# Forthcoming articles

International Journal of Dynamical Systems and Differential Equations

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 International Journal of Dynamical Systems and Differential Equations (21 papers in press)  Regular Issues  Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations   by Yupin Wang, Shurong Sun Abstract: In this paper, we study infinite-point boundary value problems for a class of \r\nhigher-order nonlinear factional differential equations involving the Riemann-Liouville derivative.\r\nBy using sum operator methods, the existence and uniqueness of solution to this kind of problems is \r\nobtained and iterative sequence of the positive solution is structured.\r\nTwo examples are provided for our new results. Keywords: fractional differential equation;\r\nKrasnoselskii\'s fixed point theorem;\r\nexistence and uniqueness;\r\npositive solution. Dynamical behaviour of miscibles fluids in Porous Media   by Fatma Zohra Nouri, Aicha Assala, Noura Djedaidi Abstract: In this paper, we are interested in studying the dynamics of miscible fluids in porous media. The model describing this issue is a system of equations, coupling the standard Navier-Stokes equations with gravity g as external force and a convective diffusion equation for a dilute concentration in the carrier fluid. By assuming that the fluids are incompressible, we first derive a new system of equations, by taking into account additional terms, due to the concentration inhomogeneties and an interfacial tension between the fluids. Then we propose a numerical approach to solve our system in order to illustrate the effectiveness of the dynamics during the fluid miscibility process. At this stage, we start by showing a stability result for our numerical scheme and then present numerical results. Keywords: Fluid Dynamics; Finite Elements; Stability; Porous media. Weak solutions for a class of generalized image restoration models   by Shuaijie Li, Peng Li Abstract: In this paper, based on some well-known restoration models, firstly we propose a general form of image restoration functional model under some conditions, and get a class of related nonlinear parabolic partial differential equations. Secondly we have established the existence and uniqueness of weak solutions for these equations. This work has supplied theoretical basis for image restoration models, and has great significance in image restoration field. Keywords: Existence; Uniqueness; Weak solution;Parabolic PDE; Image restoration. A note on homoclinic solutions for a class of semilinear fourth order differential equations without coercivity   by Adel Daouas Abstract: In this paper, we study the existence of homoclinic solutions for the following fourth order nonautonomous differential equations u(4)+wu''+a(x)u=f(x,u), x epsilon R, (F)where w is a constant, a epsilon C(R, R) and f epsilon C(R x R,R). Without coercive condition on a and under suitable assumptions on the growth of the functions f, we obtain the existence of at least one homoclinic solution for (F) by using the Mountain Pass Theorem. Some recent results in the literature are generalized. Particulary, the open problem proposed by Zhang and Yuan [Applied Mathematics and Computation 250 (2015) 280-286] is solved. Keywords: Homoclinic solutions; Mountain Pass Theorem; Fourth order differential equations. Dynamics of a discrete Leslie-Gower predator-prey model with feedback controls   by Changjin Xu, Peiluan Li Abstract: In this paper, we propose and deal with a discrete Leslie-Gower predator-prey model with feedback controls.\r\n By using the difference\r\n inequality theory, some sufficient conditions to ensure the permanence of the system are derived.\r\n The paper ends with brief conclusions. Our results are new and complement previously known results. Keywords: Leslie-Gower predator-prey model; Permanence; Feedback control; Discrete. Discrete State Space Systems of Fractional Order   by JAGAN MOHAN JONNALAGADDA Abstract: The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional Gr"{u}nwald - Letnikov type backward difference operator with the equivalent Riemann - Liouville type nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result. Keywords: Fractional order; backward (nabla) difference; state space representation; $N$-transform; system response; matrix exponential function; state transition matrix. Dynamical behaviors of an impulsive food-chain system with Hassell-Varley functional response and mutual interference   by Si Zhou, Yuanfu Shao, Qin Liu, Zhen Wang Abstract: An impulsively controlled food-chain system with Hassell-Varley functional response and mutual interference is established in this article. By applying theories and methods of ecology and ordinary differential equation, the dynamical complexity of this system is investigated. We give conditions of the extinction of prey and top predator and show that this system is uniformly bounded. By use of Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability and global stability of the prey-free and top predator-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent via impulsive comparison theorem. Finally, numerical simulations are given to substantiate our theoretical results and to illustrate various dynamical behaviors of this system. Keywords: stability; permanence; impulsive; Hassell-Varley functional response; mutual interference. Alpha-stability of fractional-order Hopfield neural networks   by Changjin Xu, Peiluan Li Abstract: This paper deals with a class of fractional-order Hopfield neural networks. Applying the contraction mapping principle and the inequality technique, some very verifiable criteria on the alpha-stability of fractional-order Hopfield neural networks are obtained. Finally, an example is given to illustrate our main theoretical findings. Our results are new and complement previously known results. Keywords: Hopfield neural networks; Fractional order; \\alpha-stability. