# Khalil EzzinbiCadi Ayyad University | UCAM · Department of Mathematics

Khalil Ezzinbi

Professor

## About

230

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Introduction

**Skills and Expertise**

## Publications

Publications (230)

In this work, we investigate the existence of mild solutions in the α \alpha -norm for a class of nonlocal integrodifferential equations. We employ the theory of resolvent operators introduced by R. Grimmer. The Leray-Schauder alternative is the principal working tool for our analysis.

In this work, we discuss the approximate controllability of some nonlinear partial functional
integrodifferential equations with nonlocal initial condition in Hilbert spaces. We assume that the
corresponding linear part is approximately controllable. The results are obtained by using fractional
power theory and α-norm, the measure of noncompactness...

The main goal of this work is to investigate the approximate controllability for a class of nonautonomous delayed integrodifferential inclusions
with unbounded delay. Our technique starts with the search for the optimal control for a linear quadratic regulator problem. The existence of such
an optimal control aids to establish sufficient conditions...

This work is related to the problem of the existence of almost periodic and almost automorphic integral solutions for some nondensely periodic linear partial functional differential equations with infinite delay in Banach space. First, we prove a variation of constants formula of integral solutions in the phase space of nonautonomous linear equatio...

The aim of this work is to study an approximation of the mild solution of a delay semi linear integro-differential equation with an initial history condition $\varphi$. Using resolvent operators theory in the sense given by R. Grimmer, we can ensure an explicit form to the mild solution $u^{\varphi}$ of our considered equation. The approximation ta...

This article establishes a reduction principle for partial functional differential equation without compactness of the semigroup generated by the linear part. Under conditions more general than the compactness of the C0-semigroup generated by the linear part, we establish the quasi-compactness of the C0-semigroup associated to the linear part of th...

In this work, we establish a new variation of constants formula for nonautonomous linear partial functional differential equations with linear part A satisfying the Hille-Yosida condition on a Banach space and is not necessarily densely defined. As well, we use this formula and the spectrum of the monodromy operator when the semigroup generated by...

The aim of this chapter is to study the regularity and the stability in the α-norm for neutral partial functional differential equations in fading memory spaces. We assume that a linear part is densely defined and generates an analytic semigroup. The delayed part is assumed to be Lipschitzian. For illustration, we provide an example for some reacti...

This chapter is devoted to the study of the existence results of local and maximal solutions on the one hand and the existence and uniqueness results of mild solutions on the second hand, for the non-autonomous evolution equation with finite delay \(\displaystyle \frac {d}{dt}u(t)=A(t)u(t)+f(t,u_{t}),\,\,\,t\in [0,T]\), subjected to the initial dat...

In this work, we study the existence of the mild solutions and the approximate controllability of some integrodifferential evolution systems with multi-valued nonlocal conditions. We use the resolvent operators theory to obtain sufficient conditions for the existence and controllability results. We present a general Kalman's controllability criteri...

The aim of this work is to study the approximate controllability for some fractional neutral inclusion system with nonlocal conditions. We establish a new variation of constant formula that helps us to formulate the problem of the approximate controllability. We assume that the linear system without the input functions is approximately controllable...

In this paper, we present a framework for investigating the mild solution of stochastic integrodifferential equations with nonlocal initial conditions in Hilbert spaces. In order to obtain the existence results, we have made use of the ideas of fixed point theorems, stochastic analysis, and the resolvent operator in the sense of Grimmer. Finally, a...

In this work, we study a class of abstract non-autonomous partial functional differential equations with infinite delay. Our main results concern the local existence of the mild solution which can blow up at the finite time. The unbounded operators associated to the non-autonomous system are assumed to be stable family which generates C 0 -semigrou...

In this paper, the optimal controls for semilinear integrodifferential evolution equations are investigated in Banach spaces. We demonstrate the existence of feasible pairs by utilizing the Cesari condition and the Filippov Theorem. Furthermore, a result on the existence of optimal control pairs for the Lagrange issue is presented.

In this work, we use measure theory to provide new sufficient conditions for the existence, uniqueness and global exponential stability of piecewise pseudo almost periodic solutions of neutral-type inertial neural networks with mixed delay and impulsive perturbations. By proposing a variable substitution, the neutral-type inertial neural networks c...

