Forthcoming articles


International Journal of Dynamical Systems and Differential Equations


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International Journal of Dynamical Systems and Differential Equations (36 papers in press)


Regular Issues


  • Dynamical behaviors of an impulsive food-chain system with Hassell-Varley functional response and mutual interference   Order a copy of this article
    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   Order a copy of this article
    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.

    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   Order a copy of this article
    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.

  • Group classification and some new periodic-like and soliton-like solutions of the generalized Fisher equation with time-variable coefficients   Order a copy of this article
    by Mohamed Abdel Latif, Entsar El-Shazly, Ahmed Elsaid, Hamed Nour 
    Abstract: In this article, we perform the group classification of the generalized Fisher equation with time-variable coefficients. Some new periodic-like and soliton-like solutions for some specific forms of the arbitrary functions are obtained. The power series solutions are obtained for some cases when the exact solutions are difficult to be obtained. Also, the convergence of these solutions is investigated.
    Keywords: Lie symmetries; Group classification; Generalized Fisher equation.

  • Global bifurcation analysis of the Kukles cubic system   Order a copy of this article
    by Valery Gaiko 
    Abstract: In this paper, we carry out the global bifurcation analysis of the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic right-hand sides and having an anti-saddle at the origin. Using our geometric approach and the Wintner-Perko termination principle, we solve the problem on the maximum number and distribution of limit cycles in this system.
    Keywords: planar polynomial dynamical system; Kukles cubic system; field rotation parameter; bifurcation; limit cycle; Wintner-Perko termination principle.

  • Unique solutions for new fractional differential equations with p-Laplacian and infinite-point boundary conditions   Order a copy of this article
    by Li Wang, Chengbo Zhai 
    Abstract: In this paper, we study the uniqueness and existence of solutions for a new fractional differential equation with p-Laplacian and infinite-point boundary conditions. The main method is a new fixed point theorem of $varphi-(h,e)-$concave operators. An example is given to illstrute the main result.
    Keywords: Riemann-Liouville fractional derivative; p-Laplacian; infinite-point boundary value problem; $varphi-(h,e)-$concave operators.

  • Behavior of Two-Dimensional Competitive System of Nonlinear Difference Equations of Higher Order   Order a copy of this article
    by Jerico Bacani, Julius Fergy Rabago 
    Abstract: We generalize the result of Mansour et. al (2012) cite{mansour} and study other related systems that deal with the dynamics of a competitive population model described by a system of nonlinear difference equations. rnMore precisely, we consider the systemrn [rn x_{n+1}=frac{x_{n-(2k-1)}}{varepsilon + delta x_{n-(2k-1)} y_{n-(k-1)}}, quad rn y_{n+1}=frac{y_{n-(2k-1)}}{rho + sigma y_{n-(2k-1)} x_{n-(k-1)}},rn ]rnwhere $varepsilon, delta, rho, sigma in {-1,1}$ and $kin mathbb{N}$ with real initial conditions $(x_n)_{n=-(2k-1)}^0$ and $(y_n)_{n=-(2k-1)}^0$ rnsuch that $varepsilon + delta x_{m-(2k-1)} y_{m-(k-1)} neq 0$ and $rho + sigma y_{m-(2k-1)} x_{m-(k-1)} neq0$ for all possible values of $m$ and $k$rnand study the form and behavior of its solutions for all values of $varepsilon, delta, rho$, and $sigma$ in ${-1,1}$. rnThis work also generalizes several other results on system of nonlinear difference equations (see cite{algham}, cite{elsayed5}, cite{ibrahim5}, cite{kurbanli} and cite{touafek1}).rnFurthermore, the one-dimensional case of the given system provides a generalization of a series of paper of E. M. Elsayed on nonlinear diffierence equations (see cite{elsayed1}, cite{elsayed2} and cite{elsayed6})
    Keywords: discrete dynamical system; nonlinear difference equation; form of solutions; convergence; periodicity; competitive system.

