Gaston M. N’Guérékata

Stepanov-like almost automorphic functions and applications to some semilinear equations

This article is concerned with the existence and uniqueness of an almost automorphic solution to the semilinear equation where is the infinitesimal generator of an exponentially stable C 0-semigroup on a Banach space and is Sp -almost automorphic for p > 1 and jointly continuous.

Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces

Abstract We prove in this paper the existence and uniqueness of mild solutions to some functional differential and functional integro-differential equations with infinite delay in Banach spaces which approach almost automorphic functions at infinity. We also discuss the existence of S-asymptotically ω - periodic mild solutions. The results are new.

S-asymptotically ω-periodic solutions to some classes of partial evolution equations

Abstract In this paper, we give some sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic (mild) solutions to some classes of partial evolution equations in Banach spaces. The main result is obtained by means of the Banach fixed point principle.

Almost automorphic solutions of second order evolution equations

This article is concerned with the existence of almost automorphic mild solutions to second order evolution equations of the form where A generates a strongly continuous semigroup and f is an almost automorphic function. Using the notion of uniform spectrum of a function and the method of sums of commuting operators in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to (*) in terms of spectrum of A and uniform spectrum of f. Moreover, we study the nonlinear perturbation of this equation and obtain an extension of results by Diagana and N’Guérékata.

Antiperiodic solutions of semilinear integrodifferential equations in Banach spaces

Abstract We are concerned in this paper with the existence of antiperiodic mild solutions of some semilinear integrodifferential equations in a Banach space. We study further properties of antiperiodic functions including a new composition theorem and introduce the new concept of asymptotically antiperiodic functions.

Paradox of enrichment: A fractional differential approach with memory

Abstract The paradox of enrichment (PoE) proposed by Rosenzweig [M. Rosenzweig, The paradox of enrichment, Science 171 (1971) 385–387] is still a fundamental problem in ecology. Most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking. Knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels. PoE resolution via these two organizational levels can be interpreted as a microscopic- and macroscopic-level solution. Fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes. The derivatives can be physically interpreted via two time scales that are considered simultaneously: the ideal, equably flowing homogeneous local time, and the cosmic (inhomogeneous) non-local time. Several mechanisms and theories have been proposed to resolve the PoE problem, but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non-local effects, which capture memory. Here we formulate the fractional counterpart of the Rosenzweig model and analyze the stability behavior of a system. We conclude that there is a threshold for the memory effect parameter beyond which the Rosenzweig model is stable and may be used as a potential agent to resolve PoE from a new perspective via fractional differential equations.

A note on the existence of positive bounded solutions for an epidemic model

Abstract In this short note, we establish an existence and uniqueness theorem about a positive bounded solution for a nonlinear infinite delay integral equation, which arises in some epidemic problems. As one can see, our main result can deal with some cases, to which many previous results cannot be applied. In addition, we show that our main result can also be applied to a Lasota–Wazewska model.

Almost Automorphic Functions of Order and Applications to Dynamic Equations on Time Scales

We revisit the notion on almost automorphic functions on time scales given by Lizama and Mesquita (2013). Then we present the notion of almost automorphic functions of order . Finally, we apply this notion to study the existence and uniqueness and the global stability of almost automorphic solution of first order to a dynamical equation with finite time varying delay.

Asymptotic Behavior of Mild Solutions to Semilinear Fractional Differential Equations

This paper is mainly concerned with the asymptotically almost automorphic mild solutions to a semilinear fractional differential equation. Some asymptotic behavior of mild solutions to this equation has been established by properties and composition theorems of asymptotically almost automorphic functions and fixed point theorems.

Existence of Antiperiodic Solutions to Semilinear Evolution Equations in Intermediate Banach Spaces

We are concerned in this paper with the antiperiodicity of mild solutions for the semilinear evolution equation \(x^{\prime}(t) = Ax(t) + f(t,x)\)where A is a sectorial operator not necessarily densely defined in X generating an hyperbolic semigroup \((T(t))_{t\geq 0}\)in a Banach space X and \(f : \mathbb{R} \times X_{\alpha } \rightarrow X\), where X α is an intermediate space. We prove the existence and uniqueness of an antiperiodic mild solution in X α , when the function \(f : \mathbb{R} \times X_{\alpha }\rightarrow X\)is antiperiodic. The result is obtained using the Banach-fixed point theorem.