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO LYAPUNOV MATRIX STOCHASTIC DIFFERENTIAL EQUATIONS   by Deekshitulu GVSR, Sastry M.V.S.S.B.B. K. Abstract: In this paper, we establish the existence and uniqueness of solutions to Lyapunov matrix stochastic di erential equations by the method of successive approximations. The continuous dependence of the solutions on parameters and initial conditions are also discussed. An example is presented to illustrate the established results. Keywords: Existence; Uniqueness; Lyapunov matrix; Stochastic differential equations; Successive Approximations. Solvability of coupled systems of hybrid fractional differential equations and inclusions   by Nana Jin, Shurong Sun Abstract: In this paper, we investigate the boundary value problems of coupled systems of hybrid fractional\r\n differential equations and inclusions with coupled boundary conditions involving Caputo fractional\r\n derivative. By means of Leary-Schauder alternative and Bohnenblust-Karlin fixed point theorem,\r\n some results concerning the existence of solutions are obtained.\r\n At the same time, we also give the relationship between the solutions and\r\n upper and lower solutions. Finally, examples are presented to illustrate our main results. Keywords: upper and lower solutions; coupled system;\r\n hybrid fractional differential equations and inclusions; existence of solutions. Special Issue on: International Meeting on Applied Mathematics in Errachidia 2016 New Trends on Dynamical Systems and Differential Equations On a non scalar damping model in micromagnetism   by Mouhcine Tilioua, Chahid Ayouch, El Hassan Essoufi Abstract: We consider a mathematical model describing magnetization dynamics with non scalar damping. The model consists on a generalized Landau-Lifshitz-Gilbert equation with a general damping tensor. We apply Faedo-Galerkin/Penalty method to show the existence of global weak solutions in one-dimensional case. Keywords: ferromagnetic materials; LLG equation; global existence; Faedo-Galerkin method. The best strategy for local mesh refinement with the PCD method   by Ahmed TAHIRI Abstract: We propose in this contribution numerical tests for local meshrnrefinement with the PCD method. We investigate two rates of localrnmesh refinement: the ratio $2$ and the ratio $3$. First we introduce arnlocal mesh refinement by subdividing the elements of the zone to bernrefined by the ratio $2$ in each direction i. e. each coarse elementrnis subdivided by $4$ fine elements. Second we subdivide the elementsrnof the zone to be refined by the ratio $3$ in each direction i. e.rneach coarse element is subdivided by $9$ fine elements. Here werninvestigate some numerical experiments to show that the optimalrnlocal refinement rate is $2$. This is in agreement with the regularityrnassumptions required for the proposed discretization. Keywords: Boundary value problem; PCD method; local mesh refinement; slave nodes,rnoptimal rate for local mesh refinement; multilevel local refinement. Periodic solutions to singular damped delay differential equations with impulses   by Fatima Dib, Naima Daoudi-Merzagui Abstract: In this paper we discuss the existence of periodic solutions for nonautonomous second-order damped delay differential equations with singular nonlinearities, in presence of impulsive effects. Simple sufficient conditions are provided that enable us to obtain positive periodic solutions.Our approach is based on a variational method. Some recent results in the literature are extended.rn Keywords: Delay differential equation; periodic solution; damped; singular nonlinearity; impulses; Mountain-Pass theorem. Weighted eigenvalue problems involving a fourth order elliptic equation with variable exponent   by Abdesslem Ayoujil Abstract: In this paper, we consider the eigenvalue problemrn$$rntriangle('triangle u'^{p(x)-2}triangle u )=lambda m(x)'u'^{q(x)-2}u~text{ in }Omega,rn$$rn$$rnu=triangle u = 0~text{ on }partialOmega,$$rn where $Omegasubsetmathbb{R}^{N}$ is a smooth bounded domain, $p, q: overline{Omega} to (1,+infty)$ are continuous functions, $lambda$ is a real parameter and $min L^{r(x)}(Omega)$ is sign-changing weight. Under appropriate conditions on the functions $p, q$ and $r$, we prove that any $lambda > 0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland'{}s variational principle. Keywords: p(x)-Biharmonic operator; eigenvalue; Ekeland'{}s variational principle; generalizedsrnSobolev spaces; Mountain Pass Theorem; weak solution. Cryptanalysis Of Farash et al.'s SIP Authentication