  • Lie Symmetry Analysis and Conservation Laws of Certain Time Fractional Partial Differential Equations   Order a copy of this article
    by Ramajayam Sahadevan, P. Prakash 
    Abstract: A method is presented to derive the Lie point symmetries of time fractional partial differential equations in the sense of Riemann-Liouville fractional derivative. The applicability of the method has been illustrated through time fractional Burgers-Korteweg-de Vries with time dependent variable coefficients, time fractional dissipative Zabolotskaya-Khokhlov equation, time fractional generalized Benjamin equation and time fractional diffusion equation with variable coefficients. Using the obtained Lie point symmetries, it is shown that each of the above mentioned time fractional partial differential equations can be transformed into a ordinary differential equations of fractional order. Exact solutions of the above mentioned time fractional equations are derived wherever possible. It is also explained how conservation laws can be derived to time fractional partial differential equations.
    Keywords: Time fractional partial differential equations; Lie group formalism; conservation laws; Riemann-Liouville fractional derivative; Erd$acute{e}$lyi-Kober fractional operators.

  • Leader following speed synchronization in multiple DC motor system using a hybrid controller   Order a copy of this article
    by Suhaib Masroor, Chen Peng, Syed Muhammad Fazal-ul-Karim 
    Abstract: In this paper, we explore an innovative approach to design a Chopper fed DC motor coupled as a Multi-agent System (MAS), predominantly leader following MAS, to achieve consensus on speed regulated by a Hybrid controller. The hybrid controller incorporates pole placement, tracking and regulation (RST) controller accompanied by adaptive model reference adaptive control (MRAC), after that incorporating MIT rule in the design analysis to endorse system stability. Leader following algorithm is fused with the system model to make the speed of following agents equivalent to that of leader. In the proposed method, every motor with its chopper circuit is treated as sole agent i-e i_th agent. In this paper, we assume that communication among leader and follower is fixed moreover, we also consider two possible scenarios of communication i-e in the presence of delay and without delay. For model simulation, MATLAB is used and the obtained results endorse effectiveness of the proposed design.
    Keywords: Leader following MAS; Consensus; DC Chopper; DC motor; Hybrid Control.

  • On Some Attractors of a Two-Dimensional Quadratic Map   Order a copy of this article
    by Mohamed Réda Ferchichi, Abla Yousfi 
    Abstract: In this paper, we study the appearance, evolution and neighborhood of two attractors of a dynamical system defined by a quadratic polynomial map T:R^2→R^2. The first is a Cantor-type attractor located on an invariant straight line. Thus, it suffices to study the restriction of the map T to this invariant line. The second is a closed curves cycle of period 2. We show, by a numerical approach, that when a parameter of the system varies, the evolution of the orbits in the region close to this second attractor is dependent on the evolution of the stable and unstable sets (homoclinic tangency) of a saddle cycle of period 2 located in this region.
    Keywords: Discrete dynamical systems; attractors; Cantor sets; invariant curves; saddle-node and homoclinic bifurcations.

  • Results on approximate controllability of second-order non-autonomous integrodifferential inclusions via resolvent operators   Order a copy of this article
    by M. Tamil Selvan, R. Murugesu 
    Abstract: In this work, we establish a set of sufficient conditions for the approximate controllability for a class of non-autonomous second-order integrodifferential inclusions in Banach spaces. We establish our main results with the help of resolvent operators and Bohnenblust-Karlin's fi xed point theorem. Then we extend our study to second-order neutral systems with nonlocal conditions. An example is given to illustrate the main result.
    Keywords: Approximate controllability; Integrodifferential inclusions; Resolvent operators; Evolution equations; Nonlocal conditions.

