International Journal of Dynamical Systems and Differential Equations (49 papers in press)
Regular Issues
 Numerical Solution of TimeDelay Systems by Hermite Wavelet
by Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati Abstract: This paper presents a direct numerical method based on Hermite wavelet to find the solution of timedelay systems. The operational matrices of integration, differentiation, production, and delay are derived and utilized to reduce the timedelay dynamical system to a set of algebraic equations. Thus, the problem is simplified greatly. The method is easy to implement. The illustrative examples with timeinvariant and timevarying coefficients demonstrate the validity of the method. Keywords: Timedelay system; Hermite wavelet; Operational matrix; Direct method.
 Solving Nonlinear Fredholm integral equations with PQWs in complex plane
by Majid Erfanian Abstract: In this article, we propose a numerical scheme to solve a kind of
nonlinear Fredholm integral equations of the second kind in the complex plane.
The periodic quasiwavelets (PQWs) constructed on [0,2pi] are utilized as a basis
of the iteration method. Using the Banach fixed point theorem, we obtain some results
concerning the error analysis. Illustrative examples are included to demonstrate
the validity and applicability of the technique. Keywords: Nonlinear Fredholm integral equation; Periodic quasiwavelet; Complex plane; fixed point theorem; error analysis.
 A Discrete Viral Infection Model with Both Modes of Transmission and Distributed Delays
by E.L. Boukari Brahim, Hattaf Khalid, E.L. Ghordaf Jalila Abstract: The aim of this work is to propose and analyze a discrete virus dynamics model with distributed delays and both modes of transmission, one by virustocell infection and the other by celltocell transfer. In the proposed model, the first distributed delays describes the time needed for infected cells to produce new virions, and the second portrays the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, we prove that the proposed discrete model has the same dynamics as the corresponding continuous model, such as positivity, boundedness and global behaviors of solutions with no restriction on the time step size. Moreover, numerical simulations are given to illustrate and confirm our main analytical results. Keywords: Viral infection; distributed delay; difference equation; global stability.
 Single controller for synchronization of coupled neural networks with distributed timevarying delays
by ChengDe Zheng, Fan Xie Abstract: This paper deals with global synchronization in arrays of delayed chaotic neural networks with nonlinear hybrid coupling. By constructing a new LyapunovKrasovskii functional, a novel synchronization criterion is presented in terms of matrix inequalities based on Chen's integral inequalities and reciprocal convex technique. These established conditions are heavily dependent on the bounds of both timedelay and its derivative. Through employing Matlab Toolbox and adjusting some matrix parameters in the derived results, the design and applications of the generalized networks can be realized. The effectiveness and applicability of the proposed methods is demonstrated by a numerical example with simulations. Keywords: synchronization; matrix inequality; hybrid coupled neural networks; reciprocal convex technique.
 Numerical approach for solving nonlinear stochastic It
by Rebiha Zeghdane Abstract: In this paper, we give a new method for solving stochastic nonlinear
Volterra integral equations by using shifted Legendre operational matrix. It is
discussed that how the stochastic differential equations (SDE) could numerically
be solved as matrix problems. By using this new operational matrix of integration
and the socalled collocation method, nonlinear Volterra integral equations is
reduced to systems of algebraic equations with unknown Legendre coefficients.
Finally, the high accuracy of approximated solutions are illustrated by several
experiment. Keywords: Stochastic Volterra integral equation; Brownian motion; Approximate solution; Best approximation; Legendre polynomials; Collocation method.
 Interval oscillation criteria for damped secondorder delay differential equation with nonlinearities given by RiemannStieltjes integral
by MUTHULAKSHMI V, MANJURAM R Abstract: The purpose of this paper is to investigate the oscillatory behavior
of certain types of damped secondorder forced delay differential equation
with nonlinearities given by RiemannStieltjes integral. By using the Riccati
transformation, some inequalitiess and integral averaging technique, interval
oscillation criteria of both ElSayed type and Kong type are established. Finally,
two examples are presented to illustrate the theoretical results. Keywords: Interval criteria; Oscillation; Delay differential equation; Damping term; RiemannStieltjes integral.
 Eventually periodicity of solutions for some discrete maxtype system of third order
by Huili Ma, Haixia Wang Abstract: This paper is concerned with the eventually periodicity of the following maxtype difference equation systemrn$$ x_{n+1}=maxleft{frac{A}{x_{n}y_{n1}},x_{n2}right},$$rn$$ y_{n+1}=maxleft{frac{A}{y_{n}x_{n1}},y_{n2}right},$$rnwhere $nin N$, $Ain R$, and the initial values $x_{2}, x_{1}, x_{0}, y_{2}, y_{1}, y_{0}$ are arbitrary nonzero numbers. Keywords: Periodic solutions; Difference equations; Maxtype system.
