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To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper we delineate in what sense the two semigroups are equivalent. In particular, we (i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, (ii) focus on large time behaviour and (iii) consider full orbits, i.e., orbits defined for time running from $-\infty $ to $+\infty $. Conceptually, the PDE formulation is by far the most natural one. It has, however, the technical drawback that the solution operators are not differentiable, precluding rigorous linearisation. For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearised Stability. Next the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation., Comment: To appear in Mathematical Models and Methods in the Applied Sciences

This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Let Bj (t) (j = 1,..., m) and B(t, τ) (t ≥ 0, 0 ≤ τ ≤ 1) be bounded variable operators in a Banach space. We consider the equation u ′ ( t ) = ∑ k = 1 m B k ( t ) u ( t - h k ( t ) ) + ∫ 0 1 B ( t , τ ) u ( t - h 0 ( τ ) ) d τ ( t ≥ 0 ) , u'\left( t \right) = \sum\limits_{k = 1}^m {{B_k}\left( t \right)u\left( {t - {h_k}\left( t \right)} \right)} + \int\limits_0^1 {B\left( {t,\tau } \right)u\left( {t - {h_0}\left( \tau \right)} \right)d\tau \,\,\,\,\left( {t \ge 0} \right),} where hk (t) (t ≥ 0; k = 1, ..., m) and h 0(τ) are continuous nonnegative bounded functions. Explicit delay-dependent exponential stability conditions for that equation are established. Applications to integro-differential equations with delay are also discussed

In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.

The objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

In 2020 Ramos et al.[5], investigated the existence and uniqueness of mild solutions for a fractional problem of the type Eqs.(1.1)-(1.3), however at the time, it was not considered the control function µ. In this paper, we impose control µ and through necessary and sufficient conditions, we investigate the results of controllability of fractional impulsive integro-differential control system in Banach spaces using the fixed point technique and the (α, θ)-resolvent operator.

We consider the well-posedness of the second-order degenerate differential equations with infinite delay (P2): (Mu′)′(t)+(Lu)′(t)=Au(t)+ ∫ −∞ta(t−s)Bu(s)ds+f(t), (0≤t≤2π) with periodic boundary conditions u(0)=u(2π), (Mu′)(0)=(Mu′)(2π), in Lebesgue–Bochner spaces Lp(𝕋;X) and periodic Besov spaces Bp,qs(𝕋;X), where A, B, L and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(L), D(A)∩D(B)≠{0} and a∈L1(ℝ+). We completely characterize the well-posedness of (P2) in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.

We study the existence and regularity of solutions for neutral integro-differential equations with nonlocal condition by applying the theory of resolvent operators established recently in the literature. Since the nonlinear terms of the systems involve spacial derivatives, we make full use of the theory of fractional power, the [math] -norm, and Schauder’s fixed point theorem to discuss the problems. An example is given to illustrate the applications of the obtained results.

This paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(A,B)$\end{document}(A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

We analyze the lumpability of linear systems on Banach spaces, namely, the possibility of projecting the dynamics by a linear reduction opera-tor onto a smaller state space in which a self-contained dynamical description exists. We obtain conditions for lumpability of dynamics defined by unbounded operators using the theory of strongly continuous semigroups. We also derive results from the dual space point of view using sun dual theory. Furthermore, we connect the theory of lumping to several results from operator factoriza-tion. We indicate several applications to particular systems, including delay differential equations. © 2017, American Institute of Mathematical Sciences. All rights reserved.

We study the existence and uniqueness of strict and $\bf C^{1+\alpha}$-strict solutions for a general class of abstract integro-differential equations with state-dependent delay. Some examples concerning partial integro-differential equations with state dependent delay are presented.

The purpose of this paper is to study a three species competition model with diffusion. It is well known that there exists a family of traveling wave solutions connecting two equilibria $(0,1,1)$ and $(1,0,0)$. In this paper, we first establish the exact asymptotic behavior of the traveling wave profiles at $\pm \infty$. Then, by constructing a pair of explicit upper and lower solutions via the combination of traveling wave solutions, we derive the existence of some new entire solutions which behave as two traveling fronts moving towards each other from both sides of $x$-axis. Such entire solution provides another invasion way of the stronger species to the weak ones.