  • An Additive Separation of Variables 3D Solution to a Dynamical BVP for Neutron Cancer Therapy   Order a copy of this article
    by Nassar Haidar 
    Abstract: We study the boundary-value problem (BVP) for irradiation of a rightrnparallelepipedal cancerous region in a (B/Gd) neutron cancer therapy (NCT)rncomposite-region setup by three mutually orthogonal, time-modulated, one-speed neutron beams. The technique of composite region coupling by a neutron source at a common boundary of different regions, that has been introduced in [1], is demonstrated to allow for an additive separation of variables (ASOV) regional neutron-density 3D wave solution to the posing four-regional boundary value problem (BVP). The beams, which may have different pulse shapes, have different modulation frequencies and variable relative time delays.
    Keywords: Accelerator Based Modulated Neutron Sources; Four-Regional Boundary-ValuernProblems; Additive Separation of Variables; One-Speed Neutron Diffusion; Neutron-Densityrn3D Wave; Dynamical NCT; Laplace Transforms; Three Mutually Orthogonal Neutron Beams.

  • Lie group analysis for heat transfer in flow of second grade fluid   Order a copy of this article
    by Tarik Amtout, Houda Biyadi, Mustapha Er-Riani, Mustapha El Jarroudi 
    Abstract: In this paper, the Lie symmetry analysis is performed for the heat transfer flow of a second grade fluid between two parallel heated plates. The symmetries for the coupled equations are given. The exact solutions and similarity reductions generated from the symmetry transformations are provided. Furthermore, translational symmetries were utilized to find a family of travelling wave solutions of the governing nonlinear problem.
    Keywords: Lie group analysis; Second grade fluid; Heat transfer flow; Similarity reduction; Travelling wave solutions.

  • Lyapunov-type inequalities on fractional q-difference Schrodinger equation with the Woods-Saxon potential   Order a copy of this article
    by Kuikui Ma, Zhenlai Han 
    Abstract: In this paper, the integer order Schr\"{o}dingerrnequation with the Woods-Saxon potential is extended to thernfractional $q$-difference field. We establish the Lyapunov-typerninequalities for nonlinear fractional $q$-difference equations, tornthe best of our knowledge, which is the first work dealing withrnLyapunov-type inequalities for nonlinear fractional $q$-differencernequations. Results in this paper even are new in integer order case.rnMoreover, we further investigate the two-point boundary valuernproblem of nonlinear fractional $q$-difference Schr\"{o}dingerrnequation with the Woods-Saxon potential. By applying thernLeray-Schauder degree theory, we get a sufficient condition of thernexistence of solutions that is relatively easy to verify comparedrnwith the result of existing literature. By utilizing thernLeggett-Williams fixed point theorem, an inequality is added to thernexistence condition of solutions of such problem studied in thernexisting literature, and we get the multiplicity of solutions ofrnthis problem. As applications, two examples are presented tornillustrate our main results.
    Keywords: Fractional $q$-difference equations; Lyapunov inequality; Boundary value problem.

  • Global Dynamics of a Cancer Stem Cell Treatment Model   Order a copy of this article
    by Kristen Abernathy, Zachary Abernathy, Robert Dougherty-Bliss, Caleb Mayer, Heidi Whiteside 
    Abstract: We provide global stability arguments for a cancer treatment model with chemotherapy and radiotherapy that accounts for the cancer stem cell hypothesis. Employing the method of localization of compact invariant sets, we resolve the global dynamics of the no-treatment, constant radiation, and combination chemotherapy and radiotherapy cases. In our analysis of the combination treatment model, we show that the presence of a chemotherapy agent lowers the required radiation strength for a globally asymptotically stable cure state.
    Keywords: cancer stem cells; global stability; cancer treatment; localization of compact invariant sets.

  • Population Dynamic Caused by War Involvement via Fractional Derivative on Time Scales   Order a copy of this article
    by Mehdi Nategh, Dumitru Baleanu, Abdolali Neamaty, Bahram Agheli 
    Abstract: In this work, a fractional derivative on time scales is discussed. Then by suggesting a new structure on the real line, we extend the objectivity of this derivative. A population dynamic problem caused by a confrontation or invasion is mentioned together with a model which led us to a non-homogeneous second order fractional PDE on time scales.
    Keywords: Time scales; Fractional dynamics; Population dynamic problem; War involvement.