This work is devoted to the study of a class of nonlocal impulsive integrodifferential equations of Volterra type. We investigate the situation when the resolvent operator corresponding to the linear part of $$x' (t) = A(t)x(t)+\int_0^t\Gamma(t,s)x(s)ds + f(t,x(t)), t ∈ I =[0,T], t̸ =t_i, i =1,2,3,...,m, x(t+ i ) =x(t_i)+J_i(x(ti)), \\x(0) = x_0+g(...

In this paper, optimal control problems for a class of stochastic functional integral-differential equations in Hilbert spaces are investigated. First, the existence of mild solutions is investigated using stochastic analysis theory, fixed point theorems, and Grimmer’s resolvent operator theory. Following that, the existence requirements of optimal...

The goal of this study is to investigate the existence and uniqueness of mild solutions, as well as controllability outcomes, for random integrodifferential equations with state-dependent delay. We prove the existence and uniqueness of mild solutions in the case where the nonlinear term is of the Carathéodory type and meets various weakly compactne...

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. We focus on a specific category of nonlocal integrodifferential equations. We apply the theory of the resolvent operator in the sense of Grimmer, as well as the fixed point strategy and the theory of the Lebesgue-Stieljes integral, in the c...

The main goal of this work is to examine the periodic dynamic behavior of some retarded periodic partial differential equations (PDE). Taking into consideration that the linear part realizes the Hille-Yosida condition, we discuss the Massera’s problem to this class of equations. Especially, we use the perturbation theory of semi-Fredholm operators...

The aim of this work is to study the existence of a global attractor for some partial neutral functional differential equations; the convergence of all the solutions to the attractor is given in terms of the alpha-norm which is more important than the classical norm. Firstly, we show some interesting properties on the semigroup solution like the as...

The aim of this work is to investigate the existence and uniqueness of μ-pseudo almost periodic (resp. μ-pseudo compact almost automorphic) weak solutions for the following partial functional differential inclusion: x′(t)+Ax(t)∋f(t,xt)for t∈R,where A:H⟶2H is a strongly maximal monotone operator on a real Hilbert space H, f:R×C⟶H is a Stepanov μ-pse...

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 (dμ(x)=ρ(x)dx+dμ1(x)), for existence, uniqueness and global exponentially stability of μ-pseudo almost periodic solutions of the consi...

In this paper, we consider a class of random partial integro-differential equations with unbounded delay. Existence of mild solutions are investigated by using a random fixed point theorem with a stochastic domain combined with Schauder’s fixed point theorem and Grimmer’s resolvent operator theory. The results are obtained under Carathéodory condit...

In this paper we give suﬃcient conditions on k ∈ L 1(ℝ) and the positive measures µ, ν such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo- almost automorphic) function spaces are invariant by the con- volution product ζf = k ∗ f. We provide an appropriate example to illustrate our convolution results. As a cons...

In this work, we study the controllability of a class of stochastic integrodifferential equations with noncompact semigroups and nonlocal condition on Hilbert spaces.We suppose that the linear part admits a resolvent operator in the sense of Grimmer. Our main results are obtained by utilizing stochastic analysis theory, Kuratowski measure of noncom...

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the me...

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution imp...

In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the...

In this work, we deal with the existence, uniqueness and controlla-bility outcomes for stochastic partial integro-differential equations with nonlocal conditions. Our analysis uses Kuratowski's measure of noncompactness, generalized Darbo's fixed point Theorem, Mönch fixed point Theorem and Grimmer's resolvent operator theory. Some appropriate cond...

In this paper, we show the existence of mild solutions for a class of neutral partial integrodifferential equations with lack of compactness. The results are obtained using noncompact resolvent operators and a new fixed point theorem of Monch-Krasnosel’skii type. Our results are applied to a large variety of partial differential equations in which...

The aim of this work is to give sufficient conditions ensuring that the space PAP(, X, µ) of µ-pseudo almost periodic functions and the space PAA(, X, µ) of µ-pseudo almost automorphic functions are invariant by the convolution product f = k * f, k ∈ L1(). These results establish sufficient assumptions on k and the measure µ. As a consequence, w...

