# Amar DebboucheUniversité 8 mai 1945 - Guelma · Department of Mathematics

Amar Debbouche

PhD & Habilitation

## About

118

Publications

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Introduction

## Publications

Publications (118)

In this work, the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractional calculus, fixed point technique and also we introduced a new concept called (α,u)-resolvent family. As an application that illustrates...

In this paper, Schauder fixed point theorem, Gelfand-Shilov principles combined with semigroup theory are used to prove the existence of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. To illustrate our abstract results, an example is given.
MSC:
35A05, 34...

We study the existence and approximate controllability of a class of
fractional nonlocal delay semilinear differential systems in a Hilbert space.
The results are obtained by using semigroup theory, fractional calculus, and
Schauder's fixed point theorem. Multi-delay controls and a fractional nonlocal
condition are introduced. Furthermore, we prese...

This paper investigates the existence and uniqueness of mild solutions for a class of Sobolev type fractional nonlocal abstract evolution equations in Banach spaces. We use fractional calculus, semigroup theory, a singular version of Gronwall inequality and Leray–Schauder fixed point theorem for the main results. A new kind of Sobolev type appears...

We study the well-posedness for solutions of an initial-value boundary problem on a two-dimensional space with source functions associated to nonlinear fractional diffusion equations with the Riemann-Liouville derivative and nonlinearities with memory on a two-dimensional domain. In order to derive the existence and uniqueness for solutions, we mai...

The brain is one of the most complex dynamic systems owing to its intricate structure composed of a network of actions and reactions working in a coordinated effort to execute and control several processes in the body. These networks incorporate a mixture of integration, differentiation, feedback loops, and other regulatory mechanisms that enable a...

The concept of a β-integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence...

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter Ĥ ∈ (1/2, 1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system....

We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, using the standard Laplace transform and Schaud...

We investigate the controllability analysis of nonlinear fractional order neutral-type stochastic integro-differential system with non-Gaussian process. We stress out the stochastic term of our system driven by the uncomplicated non-Gaussian Hermite process known as the Rosenblatt process, which is named after by Murray Rosenblatt who first devised...

We establish several qualitative properties of a neutral switched impulsive evolution system on an arbitrary time domain by using the theory of time scales. This is the first attempt for switched evolution systems with impulses in abstract spaces. First, we investigate the existence of a unique solution and Ulam’s type stability results. After that...

A heroin epidemic mathematical model with prevention information
and treatment, as control interventions, is analysed,
assuming that an individual’s behavioural response depends
on the spreading of information about the effects of heroin.
Such information creates awareness, which helps individuals
to participate in preventive education and self-pro...

This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI...

This paper is concerned with the relative controllability for a class of fractional differential equations with multiple time delays. The solution representation is introduced for this system via multiple delayed perturbations of Mittag-Leffler function. Necessary and sufficient conditions for the indicated problem to be relatively controllable are...

We study the controllability criteria for a class of fractional integro-differential damped systems with impulsive perturbations. The solution representation is derived for both linear and non-linear damped systems, and the introduced formulations were constructed by employing Laplace transformation with Mittag-Leffler matrix function. We present n...

The aim of this work is to investigate the controllability of a class of switched Hilfer neutral fractional systems with non-instantaneous impulses in the finite-dimensional spaces. We construct a new class of control function that controls the system at the final time of the time-interval and controls the system at each of the impulsive points i.e...

Cover Story - This paper (title on cover page) explores a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. Homogeneous and nonhomogeneous solutions are derived. A numerical simulation concerning some of the proposed fractional solutions is put forth to prov...

We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some of the proposed fractional solutions to p...

The goal of this paper is to consider a new class of φ-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss...

This paper deals with a set of three partial differential equations involving time-fractional derivatives and nonlinear diffusion operators. This model helps us to understand the HIV spread and transmission into the patient. First, we prove the existence and uniqueness of weak solutions to the mathematical model. Then, the Galerkin finite element s...

Special issue in light of: 2nd International Conference on Mathematical Modelling in Applied Sciences August 20–24, 2019, Belgorod, Russia http://icmmas19.
alpha-publishing.net.
There were 171 received submissions in this special issue. Following CAM standards and having in mind the limited SI pages, a set of these papers have been rejected in ini...

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associate...

In this work, we present and discuss the approaches, that are used for modeling and surveillance of dynamics of infectious diseases by considering the early stage asymptomatic and later stage symptomatic infections. We highlight the conceptual ideas and mathematical tools needed for such infectious disease modeling. We compute the basic reproductio...

In this paper, we prove the existence, stability and controllability results for fractional damped differential system with non-instantaneous impulses. The results are obtained by using Banach fixed-point theorem, nonlinear functional analysis, Mittag-Leffler matrix function and controllability Grammian matrix. At last, some numerical examples are...