We prove the existence and uniqueness of the solutions for the following impulsive semilinear evolution equations with delays and nonlocal conditions: $$\left\{ \begin{array}{lr} \acute{z} =-Az +F(t,z_{t}), & z\in Z, \quad t \in (0, \tau], t \neq t_k, \\ z(s)+(g(z_{\tau_1},z_{\tau_2},\dots, z_{\tau_q}))(s) = \phi(s), & s \in [-r,0],\\ z(t_{k}^{+}) = z(t_{k}^{-})+J_{k}(z(t_{k})), & k=1,2,3, \dots, p. \end{array} \right.$$ where $0 \lt t_1 \lt t_2 \lt t_3 \lt ··· \lt t_p \lt \tau, 0 \lt \tau_{1} \lt \tau_{2} \lt ··· \lt \tau_{q} \lt r \lt \tau, Z$ is a Banach space $Z$, $z_t$ defined as a function from $[−r, 0]$ to $Z^ \alpha$ by $z_{t}(s) = z(t + s),−r \le s \le 0, g : C([−r, 0];Z^{\alpha}_{q}) \rightarrow C([−r, 0];Z^{\alpha})$ and $J_{k} : Z^{\alpha} \rightarrow Z^{\alpha}, F : [0,\tau] ×C(−r,0;Z^{\alpha}) \rightarrow Z$. In the above problem, $A : D(A)\subset Z \rightarrow Z$ is a sectorial operator in $Z$ with $−A$ being the generator of a strongly continuous compact semigroup $\{T(t)\}_{t \ge 0}$, and $Z^{\alpha}= D(A^{\alpha})$. The novelty of this work lies in the fact that the evolution equation studied here can contain non-linear terms that involve spatial derivatives and the system is subjected to the influence of impulses, delays and nonlocal conditions, which generalizes many works on the existence of solutions for semilinear evolution equations in Banch spaces. Our framework includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mohany equation with impulses, delays and nonlocal conditions.

This paper is concerned with the nonlinear stability of traveling waves of a delayed monostable epidemic model with quasi-monotone condition. We prove that the traveling wave front is exponentially stable by means of the weighted-energy method and the comparison principle to perturbation in some exponentially weighted $L^{\infty}$ spaces, when the difference between initial data and traveling wave front decays exponentially as $x \to -\infty$, but the initial data can be suitable large in other locations. Finally, we present two examples to support our theoretical results.

The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild solutions is introduced through a new type of variation of constants formula and the well-posedness is established under a semilinear stability condition with respect to a metric-like functional and a subtangential condition. The abstract result is applied to a size-structured model with birth delay.

In this article we study the existence of positive periodic solutions for a fourth-order nonlinear neutral differential equation with variable delay. The main tool employed here is the Krasnoselskii's fixed point theorem dealing with a sum of two mappings, one is a contraction and the other one is completely continuous.

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space L^p(I,X), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in L^p(I,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces., Comment: 40 pages

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

In this paper, we consider a class of nonlocal dispersal equation with nonlocal time-delayed reaction. We prove that all noncritical wavefronts are globally exponentially stable by the weighted energy method and comparison principle. However, for the critical wavefronts, we prove that they are globally asymptotically stable.

In this paper, we consider with the stability of traveling wave fronts for the nonlocal diffusion equation with delay and global response. We first establish the existence and comparison theorem of solutions for the nonlocal reaction-diffusion equation by appealing to the theory of abstract functional differential equation. Then we further show that the traveling wave fronts are asymptotical stability with phase shift. Our main technique is the super and subsolution method coupled with the comparison principle and squeezing method.

The aim of this work is to present new approach to study weighted pseudo almost periodic and automorphic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We study the existence and uniqueness of $(\mu,\nu)$-pseudo almost periodic and automorphic solutions of class $r$ for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed in Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition, the delayed part are assumed to be pseudo almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument.

Letube a given bounded uniformly continuous mild solution of a higher-order abstract functional differential equation of delay or advance type. We give a so-called Massera-type criterion for the existence of a mild solution, which is a “spectral component” ofuwith spectrum similar to the one of the forcing termf. Various spectral criteria for the existence of almost periodic and quasiperiodic mild solutions are given.