  • Front transition in higher order diffusion equations with a general reaction nonlinearity   Order a copy of this article
    by Samir Shamseldeen 
    Abstract: In this paper, we investigate the wave front solutions of a class of higher order reaction-diffusion equations with a general reaction nonlinearity. Linear stability analysis with a modulated traveling wave perturbation is used to prove the existence of wave front solutions. We proved that the studied equation supports both monotonic translating front and patterned front solutions. Also, a minimal front speed and the condition for a transition between these front types (monotonic and patterned) are determined. Two numerical examples are discussed (the extended Fisher-Kolmogorov equation with two different reaction nonlinearities) to support the obtained results.
    Keywords: reaction-diffusion equations; traveling waves; Minimal front speed; pulled fronts.

  • Role of harvesting in controlling chaos and disease propagation in predator-prey system with disease in prey   Order a copy of this article
    by Prodip Roy, Krishna Pada Das, Partha Karmakar, Seema Sarkar (Mondal) 
    Abstract: The paper explores an eco-epidemiological model with harvesting in the species and the disease is circulating in the prey population. The curiosity of this paper is to consider the role of harvesting on chaotic dynamics. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the interior equilibrium point. Further we pay attention to the direction of Hopf Bifurcation. Our numerical simulations reveal that the three species eco-epidemiological system shows chaos in low level of infection. It is observed that when force of infection increases chaos becomes stable. We conclude that chaotic dynamics can be controlled by the harvesting parameter as well as the force of infection. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system.
    Keywords: Disease in prey; stable; permanence; impermanence; chaos; Lyapunov exponent.

  • Disappearance of limit cycle oscillations in a predator-prey model: role of mortality due to predation of infected prey   Order a copy of this article
    by Krishna Pada Das, Subhabrata Ghosh, Somnath Maiti 
    Abstract: This present work establishes an eco-epidemiological model byrnthree element, namely a susceptible prey,an infected prey and preda-rntor.In this situation the prey population shows its migratory nature.rnIt is observed in real life situation that a predator dies due to preda-rntion of an infected prey or they can even survive after consumptionrnof an infected one. We make a general consideration on the growth ofrnthe predator population by consuming the infected prey population.rnWe found that if the predation of the infected prey has a negativerneffect on the growth of the predator population, then the predatorsrnmay be washed away from the system but it will survive if the intakernof the infected prey has a positive effect which boost up its growth.rnWe found specific conditions under the influence of predational effectsrnfor the stability of the system around the coexistence of steady state.rnHopf-bifurcation and persistence conditions are also worked out. We observe that simultaneously with an increase in the degree of infectionrnthe system loses its stability and indicate limit cycle oscillation. Wernobserve that the oscillatory coexistence of the species disappears andrnthe system reaches its steady state position regulated by the conver-rnsion efficiency of the predator. Finally we perform extensive numericalrnsimulations to establish our analytical approach.
    Keywords: Disease in prey; Malthus growth; non positive restricted parameter;rnstability; Hopf-bifurcation; permanence.

  • First order fifth degree Fuchs differential equation with fixed critical points   Order a copy of this article
    by Khaled M'hamed-messaoud, Toufik Laadj, Arezki Kessi 
    Abstract: In this paper, we are interested in studying the nonlinear differential equations of order one and fifth degree, whose general integral is uniform. We will give sufficient conditions, for the considered equations to be with fixed critical points.
    Keywords: Fuchs differential equations; fixed critical points; painlevé property.
    DOI: 10.1504/IJDSDE.2019.10013708
  • On the global existence of solutions to a class of semilinear fractional order system with delay   Order a copy of this article
    by Mengrui Xu, Shurong Sun 
    Abstract: In this paper, we supplement the global existence of solutions to a class of nonlinear fractional order system with time delay by using the Leray-Schauder fixed point theorem, and improve conditions to make the estimate value of the above solution hold. An example as an application of our results is also given.
    Keywords: Fractional differential system; Initial value problem; Delay; Existence.

  • Permanence and extinction for a stochastic two-species competitive system   Order a copy of this article
    by Yizhong Liu 
    Abstract: This paper deals with a stochastic two-species competitive system. Some very verifiable criteria on the uniformly weakly persistent in the mean almost surely (a.s.) and extinction for each species are obtained. Some examples are given to illustrate our main theoretical findings. Our results are new and complement previously known results.
    Keywords: Stochastic Competitive model; Permanence;rnExtinction; Time delay.