In light of: 2nd International Conference on Mathematical Modelling in Applied Sciences August 20–24, 2019, Belgorod, Russia http://icmmas19. alpha-publishing.net.
This Focus Point covers twelve original papers obtained from advanced theoretical analysis, experimental, and numerical simulations in Cancer and HIV/AIDS research. Results include a ra...

In this article, we establish some sufficient conditions for total controllability of a neutral fractional differential system with impulsive conditions in the finite-dimensional spaces. This type of controllability concerns the controllability problem not only at the final time but also at the impulse time. We use Mittag-Leffler matrix function, n...

We prove a sufficient condition for the stability of a stationary solution to a system of nonlinear partial differential equations of the diffusion model describing the growth of malignant tumors. We also numerically simulate stable and unstable scenarios involving the interaction between tumor and immune cells.

A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space has dimension d∈{2,3} and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of the Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the...

We investigate a class of nonlinear time-partial differential equations describing the growth of glioma cells. The main results show sufficient conditions for the stability of stationary solutions for these kind of equations. More precisely, we study different spatial variables involving radial or axial symmetries. In addition, we also numerically...

This paper deals with cervical cancer which is the second most common cancer in females and a major cause of death in the world now a days. Detection of symptoms of cervical cancer basically focused at an advance stage. But it is possible to detect cancer at an early stage through diagnosis and screening. The detection through diagnosis at early st...

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter-Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappi...

In this manuscript, we investigate existence theory as well as stability results to the biological model of HIV (human immunodeficiency virus) disease. We consider the proposed model under Caputo-Fabrizio derivative (CFD) with exponential kernel. We investigate the suggested model from other perspectives by using fixed point approached derive its e...

This manuscript introduces the square-mean doubly weighted pseudo almost automorphy and also square-mean doubly weighted pseudo almost automorphy in the sense of Stepanov (S^{2}_{ l} ) over time scales. We derive results for a general stochastic dynamic system on time scales which can model a stochastic cellular neural network with time shifting de...

This article deals with a mathematical model on cervical cancer dynamics at cellular level to describe the interactions between Cancerous cell, Natural killer cell (NK), Effector T cell and Human Papilloma virus. Our body immune system is capable to kill the cancer cells and Natural killer (NK), Effector T cells, are ultimately responsible for erad...

In this article, exact traveling wave solutions of a Wick-type stochastic nonlinear Schrödinger equation and of a Wick-type stochastic fractional Regularized Long Wave-Burgers (RLW-Burgers) equation have been obtained by using an improved computational method. Specifically, the Hermite transform is employed for transforming Wick-type stochastic non...

This paper is concerned with some mathematical and numerical aspects of a Lotka‐Volterra competition time‐fractional reaction‐diffusion system with cross‐diffusion effects. First, we study the existence of weak solutions of the model following the well‐known Faedo‐Galerkin approximation method and convergence arguments. We demonstrate the convergen...

In this paper, we investigate approximate controllability of Hilfer fractional Sobolev type differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus and multivalued analysis. An example is provided to illustrate the obtained results.

We study optimal control problems for a class of second‐order stochastic differential equation driven by mixed‐fractional Brownian motion with non‐instantaneous impulses. By using stochastic analysis theory, strongly continuous cosine family, and a fixed point approach, we establish the existence of mild solutions for the stochastic system. Moreove...

In this paper, we examine an optimal control problem of a coupled nonlinear parabolic system with cross-diffusion operators. The system describes the density of tumor cells, effector-immune cells, circulating lymphocyte population and chemotherapy drug concentration. The distributed control has been taken for drug concentration to control the amoun...

We study a class of final value problems for time fractional wave equations involving Caputo’s fractional derivative of order...

We investigate time optimal control of a system governed by a class of non-instantaneous impulsive differential equations in Banach spaces. We use an appropriate linear transformation technique to transfer the original impulsive system into an approximate one, and then we prove the existence and uniqueness of their mild solutions. Moreover, we show...

The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square...

This work deal with asymptotic almost-periodicity of mild solutions for a class of difference equations with a Weyl-like fractional difference in Banach space. Based on a combination of a decomposition technique and the Krasnoselskii’s fixed point theorem, we establish some new existence theorems of mild solutions with asymptotic almost-periodicity...

This article studies the existence and uniqueness of a weak solution of the time-fractional cancer invasion system with nonlocal diffusion operator. Existence and uniqueness results are ensured by adapting the Faedo-Galerkin method and some a priori estimates. Further, finite element numerical scheme is implemented for the considered system. Finall...

