic solutions of the following nonlinear differential equation: where is a Banach space and is almost automorphic with a parameter. We give sufficient conditions for the existence of almost automorphic solutions when the differential equation admits at least a solution with a relatively compact range defined on the half-line of positive numbers. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: This work aims to investigate the existence and uniqueness of pseudo-almost-automorphic solutions for some neutral partial functional differential equations in Banach spaces. Recall that the new concept of pseudo-almost-automorphy generalizes the one of the pseudo-almost-periodicity and it has been recently introduced in the literature. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed parts are assumed to be pseudo-almost-automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of -almost-periodic 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 reduction principle developed recently in [M. Adimy, K. Ezzinbi, A. Ouhinou, Variation of constants formula and almost-periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications 317 (2006) 668–689] to prove the existence of a -almost- periodic solution when there is at least one bounded solution in . We give an application to the Lotka–Volterra model with diffusion. [Copyright &y& Elsevier]

We study some properties of bounded and C(1)-almost automorphic solutions of the following Liénard equation: x″ + f (x) x′ + g(x) = p(t), where p : R → R is an almost automorphic function, f, g : (a, b) → R are continuous functions and g is strictly decreasing. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence and uniqueness of an almost automorphic solution for some dissipative ordinary and functional differential equations in Banach spaces. We prove the existence and uniqueness of an almost automorphic solution for dissipative differential equations and then we apply this result to show the existence of almost automorphic solutions for some functional differential equations. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of Cn-almost periodic solutions and Cn-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 Cn-almost periodic and Cn-almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of Cn-almost periodic and Cn-almost automorphic strict solutions. [ABSTRACT FROM AUTHOR]

We study the convolution of Stepanov-like almost automorphic functions and L¹ functions. Also we consider nonautonomous evolution equations, with a periodic operator coefficient and Stepanov-like almost automorphic forcing, and show that, under certain assumptions, any bounded mild solution is almost automorphic. [ABSTRACT FROM AUTHOR]

In this work, we use recent results on pseudo almost periodicity to study a class of non-autonomous cellular neural networks with mixed delays. Sufficient conditions are obtained in context of general measure theory (d μ (x) = ρ (x) dx + d μ 1 (x)), for existence, uniqueness and global exponentially stability of μ -pseudo almost periodic solutions of the considered model. As a consequence, we establish the exponential attractiveness of μ -pseudo almost periodic solutions of the addressed model as t ⟶ + ∞. Some examples and numerical simulations are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we study the existence and uniqueness of a pseudo-almost automorphic solution for some partial functional differential equations of neutral type. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition. The delay part is assumed to be pseudo-almost automrophic in time. We prove if the homogeneous linear equation has an exponential dichotomy, then the nonhomogeneous linear equation has a unique pseudo-almost automorphic solution. An application is given for some nonlinear equations. We present also a new concept of generalized pseudo-almost automorphy. [Copyright &y& Elsevier]

We study the existence of compact almost automorphic weak solutions for the differential inclusion u ′ (t) + A u (t) ∋ f (t) for t ∈ R , where A : D (A) ⊂ H ⟶ 2 H is maximal monotone and the forcing term f is compact almost automorphic. We prove that the existence of a uniformly continuous weak solution on R + having a relatively compact range over R + implies the existence of a compact almost automorphic weak solution. For that goal, we use Amerio's principle. We prove also the existence, uniqueness, and global attractivity of a compact almost automorphic weak solution where A is strongly maximal monotone. For illustration, some applications are provided for parabolic and hyperbolic equations. [ABSTRACT FROM AUTHOR]