 Optimal Control of Behaviour and Treatment in a Nonautonomous SIR Model
by Samhita Das, Pritha Das, Parthasakha Das Abstract: In this paper we have considered a nonautonomous SIR (susceptible, infected, removed) model with saturation incidence rate for disease transmission. The global dynamical properties like permanence and global stability of the system as well as extinction of disease are analytically and numerically studied. The impact of behavioural patterns of individuals on disease control is validated along with possible applications. Further, Pontryagin's Maximum Principle is used to characterize optimal level of the two controls, treatment and awareness level. Our objective is to minimise the infected population as well as the cost of applied control. The controls at optimal level are found to achieve different levels of impact on infection. It is observed that the combined impact of treatment and awareness exhibits more effective result in disease control compared to their single application. Based on observation, the strategy regarding the implementation of awareness and treatment is suggested. Keywords: Nonautonomous SIR model ; Saturation incidence rate; Permanence; Extinction; Optimal control; Awareness; Treatment.
 Existence results on impulsive stochastic semilinear differential inclusions
by Mustapha Meghnafi, Mohamed Ali Hammami, Tayeb Blouhi Abstract: In this paper, we present some existence results of mild solutions and studyrnthe topological structure of solution sets for the following firstorder impulsivernstochastic semilinear differential inclusions driven by L ́vy noise with periodicrnernboundary conditions. We consider the cases in which the right hand side can berneither convex . The results are obtained by using fixed point theorems for multirnvalued mappings, more precisely, the technique is based on fixed point theoremrna nonlinear alternative of LeraySchauders fixed point theorem in generalizedrnmetric and Banach spaces.rn Keywords: Mild solutions; Periodic solutions; impulses; matrix converrngent to zero; generalized Banach space; Poisson jumps; fixed point; setvalued analysis,rndifferential inclusions.rn.
 Oscillation Criteria for First Order Forced Delay Dynamic Equations with Maxima on Time Scales
by H.A. Agwa, Heba A. Hassan, Esraa Magdy Abstract: In this work, we establish some new oscillation criteria for forced first order dynamic equations with maxima. Our results not only complement and generalize some existing results, but also can be applied to some oscillation problems that were not covered before, we also give some examples to illustrate our main results.
Keywords: Oscillation; forced dynamic equations; time scales; maxima.
 Oscillation theorems and asymptotic behavior of certain thirdorder neutral differential equations with distributed deviating arguments
by Yibing Sun, Yige Zhao Abstract: The purpose of this paper is to study the oscillation criteria for a class of thirdorder neutral differential equations with distributed deviating arguments
$$
big[b(t)((a(t)(z'(t))^{alpha_1})')^{alpha_2}big]'+int^d_c q(t,xi)f(x(sigma(t,xi)))dxi=0, tgeq t_0
$$
where $z(t)=x(t)+int^n_m p(t,xi)x(tau(t,xi))dxi$ and $alpha_i$ are ratios of positive odd integers, $i=1, 2$. By using a generalized Riccati transformation and an integral averaging technique, we establish some new theorems, which ensure that all solutions of this equation oscillate or converge to zero. Some examples are given to illustrate our main results. Keywords: thirdorder neutral differential equations; distributed deviating arguments; oscillation; asymptotic behavior; generalized Riccati transformation.
 Explosive tritrophic food chain model with herd behaviour of prey and finite time blowup of the top predator
by Debaldev Jana, G.P. Samanta, Ashok Mondal, Sudeshna Mondal, A.K. Pal, Debasis Manna Abstract: In this work, we have discussed the dynamical behaviours of a three species food chain model where the prey species exhibits herd behaviour and sexually reproductive top predator are of generalist type. Positivity and uniform boundedness of the system are studied to verify its wellposedness. Some conditions for extinction of prey and predators are derived. Feasibility criteria and stability analysis of all the equilibrium points are discussed here. Hopfbifurcation condition for interior equilibrium point is carried out analytically. Mathematical conditions for finite time blowup of top predator are established. Numerical simulations are carried out to validate our analytical findings. Keywords: Square root functional response; generalist predator; sexual reproduction; Hopfbifurcation; finite time blowup.
 Global stability of virus dynamics models with capsids and two routes of infection
by Ahamed Elaiw, Sami Almalki Abstract: We study the global dynamics of withinhost viral infection models with virus DNA containing capsids. The effect of antibody immune response has been considered. The uninfected cell become infected due to its contacts with a virus or an infected cell. In the second model, the incidence rate is given by saturation. The wellposedness of the model is establised. We utilize Lyapunov method to prove the global stability of the equilibria. We support our theoretical results by numerical simulations. Keywords: Viral infection; global stability; Lyapunov function; capsids.
 On the initial value problem of impulsive differential equation involving CaputoKatugampola fractional derivative of order q(1, 2)
by Xianmin Zhang Abstract: This paper mainly focuses on the nonuniqueness of solution to the initial value problem (IVP) of impulsive fractional differential equation (IFrDE) with CaputoKatugampola derivative (of order q (1, 2)). This impulsive higher order fractional differential equation may involve two inhomogeneous impulses, and the obtained result show that their equivalent integral equation include two arbitrary constants, which means that its solution is nonunique. Next, a numerical example is used to show the nonuniqueness of solution for the IVP for IFrDE. Keywords: fractional differential equation; impulsive fractional differential equation; impulse; CaputoKatugampola fractional derivative.