In this work, we prove the existence and uniqueness of $\mu$-pseudo almost automorphic solutions for some class of semilinear nonautonomous evolution equations of the form: $ u'(t)=A(t)u(t)+f(t,u(t)),\; t\in\mathbb{R} $ where $ (A(t))_{t\in \mathbb{R}} $ is a family of closed densely defined operators acting on a Banach space $X$ that generates a s...

The aim of this work is to study the stability for some integro-differential equations with impulses driven by fractional Brownian motion with noncompact semigroup in Hilbert spaces. We assume that the linear part has a resolvent operator not necessary compact but is operator norm continuous. Sufficient conditions for the existence of mild solution...

In this work, we study the existence and uniqueness of almost automorphic solutionsfor semilinear nonautonomous parabolic evolution equations with inhomogeneousboundary conditions using the exponential dichotomy. We assume that the homoge-neous problem satisfies the “Acquistapace–Terreni” conditions and that the forcingterms are Stepanov-like almos...

In this work, we study the existence and regularity of solutions for a class of nondensely defined partial functional integrodifferential equations. We suppose that the undelayed part admits an integrated resolvent operator in the sense given by Oka [J. Integral Equations Appl. 7(1995), 193–232.]. We give some sufficient conditions ensuring the exi...

We study the existence of compact almost automorphic weak solutions for the differential inclusion u′(t)+Au(t)∋f(t) for t∈R, where A:D(A)⊂H⟶2H 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 exist...

This work is a survey of many papers dealing with new methods to study partial functional differential equations. We propose a new reduction method of the complexity of partial functional differential equations and its applications. Since, any partial functional differential equation is well-posed in a infinite dimensional space, this presents many...

The paper is concerned with the study of some partial neutral functional integrodifferential equations with nondense domain. Using the integrated resolvent operator theory, we derive some results concerning the existence and regularity of solutions. Finally, an example is given to illustrate our theory.

This work concerns the study of the approximate controllability for
some nonlinear partial functional integrodifferential equation with infinite
delay arising in the modelling of materials with memory, in the framework
of Hilbert spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its li...

The aim of this work is to prove some results about the existence and regularity of solutions for some partial integrodifferential equations with nonlocal conditions. We suppose that the linear part has a resolvent operator in the sens given by Grimmer. The non linear part is assumed to be continuous and Lipschitzian with respect to the second argu...

This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion {S_{Q}^{H}(t)} , with Hurst parameter {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions,...

In this work, we study the asymtotic behavior of the mild solutions of a class of stochastic partial functional integrodifferential equation on Hilbert spaces. Using the stochastic convolution developped in [12], we establish the exponential stability in mean square with p≥2. Also pathwise exponential stability is proved for p>2. We extend the resu...

In this paper, we prove the existence of mild solutions for a class of nonlinear impulsive integrodifferential equations with a nonlocal initial conditions. Sufficient conditions for the existence are derived with the help of the resolvent operator. In the end, an example is given to show the application of our result.

In this work, we provide a new approach to address the solvability of a class of partial integrodifferential equations with nonlocal conditions and two linear parts, the first one being the generator of a strongly continuous semigroup, while the second one is of integral type. We introduce new ideas to handle the measure of noncompactness of a clas...

In this work, we study the existence and uniqueness of an almost automorphic solution to
semilinear nonautonomous parabolic evolution equations with inhomogeneous boundary conditions using the exponential dichotomy. We assume that the homogeneous problem satisfies the "Acquistapace–Terreni" conditions and that the forcing terms are Stepanov-like al...

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.

This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the fam...

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...

In this work, we study the existence, uniqueness and stability in the α-norm of solutions for some stochastic partial functional integrod-ifferential equations. We suppose that the linear part has an analytic re-solvent operator in the sense given in Grimmer [8] and the nonlinear part satisfies a Hölder type condition with respect to the α-norm ass...

In this work, we deal with the positivity and stability study for some partial functional differential equations with infinite delay and Hille–Yosida operators. We first show the positivity of solutions in a suitable phase space. This enables us to establish a stability criterion independently of delay.