Fractional control dynamic systems automatically appear in various scientific fields such as physics, chemistry, electricity, engineering, medicine, and so on. Degenerate fractional differential equations have also begun to appear very efficiently in the modeling of several real phenomena. They motivated many researchers to study their quantitative...

We investigate an optimal control problem involving a class of fractional evolution equations in separable Hilbert spaces. The strategy of this paper is establishing low dimensional approximations for this type of equations by using approximation methods. We derive three kinds of convergence results of mild solutions under appropriate assumptions....

We have focused within this Thesis on some new problems involving fractional differential equations and inclusions which are applied in real life. In particular, we firstly started to review some essential facts from fractional calculus, abstract differential equations, multi-valued analysis, fixed point techniques and control theory that are used...

We introduce a time-fractional Keller–Segel model with Dirichlet conditions on the boundary and Caputo fractional derivative for the time. The main result shows the existence theorem of the proposed model using the Faedo–Galerkin method with some compactness arguments. Moreover, we prove the Mittag–Leffler stability of solutions of the considered m...

In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram-type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results.

The purpose of this work is to investigate the problem of solutions to the time-fractional Navier-Stokes equations with Caputo derivative operators. We obtain the existence and uniqueness of the solutions to each approximate equation, as well as the convergence of the approximate solutions. Furthermore, we present some convergence results for the F...

In this paper, we utilize fractional calculus, theory of semigroup and fixed point approach to prove existence and approximate controllability results for a class of fractional differential equations (FDEs) with noninstantaneous impulses by constructing a suitable composite control function, imposing that the associated linear problem is approximat...

There were 171 received submissions in this special issue. Following CAM standards and having in mind the limited SI pages, a set of these papers have been rejected in initial stage within pre review, others moved to review processes resulting in 32 accepted papers.
We take this opportunity to thank all the contributors of MFDSA 2017, more precisel...

The Cauchy problem for a distributed order equation in a Banach space
with the fractional Gerasimov–Caputo derivative and a linear bounded operator in the
right-hand side is studied. Existence and uniqueness conditions for the problem solution
in the space of exponentially growing functions are found by the methods of the Laplace
transformation the...

This paper is devoted to study a class of abstract fractional evolution equation in a Banach space X: $$\begin{aligned} \hbox {D}_{+}^{\alpha }x(t)+Ax(t)=F(t,x(t)), \quad t\in \mathbb {R}, \end{aligned}$$
(1)
where \(0<\alpha <1\), \(-A\) is the infinitesimal generator of a \(C_{0}\)-semigroup on X, and F(t, x) is an appropriate function defined on...

We study the solvability and optimal controls of an impulsive nonlinear Hilfer fractional delay evolution inclusion in Banach spaces. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. We introduce the notion of Clarke delay subdifferential. To continue and extend previous works in th...

We establish existence, approximate controllability and optimal control of a class of impulsive non-local non-linear fractional dynamical systems in Banach spaces. We use fractional calculus, sectorial operators and Krasnoselskii fixed point theorems for the main results. Approximate controllability results are discussed with respect to the inhomog...

The purpose of the study is to analyze the time-fractional reaction-diffusion equation with nonlocal boundary condition. The proposed model is used to predict the invasion of tumor and its growth. Further, we establish the existence and uniqueness of a weak solution of the proposed model using the Faedo-Galerkin method and compactness arguments.

We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the p...

In this paper, we study P-type, PIα-type, and D-type iterative learning control for fractional impulsive evolution equations in Banach spaces. We present triple convergence results for open-loop iterative learning schemes in the sense of λ-norm through rigorous analysis. The proposed iterative learning control schemes are effective to fractional hy...

This special issue considered substantially extended versions of papers presented at the conference ICMCMST 2015 as well as external submissions. There were about 150 colleagues in attendance at the International Conference, who gave 132 presentations (plenary lectures, oral talks, and posters), including the 20 papers published in this special iss...

In this paper, we investigate general fractional derivatives with a nonsingular power-law kernel. The anomalous diffusion models with nonsingular power-law kernel are discussed in detail. The results are efficient for modelling the anomalous behaviors within the frameworks of the Riemann-Liouville and Liouville-Caputo general fractional derivatives...

In view of the complex behaviors in the heat and fluid flow involving the different differential and integral operators, e. g., classical, fractional-order, local fractional (fractal), and other operators, our chief aims of the special issue is to propose the new methods for find-ing the approximate or exact, or analytical or, numerical solutions f...

We investigate non-differentiable analytical solutions of diffusion equations arising in fractal heat transfer. The originality of this contribution is being to extend previous recent works to n- dimensional spaces. For the main results, we use both Adomian decomposition method & variational iteration method in the sense of local fractional operato...