In the present work, we are concerned with approximations of solutions to a retarded type fractional differential equation with a deviated argument in a separable Hilbert space~$H$. We consider an integral equation associated with a given problem and then consider a sequence of approximate integral equations. We prove the existence, uniqueness and convergence to each of the approximate integral equations by using analytic semigroup theory and the fixed point method. We also prove that the limiting function satisfies the associated integral equation. Finally, we consider Faedo-Galerkin approximations of solutions and prove some convergence results.

In this paper we prove the global existence and attractivity of mild solutions for neutral semilinear evolution equations with state-dependent delay in a Banach space.

International audience; We use a new concept of weighted ergodic function based on the measure theory to investigate the existence and uniqueness of weighted pseudo almost periodic solution for a class of partial functional differential equations with infinite delay in fading memory spaces. We illustrate our theoretical results by studying some Lotka-Voltera reaction-diffusion systems with infinite delay.

Under suitable assumptions on the delay operator $F$, we give necessary and sufficient conditions for the inhomogeneous abstract degenerate delay equations: $ (Mu)'(t)=Au(t)+Fu_{t}+f(t), \ (t\in \T)$ to have $L^p$-maximal regularity.

Toka DiaganaHoward University,2441 6th Street N.W.,Washington, D.C. 20059,USA.tdiagana@howard.eduKhalil EzzinbiUniversit´e Cadi Ayyad,Faculte´ des Sciences Semlalia,D´epartement deMath´ematiques,BP 2390, Marrakesh, MarocMohsen MiraouiInstitut sup´erieur des Etudestechnologiques de Kairouan,Rakkada-3191 Kairouan,Tunisie.ABSTRACTMotivated by the recent works by the first and the second named authors, in this paperwe introduce the notion of doubly-weighted pseudo-almost periodicity (respectively,doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basicproperties of these new spaces are studied. To illustrate our work,westudy,underAcquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions tosome nonautonomous partial evolution equations in Banach spaces. A few illustrativeexamples will be discussed at the end of the paper.RESUMENMotivado por los trabajos recientes del primer y segundo autor, en este art´iculo intro-ducimos la noci´on de seudo-casi periodicidad con doble peso (seudo-casi automorf´ia condoble peso respectivamente) usando Teor´ia de la Medida. Se estudian las propiedadesbasicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones deAcquistapace-Terreni y dicotom´ia exponencial estudiamos la existencia de soluciones(respectivamente, (µ,ν) seudo-casi peri´odicas (µ,ν) seudo-casi automorficas) para al-gunas ecuaciones parciales de evoluci´on autonomas en espacios de Banach. Algunosejemplos ilustrativos se discutiran al final del art´iculo.Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni con-ditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomousequation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy;(µ,ν)-pseudo-almost periodicity; (µ,ν)-pseudo-almost automorphy; neutral systems; positive mea-sure.2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;47D06; 28Axx; 58D25; 65J08.

The objective of this work is to give some relationship between the Favard spaces and the $p$-admissibility (resp. $(p,q)$-admissibility) of unbounded control operators for linear (resp; bilinear) systems in Banach spaces. For linear case, this enables to give a simple identification of the space of the $1-$admissible control operators in Banach space and it enables us to extend the result of Weiss [29] (for $p=1$) on reflexive Banach spaces to a general situation. This result is applied to boundary control systems. The results obtained for bilinear systems generalize those given in Idrissi [16] and Berrahmoune [2] and are applied to diffusion equations of fractional order time distributed order.

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form(*) x′=f(t,x,xt).To this end, we first associate with every almost periodic semilinear equation(**) x′=F(t,x).a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the functionfin terms ofxand Lipschitz conditions with respect toxt.

In this article, we study the existence of mild solutions for the nonlocal Cauchy problem for a class of abstract fractional neutral integro-differential equations with infinite delay. The results are obtained by using the theory of resolvent operators. Finally, an application is given to illustrate the theory.

We study the $(W^{2,p},W^{1,p})$-mild well-posedness of the second order differential equation $(P_2): u'' = Au+f$ on the real line $\mathbb{R}$, where $A$ is a densely defined closed operator on a Banach space $X$. We completely characterize the $(W^{2,p}, W^{1,p})$-mild well-posedness of $(P_2)$ by $L^p$-Fourier multiplier defined by the resolvent of $A$.

