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This paper investigates the existence of Eberlein weakly almost periodic solutions for differential equations of the form u ′ = A u + f (t) and u ′ = A (t) u + f (t) . In the first scenario, when A generates a strongly asymptotically semigroup, we establish the existence of Eberlein-weakly almost periodic solutions, thereby extending and improving a previous result in Zaidman(Ann Univ Ferrara 14(1): 29–34, 1969). In the second case, we consider a more general situation where A(t) is a (possibly nonlinear) operator satisfying a monotony condition. Unlike most existing works in the literature, our approach does not rely on tools of exponential dichotomy and Lipschitz nonlinearity. Lastly, we illustrate the practical relevance of our findings by presenting real-world models, including a hematopoiesis model, that exemplify the key findings. A numerical simulation is also provided. [ABSTRACT FROM AUTHOR]

In this work, we prove some new results concerning the class of Eberlein weakly almost periodic functions in Stepanov’s sense. We prove that if the forcing term of a partial functional differential equation with infinite delay is Eberlein-weakly almost periodic in Stepanov’s sense, then the solution is even Eberlein-weakly almost periodic. This shows that a less regular almost periodic behavior in the forcing term yields a more regular almost periodic behavior in the solution. The theoretical results are illustrated in the Lotka–Volterra model. [ABSTRACT FROM AUTHOR]

In this paper via the so-called Fenchel–Lagrange duality, we provide necessary local optimality conditions for a reverse convex programming problem (P) . As is well known, this duality has been first defined for convex programming problems. So, since in general problem (P) is not convex even if the data is, we first proceed to a decomposition of an equivalent problem of (P) into a family of convex minimization subproblems. Then, by means of the decomposition and the Fenchel–Lagrange duality applied to the subproblems we provide necessary local optimality conditions for the initial problem (P) . [ABSTRACT FROM AUTHOR]

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 thiswork, we provide sufficient conditions ensuring the existence and uniqueness of an Eberlein weakly almost periodic solutions for some semilinear integro-differential equations with infinite delay in Banach spaces. For illustration, we provide an example arising in viscoelasticity theory. [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]

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]

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]