Protocol   by Mourade AZROUR, Yousef FARHAOUI, Mohammed OUANAN Abstract: Session Initiation Protocol (SIP) is the most popular signaling protocol used in order to establish, maintain and terminate a multimedia sessions between different participants. Nowadays, the security of SIP is becoming more and more important. Authentication is the most important security service required for SIP. To provide secure communication, many SIP authentication schemes have been proposed. Very recently, Farash et al. proposed a new SIP authentication protocol based on Elliptic Curve Cryptography. They proved that their scheme is secured against different attacks. However, in this paper we show that Farash et al.'s protocol suffers from Denning-Sacco attacks and Denial of Service (DoS) attacks. Moreover, we propose our solution to solve the problem. The security analysis shows that our proposed solution is more secure and can resist to various attacks. Keywords: Session Initiation Protocol; SIP; security; attack; authentication protocol; Elliptic Curve Cryptography ; Denning-Sacco; Denial of Service (DoS);. Vector extrapolation method for non-overlapping Schwarz iterations   by Nabila Nagid, Hassan Belhadj, Mohamed Ridouan Amattouch Abstract: In this paper, we propose a vector extrapolation method for accelerating the non-overlapping Schwarz iterations in the case of the nonlinear reaction diffusion equations. The acceleration occurs at two levels: the acceleration of sequences produced by the fixed point algorithm, and the acceleration of sequences generated by Schwarz method. Specifically, the acceleration occurs in the internal Dirichlet boundary conditions. In order to illustrate the interest of the proposed algorithm, we have performed different test-cases of analytical examples, all results show the efficiency of the proposed approach in terms of CPU-Time, number of iterations and the rate of convergence. Keywords: nonlinear reaction diffusion problems; fixed point method; finite element method; domain decomposition method; non-overlapping Schwarz algorithm; extrapolation method; vector $\varepsilon$-algorithm. Nonlinear Steklov eigenvalue problem with variable exponents and without Ambrosetti-Rabinowitz condition   by ABDELLAH ZEROUALI, BELHADJ KARIM, OMAR CHAKRONE Abstract: In this paper, we study a nonlinear Steklov eigenvalue problem involving the p(x)- Laplacian on a bounded domain. We introduce a new variational technic that allows us to investigate this problem without need of the Ambrosetti-Rabinowitz condition on the nonlinearity. Keywords: $p(x)$-Laplacian; Steklov problem; variable exponent Lebesgue-Sobolev spaces; critical point. Content Frequency and Shape Features based on CBIR: Application to color images   by Youness CHAWKI Abstract: Due to the diversity of the image content, we propose in this study a new technique for Content-based Image Retrieval (CBIR) to characterize the image. In this scenario, all images are characterized by their frequency content and their shape information. Indeed, using the High Resolution Spectral Analysis methods, especially the 2-D ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques), we extract from the image its content frequency and with the ART (Angular Radial Transform) its shape information in order to construct a new vector descriptor. The experimental results applied to the Coil_100 database show the effectiveness and the robustness of our approach compared to the method based on the shame information only, the precision average can be reaches 74.49%. Keywords: CBIR; Frequency Content; High Resolution; Spectral Analysis; 2-D ESPRIT; Shape feature; ART. Numerical Analysis of Some Time Fractional Partial Differential Equations for Noise Removal   by Moulay Rchid Sidi Ammi, Ismail Jamiai Abstract: In this paper, we consider the numerical resolution of two time fractional partial differential equations for processing image denoising, which are obtained from some classical partial differential equations for processing image denoising, edge preservation and compression by replacing the first order time derivative with a fractional derivative of order $alpha$, with $0 Keywords: Fractional calculus; Image processing; Finite differences method. Regional boundary controllability of semilinear parabolic systems with state constraints by Ali Boutoulout, Touria Karite Abstract: This work focuses on controllability of semilinear parabolic systems with state constraints. Subdifferential techniques are used to compute the control u that steers the system (S) from the inial state y0 to a final one between two prescribed functions, only on a boundary subregion Γ of the system evolution domain Ω. Keywords: Distributed systems; parabolic systems; regional controllability; constrained controllability; sub-differential; optimal control; boundary. POSITIVE SOLUTIONS TO A NONHOMOGENEOUS ELLIPTIC SYSTEM OF FOURTH ORDER by Mohamed Talbi Abstract: In this work, the existence of positive solutions to a nonlinear elliptic system of fourth order with positive parameters$lambda$and$mu$is studied. We establish an existence of a value$hat{lambda}(mu)$given under variational form such that forrn$0 Keywords: Ekeland’s variational principle; Palais-Smale condition; p-biharmonic operator.