This paper is concerned with the existence of mild solutions, strong solutions and strict solutions for a class of neutral integrodifferential equations with nonlocal conditions in Banach space. Since the nonlinear terms of the systems involve spacial derivatives, the theory of fractional power and $\alpha$-norm is used to discuss the problem. In the end an example is provided to illustrate the applications of the obtained results.

We establish new and general existence theorems of Lipschitz continuous solutions to integral equations with infinite delay in Banach spaces, as well as of strong and classical solutions to the Cauchy problem for some evolution equations with infinite delay. An example is given to illustrate the abstract result.

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic equilibria for nonlinear partial neutral functional differential equations. We suppose that the linear part $A$ satisfies the Hille-Yosida condition on a Banach space and is not necessarily densely defined; the delayed part is assumed to be Lipschitz. We show the existence of stable and unstable manifolds near hyperbolic equilibria when the neutral operator is stable and the semigroup generated by the part of $A$ in $\overline{D(A)}$ is compact. Local stable and unstable manifolds are also obtained when the undelayed part is a C$^{1}$ function in a neighborhood of the equilibria.

In this paper, we investigate existence for a class of partial neutral functional integrodifferential equations with infinite delay. Using the integrated semigroup theory and the well known Banach fixed point theorem, we establish local existence and uniqueness of integral solutions to these equations. To illustrate our abstract results, we conclude this work by an example. .

In this paper, we use the extrapolation method combined with a fixed-point theorem for the sum of completely continuous and contraction operators, to establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some classes of nondensely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces with infinite delay.

The paper will investigate the local existence, uniqueness of solution to integrodifferential equations with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies a Hille-Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied.

This paper deals with the approximation of the spectrum of linear and nonautonomous delay differential equations through the reduction of the relevant evolution semigroup from infinite to finite dimension. The focus is placed on classic collocation, even though the requirements that a numerical scheme has to fulfill in order to allow for a correct approximation of the spectral elements are recalled. This choice, motivated by the analyticity of the underlying eigenfunctions, allows for a convergence of infinite order, as rigorously demonstrated through a priori error bounds when Chebyshev nodes are adopted. Fundamental applications such as determination of asymptotic stability of equilibria (autonomous case) and limit cycles (periodic case) follow at once., 31 pages

Under suitable assumptions on the Fourier transform of the delay operator $F$, we give necessary and sufficient conditions for the inhomogeneous abstract delay equations: $ u'(t)=Au(t)+Fu_{t}+f(t), \ (t\in \mathbf T)$ to have maximal regularity in Besov spaces $B_{p,q}^s(\mathbf T, X)$ and Triebel-Lizorkin spaces $F_{p,q}^s(\mathbf T, X)$. .

We study the controllability problem for a system governed by a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. Notice that we are not assuming that the generalized semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows to obtain the general controllability principle in the terms of the topological degree theory for condensing multivalued operators. Two realizations of this principle are considered.

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some non-densely defined impulsive semilinear functional differential equations in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations are also considered.

In this paper, we study the existence of asymptotically almost periodic solutions to some classes of second-order partial functional-differential equations with unbounded delay. Our results are, subsequently, applied to studying the existence of asymptotically almost periodic solutions to some integrodifferential equations, which arise in the theory of heat conduction within fading memory materials.

In this paper, we establish some results to show the existence and uniqueness of classical solution of the non-autonomous second-order integrodifferential equation of the type: \begin{eqnarray*} u''(t) & = & A(t)\;[u(t)+\int_0 ^{t} k_{1}(t,s)u(s)ds]'+B(t)\;[u(t)+\int_0 ^{t} k_{2}(t,s)u(s)ds]\\ & & +f(t),\; 0\leq t \leq T, \\ u(0) & = & u_{0} \in E,\quad u'(0)=u_{1} \in E, \end{eqnarray*} on a Banach space $E$, by means of the matrix operator method.

In this work, we discuss the existence, regularity and stability of solutions for some partial functional differential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space $X$ and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.