  • Existence and multiplicity of solutions for the fractional Schr   Order a copy of this article
    by Peiluan Li 
    Abstract: In the present paper, we investigate the solutions of a fractional Schr
    Keywords: Fractional Schrodinger equations; Steep potential well; Infinitely many solutions.

  • An Antiplane Electro-Elastic Contact Problem: Case Regularized Friction Law   Order a copy of this article
    by Mohamed Dalah 
    Abstract: In this paper, we study an antiplane electro-elastic contact problem with regularized friction law. We suppose that the material is assumed to be electro-elastic and the foundation is assumed to be electrically conductive. In the first step, we write the mathematical model and its well-posedness. In the second step, we try to give the variational formulation and main results. Finally, we prove that the weak solution of the antiplane electro-elastic contact problem with regularized friction law is converge to the weak solution of problem P in [3] when the parameter of regularization $rho$ is very small.
    Keywords: Regularized friction law; Antiplane; Friction law; Variational formulation; Variational inequality; Weak solution.

  • Intrinsic growth rates influence on the net economic rents of fishermen   Order a copy of this article
    by Meriem BENTOUNSI, Imane AGMOUR, Naceur ACHTAICH, Youssef EL FOUTAYENI 
    Abstract: In this paper, a mathematical model consisting of Sardina pilchardus, Xiphias gladius and Bluefin tuna is considered. The differential system is based on the Lotka-Volterra scheme, considering a logistic growth for each population. Biological and bioeconomic equilibria of the system are obtained, and criteria for local stability and instability of the system are derived. This work is an attempt to study the effect of intrinsic growth coefficients changes on the net economic rents of several fishermen exploiting the three marine populations in the Atlantic coast of Morocco based on the parameters given by INRH (Institut National de Recherche Halieutique). To achieve this objective, we have to study the bioeconomic model. Numerical simulations are given to support the theoretical results.
    Keywords: Bioeconomic model; intrinsic growth coefficients; Stability analysis; Net economic rent; Generalized Nash equilibrium problem.

  • Some Oscillation Theorems for Generalized Lienard Type System   Order a copy of this article
    by Tohid Kasbi, Vahid Roomi, Aliasghar Jodayree Akbarfam 
    Abstract: In this work a generalized Li
    Keywords: Oscillation; Liénard system; Differential Equations.

    by Kishor Gaikwad 
    Abstract: The aim of this work is to determine the temperature, displacement function, thermal stresses and thermal deflection of a thin circular plate defined as 0  r  a, 0  z  h under an unsteady temperature field due to internal heat generation within it. Initially, the plate is kept at an arbitrary temperature F(r, z). For times t > 0, heat is generated within the thin circular plate at a rate of g(r, z, t) W.m−3. The governing heat conduction equation has been solved by generalized finite Fourier transform and finite Hankel transform technique. The results are obtained in a series form in terms of Bessels functions. The results for temperature, displacement function, thermal stresses and thermal deflection have been computed numerically and are illustrated graphically.
    Keywords: Inverse thermoelastic problem; Thermal deflection; Circular plate; Heat generation; Thermal stresses; Axi-symmetric.

    by Fatima Dib, Naima Daoudi-Merzagui 
    Abstract: In this paper we discuss the existence of periodic solutions for an impulsive nonautonomous second order delay differential equation. We obtain multiple periodic solutions by a variational approach.
    Keywords: Second order delay differential equation; periodic solution; impulses; mountain pass theorem.

    by Hadi Parham, Fateme Helen Ghane, Esmaeel Rezaali 
    Abstract: The present work is concerned with the ergodic shadowing property of non-autonomous discrete-time dynamical systems. The concept of the ergodic shadowing was introduced by Fakhari and Ghane (2010) for a continuous onto map. Here, we will generalize the notion of ergodic shadowing to non-autonomous discrete-time dynamical systems. We show that every uniformly equicontinuous non-autonomous discrete-time system with the ordinary shadowing and topologically mixing properties enjoys the ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property. Finally, we prove that every uniformly expanding non-autonomous discrete-time system satisfies the ergodic shadowing property.
    Keywords: non-autonomous dynamical systems; ergodic shadowing; pseudo orbital pecification.