The aim of this work is to establish several results on the existence of mild solutionse nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We show the local existence of the mild solutions which...

In this work, we study the existence, uniqueness and stability in the α-norm of solutions for some stochastic partial functional integrodifferential equations. We suppose that the linear part has an analytic resolvent operator in the sense given in Grimmer [7] and the nonlinear part satisfies a Hölder type condition with respect to the α-norm assoc...

This paper is a continuation of the investigations done in the literature regarding the so called Bohr-Neugebauer property for almost periodic differential equations in Hilbert spaces. The aim of this work is to extend the investigation of this property to almost automorphic functional partial differential equations in Banach spaces. We use a compa...

The purpose of this work is to give sufficient conditions which guarantee the existence and the uniqueness of positive μ-pseudo almost periodic solutions for the nonlinear infinite delay integral equation . We improve the original work of [H. S. Ding, Y. Y. Chen and G. M. N’Guérékata, Existence of positive pseudo almost periodic solutions to a clas...

In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$without the assumption of equico...

In this work, we develop the necessary conditions of optimality for partially observed control problems governed by stochastic integrodifferential equations of neutral type on Hilbert spaces driven by relaxed controls. Under suitable conditions, we establish the existence and the uniqueness of mild solution which has a continuous modification. We p...

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.

The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. T...

The aim of this work is to establish a Perron type theorem for some nondensely defined partial functional differential equations with infinite delay. More specifically, we show that if the nonlinear delayed part is "small" (in a sense to be made precise below), then the asymptotic behavior of solutions can be described in terms of the dynamic of th...

We prove that a partial functional differential equation with infinite delay has the Bohr–Neugebauer property, when the semigroup generated by the differential operator is immediately compact and when the phase space has the uniform fading memory property. To illustrate our main result, we propose an application to a reaction–diffusion equation wit...

This work concerns the study of the controllability for some nonlinear partial functional integrodifferential equation with finite delay arising in the modelling of materials with memory in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in t...

In this work, we study the existence and stability of solutions for an optimal control problem governed by an integrodifferential equation with compact control set in the space L1([0,b];X). The stability results, in the sense of Baire category theory, for optimal control problems are obtained by the theory of set-valued mapping and the notion of es...

In this work, we consider a class of partially observed stochastic integrodifferential equations on Hilbert spaces subject to measurement uncertainty. We prove the existence of optimal feedback control law from a class of operator valued functions furnished with the product topology. This work is an extension of [2] for uncertain systems governed b...

In this work, we use an approach due to Favard (Acta Math 51:31–81, 1928) to study the existence of weakly almost periodic and almost automorphic solutions for some evolution equation whose linear part generates a (Formula presented.)-group satisfying the Favard condition in uniformly convex Banach spaces. When this (Formula presented.)-group is bo...

.In this work, we discuss the existence and uniqueness of µ-pseudo almost periodic solutions in the α−norm for some neutral partial differential equations with finite delay.

In this work, we give sufficient conditions for the existence of a mild solution for some impulsive integro-differential equations in Banach spaces. We study the existence without assuming the Lipschitz condition on the nonlinear term f. The compactness on the C0-semigroup(T(t))t≥0 in a Banach space is not needed. We use Hausdorff’s measure of nonc...

In this work, we study the existence of mild solutions for a class of partial integrodifferential equations with nonlocal conditions. Our analysis uses the resolvent operator theory and relies on a new fixed point theorem of Sadovskii-Krasnosel’skii type. Our results improve and complement several earlier related works. Some examples are provided t...

This work aims to investigate the existence and uniqueness of almost periodic solution for partial functional differential equations with delay. 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 almost periodic with respect to the first...

In this work, we consider the control system governed by some partial functional integrodifferential equations with finite delay in Banach spaces. We assume that the undelayed part admits a resolvent operator in the sense of Grimmer. Firstly, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a...

In this work, we study the existence and exponentially stability in mean square for some stochastic integrodifferential equation with delays. Also pathwise exponential stability is proved for . We assume that the linear part has a resolvent operator in the sens given by Grimmer (1982). The delayed part is assumed to be continuous. Our results are p...