We introduce the optimality question to the relaxation in multiple control
problems described by Sobolev type nonlinear fractional differential equations
with nonlocal control conditions in Banach spaces. Moreover, we consider the
minimization problem of multi-integral functionals, with integrands that are
not convex in the controls, of control sys...

In this paper, a new class of stochastic impulsive differential equations involving Bernoulli distribution is introduced. For tracking the random discontinuous trajectory, a modified tracking error associated with a piecewise continuous variable by zero-order holder is defined. In the sequel, a new random ILC scheme by adopting global and local ite...

We investigate the unique solvability of a class of nonlinear nonlocal differential equations
associated with degenerate linear operator at the fractional Caputo derivative. For the main
results, we use the theory of fractional calculus and (L; p)-boundedness technique that based
on the analysis of both strongly (L; p)-sectorial operators and stron...

This paper introduces a new concept called impulsive control inclusion condition, i.e., the impulsive condition is presented, in the first time, as inclusion related to multivalued maps and controls. The notion of approximate controllability of a class of semilinear Hilfer fractional differential control inclusions in Banach spaces is established....

We try to seek a representation of solution to an initial value problem for impulsive fractional differential equations (IFDEs for short) involving Riemann–Liouvill (RL for short) fractional derivative, then prove an interesting existence result, and introduce Ulam type stability concepts of solution for this kind of equations by introducing some d...

We investigate nonexistence results of nontrivial solutions of fractional differential inequalities of the form $$\bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i}-\Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq {|\eta|}^{\alpha_{i+1}} {| x_{i+1} |}^{\beta_{i+1}}, \quad (\eta ,t) \in{\mathbb{H}}^{...

In this paper, we investigate the approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. In particular, we obtain a new set of sufficient conditions for the approximate controllability of nonlinear fractional stochastic differential inclusions under the assumption that the corresponding linear system...

We introduce a class of operator pairs for linear homogeneous fractional differential equations in Banach spaces. Reflexive Banach space was decomposed into direct sum of the phase space of the equation and of the kernel of the operator under the fractional derivative. Analytic in a sector of the complex plane containing the positive semiaxis famil...

This thesis is devoted to study the
approximate controllability for two
class of nonlocal fractional stochastic
control systems of Sobolev type in
Hilbert spaces.
A new set of sufficient conditions for
approximate controllability of Sobolev
type nonlocal fractional stochastic dynamic systems are formulated and
proved.
Also, The approximate...

In the article, the fractal heat-transfer models are described by the local
fractional integral equations. The local fractional linear and nonlinear
Volterra integral equations are employed to present the heat transfer
problems in fractal media. The local fractional integral equations are
derived from the Fourier law in fractal media.

The objective of this paper is
to investigate the complete controllability property of a nonlinear nonlocal fractional stochastic control
system with poisson jumps in a
separable Hilbert space. By employing a fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from
deterministic control problems for
the...

A control system described by fractional evolution integro-differential
equations and fractional integral nonlocal control conditions is investigated.
This posed system is subjected to mixed multivalued control constraints
whose values are nonconvex closed sets. Along with the original system,
we consider the system in which the constraints on the...

We prove existence and uniqueness of mild solutions to Sobolev type
fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal
condition is considered in terms of a Riemann-Liouville fractional derivative.
A Lagrange optimal control problem is considered, and existence of a
multi-integral solution obtained. Main tools include frac...

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Holder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from d...

We introduce a nonlocal control condition and the notion of approximate
controllability for fractional order quasilinear control inclusions.
Approximate controllability of a fractional control nonlocal delay quasilinear
functional differential inclusion in a Hilbert space is studied. The results
are obtained by using the fractional power of operato...

Nonexistence results for a class of two-times differential equations with fractional derivatives of orders between zero and one are presented. Furthermore, the result is extended to a two-times system of two differential equations with fractional derivatives of orders between zero and one.

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.

We introduce a new tool called alpha-Sobolev resolvent family to establish the existence and uniqueness of solutions for a class of some fractional delay quasilinear integro-differential equations with nonlocal and impulsive conditions of Sobolev type. We use fractional calculus, resolvent family and Banach fixed point theorem for the main result....

## Questions

Question (1)

*=====Begin Question=====*

*Proposition*

*Assume that covid-19 is an organism which composed (naturally, automatically or by construction) via sub-organisms under appropriate environmental and climatic factors which allow it to grow, mature and develop in human body A.*

*Diffusion Problem*

*Is it possible that this virus can be diffused electronically (electronic effect such as social media) via discrete networks which permit providing the same conditions to be infected in human body B? I mean not necessary direct contacts like touch hands.*

*=====End=====*