  • Global exponential convergence of CNNs with multi-proportional delays and leakage delays   Order a copy of this article
    by Changjin Xu 
    Abstract: In this article, we deal with cellular neural networks (CNNs) with multi-proportional delays and leakage delays. Applying the differential inequality techniques, a new sufficient criterion which ensures the global exponential convergence of CNNs with multi-proportional delays and leakage delays is obtained. Numerical simulations are given to illustrate the obtained results. The theoretical predictions of this article are novel and complement some earlier publications.
    Keywords: Cellular neural networks; Leakage delay; Exponential convergence;rnProportional delay.

  • Multivariate vector sampling expansion in shift-invariant subspaces   Order a copy of this article
    by Qingyue Zhang 
    Abstract: Sampling theorems on a shift-invariant subspace are having a significant impact, since they avoid most of the problems associated with classical Shannon's theory. Vector sampling theorems on a shift-invariant subspace which are motivated by applications in multi-channel deconvolution and multi-source separation are active field of study. In this paper, we consider vector sampling theorems on a multivariate vector shift-invariant subspace. We give a multivariate vector sampling expansion on a multivariate vector shift-invariant subspace. Some equivalence conditions for the multivariate vector sampling expansion to hold are given. We also give several examples to illustrate the main result.
    Keywords: sampling theorems; vector sampling theorems; shift-invariant subspaces; super Hilbert space; frames.

  • On invariant analysis, group classification and conservation laws of two component Novikov equation}   Order a copy of this article
    by Manjit Singh 
    Abstract: The two-component Novikov equation is investigated for group classification and non-trivial local conservation laws. In addition to Lie group analysis, the existing classification of 4-dimensional Lie algebra is used to improve the classifications of Lie algebra of Novikov equations. Apart from this, the direct method is used in the construction of conservation laws using multipliers.
    Keywords: Lie symmetries; Group classifications; Conservation laws.

  • A Convergence Computational Scheme for System of Integral Equation Using Finite Element Method   Order a copy of this article
    by Hamed Zeidabadi, Mohammad Heidari 
    Abstract: In this paper, a computational scheme for extracting approximate solutions of system of integral equations is proposed. For this purpose, by considering the variational form of the problem and using finite element method, the system of integral equations are reduced to a system of algebraic equations, that are solved by an efficient algorithm. Also, the existence and uniqueness of the system of integral equations are illustrated and the convergence of the approximate solution to the exact solution is investigated. Finally, the effectiveness of the proposed method is discussed by comparing with the results of the given approaches in Babolian et al. [1] and Jafarian et al. [2].
    Keywords: System of Volterra Integral Equation; Finite Element Method; Existence; Uniqueness.

  • Approximate Controllability of Hilfer Fractional Sobolev Type Integrodifferential Inclusions with Nonlocal Conditions   Order a copy of this article
    by JinRong Wang 
    Abstract: 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 interesting example is provided to illustrate the obtained results.
    Keywords: Approximate controllability; Hilfer fractional derivative; Sobolev type differential inclusions; Semigroup theory.

    by Azam Ehsani, Fateme Helen Ghane, Marzie Zaj 
    Abstract: We consider finitely generated semigroup actions on a compact manifold and discuss their ergodic properties. We introduce Markovian mostly expanding semigroups and show that each $C^{1+\alpha}$ Markovian mostly expanding semigroup action is ergodic (with respect to the Lebesgue measure) whenever it is strongly transitive. Moreover, it is proved that each Markovian mostly expanding semigroup is non uniformly expanding. Our approach provides a large class of non-uniformly expanding semigroups.
    Keywords: finitely generated semigroup action; backward minimality; strong transitivity; ergodicity of semigroup actions; non-uniformly expanding property.