The aim of this work is to study the existence of a periodic solution for some nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We use Massera’s approach (Duke Math 17:457–475, 1950), we prove...

In this work, we study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Fréchet spaces. We assume that the linear part has a resolvent operator in the sense given by Grimmer (Trans Am Math Soc 273: 333–349, 1982). The nonlinear part is a sum of a Lipschitzian function and another satis...

In this work, we introduce the concept of μ-pseudo almost automorphic processes in distribution. We use the μ-ergodic process to define the spaces of μ-pseudo almost automorphic processes in the square mean sense. We establish many interesting results on the functional space of such processes like a composition theorem. Under some appropriate assum...

The aim of this work, is to study the existence and regularity of mild solutions for a class of abstract partial functional integrodifferential equations with infinite delay under the alpha-norm. We assume that the linear part generates an analytic semigroup, the nonlinear part is assumed to be continuous with respect to the alpha norm associated t...

In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient...

We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More...

The aim of this work was to study the existence and uniqueness of \({\mu}\)-pseudo almost periodic solutions in the \({\alpha}\)-norm for some partial
differential equation with infinite delay. In the last section we give an application.

In this work, we study of the solvability and the existence of optimal controls of some partial functional
integrodifferential equations with classical Cauchy initial condition in Banach spaces. We assume that the linear
part admits a resolvent operator in the sense of Grimmer. Firstly, we investigate the existence and uniqueness
of mild solutions....

This work concerns the study of the controllability for some partial functional integrodifferential equa-
tion with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system
by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the
measure of...

In this work, we establish a new concept of square-mean pseudo almost periodic and automorphic processes using the measure theory. We use the μ−ergodic process to define the spaces of μ−pseudo almost periodic and automorphic processes in the square-mean sense. We establish many interesting results on those spaces like completeness and composition t...

The goal of this paper is to study, in the $$\alpha $$α-norm the existence of solutions for a class of neutral partial functional integrodifferential equations with finite delay. We assume that the linear part generates an analytic and compact semigroup and the nonlinear part is continuous and involves spatial partial derivatives in the second argu...

This article presents the result on existence and stability of mild solutions of stochastic partial differential equations with infinite delay in the phase space \(\mathcal {B}\) with non-lipschitz coefficients. We use the theory of resolvent operator devolopped in Grimmer (Trans Am Math Soc 273(1):333–349, 1982) to show the existence of mild solut...

In this work, we introduce the concept of a convex-power condensing mapping T with respect to another mapping S as a generalization of condensing and convex-power condensing mappings. Some fixed point theorems for the sum T + S with S is a strict contraction and T is convex-power condensing with respect to S are established. The cases where S is no...

In this work, we give sufficient conditions for the existence and uniqueness of μ−pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ−pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase...

The aim of this work is to study the new concept of the (μ, ν)-pseudo almost automorphic functions for some non-autonomous differential equations. We suppose that the linear part has an exponential dichotomy. The nonlinear part is assumed to be (μ, ν)-pseudo almost automorphic. We show some results regarding the completness and the invariance of th...

In this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic
functions which is more generale than the classical one, and we obtain a new existence result of
μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution
equations with Stepanov μ-pseudo almost periodic...

We give sufficient conditions for the existence of integral solutions for a class of neutral functional differential inclusions. The assumptions on the generator are reduced by considering nondensely defined Hille-Yosida operators. Existence and controllability results are established by combining the theory of addmissible multivalued contractions...

In this work, we prove the existence and uniqueness of μ-pseudo almost periodic and μ-pseudo almost automorphic solutions in the α −norm for some class of partial differential equations in Banach spaces. For illustration, we propose to study some model arising in physical systems.

In this work, we present a new composition theorem of \(\mu \) -pseudo almost automorphic functions in the sense of Stepanov satisfying some local Lipschitz conditions. Using this results, we establish an existence result of \(\mu \) -pseudo almost automorphic solutions for some nonautonomous neutral partial evolution equation with Stepanov \(\mu \...

In this paper we propose to study, in the \(\alpha \)-norm, a class of neutral partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic and compact semigroup and the nonlinear part of the system involve spatial derivatives. At the end, an example is provided to illustrate the applica...