 On a coupled nonlinear fractional integrodifferential equations with coupled nonlocal generalized fractional integral boundary conditions
by Subramanian Muthaiah Abstract: We investigate a coupled LiouvilleCaputo fractional integrodifferential equations (CLCFIDEs) with nonlinearities that depend on the lower order fractional derivatives of the unknown functions, and also fractional integrals of the unknown functions supplemented with the coupled nonlocal generalized RiemannLiouville fractional integral (GRLFI) boundary conditions. The existence and uniqueness results have endorsed by LeraySchauder nonlinear alternative, and Banach fixed point theorem respectively. Sufficient examples have also been supplemented to substantiate the proof, and we have discussed some variants of the given problem.
Keywords: Fractional differential equations; LiouvilleCaputo derivatives; Coupled system; Generalized fractional integrals; Nonlocal; Existence; Fixedpoint.
 A new hybrid collocation method for solving nonlinear twopoint boundary value problems
by R. Delpasand, Seyed Mohammad Mehdi Hosseini, F.M. Maalek Ghaini Abstract: In this paper, numerical solution of boundary value problems of nonlinear ordinary differential equations by the collocation method is considered. Of course, to avoid solving systems of nonlinear algebraic equations resulting from the method, residual function of the boundary value problem is considered and an unconstrained optimization model is suggested. Particle Swarm Optimization algorithm is used for solving the unconstrained optimization problem. In addition, convergence properties of the Chebyshev expansion are studied. The scheme is tested on some interesting examples and the obtained results demonstrate reliability and efficiency of the proposed hybrid method. Keywords: Nonlinear boundary value problems; Pseudospectral method; Chebyshev polynomials; Particle Swarm Optimization; Convergence analysis.
 Qualitative analysis of a fractional model for HBV infection with capsids and adaptive immunity
by Moussa Bachraoui, Khalid Hattaf, Noura Yousfi Abstract: This paper presents a mathematical model governed by fractional differential equations (FDEs) that describes the dynamics of hepatitis B virus (HBV) infection in within human body. The FDE model takes into account the HBV DNAcontaining capsids, and the adaptive immunity mediated by cytotoxic T lymphocytes (CTL) cells and antibodies. Also, the incidence of infection is presented by HattafYousfi functional response that includes various forms existing in the literature. Moreover, the qualitative properties of the FDE model is rigorously established. Finally, numerical simulations are presented to support the theoretical results. Keywords: HBV infection; adaptive immunity; fractional differential equations; global dynamics.
 Stable RBFRA method for solving fuzzy fractional kinetic equation
by H. Jafari, F. Fakhr Kazemi Abstract: The direct method based on the flat radial basis functions for obtaining numerical solution of differential equations is highly illconditioned. Therefore,
many studies have been dedicated to overcome this illconditioning by using different techniques.\
Here, the radial basis function algorithm based on vectorvalued rational approximations is utilized to obtain the numerical solution of fuzzy fractional differential equations. This stable method can be applied with any sort of smooth radial basis function easily and accurately.
To illustrate the accuracy and stability of the presented algorithm, we focus on solving the kinetic model with fuzzy fractional derivative. Keywords: Radial basis functions; Rational approximation; Kinetic fuzzy fractional model; Shape parameter; Caputofuzzy fractional derivative.
 A twoechelon supply chain model with deterioration and stockdependent
demand via forward and backward stocking policies
by GANESH KUMAR M Abstract: We have developed an integrated inventory model for deteriorating items in a two echelon supply chain. In this model, we have assumed that the vendor produced a single product at a constant rate and transferred it in equalsized batches to the buyers warehouse. Some of the products are presented to the customer in the buyer display area and the demand is assumed to be positively dependent on the products displayed. Shortages are not permitted, and instantaneous replenishment is made when the inventory level reaches zero. Due to deterioration, the vendor incurs a warranty cost for each deteriorated item produced. In this model, we incorporated unit time production costs. We compared the total profit for both forward and backward stock policy, and we show that the holding cost decreases as the stock moves downstream, the vendor has to adhere to the forward stock policy. The aim is to determine the number of deliveries needed to transfer the items from the vendor to the buyers warehouse and from the buyers warehouse to display area, lot size such that the average profit of the system attains its maximum. Numerical examples are provided for illustrating the model. Keywords: Discrete optimisation; inventory control; lot sizing; supply chain; stock
dependent demand.
 Coronary Artery Disease Classification from Clinical Heart Disease Features using Deep Neural Network
by RAJESWARI D, THANGAVEL K Abstract: Coronary artery disease (CAD) is the most dreadful clinical syndrome affecting a multitude of people globally and it increases the morbidity rate every year. Early detection of CAD is very important for appropriate treatment which can stop complications like heart failure. The clinical health data can effectively be used for the noninvasive detection of CAD. In this work, we employ Deep Neural Network (DNN) for developing a heart disease prediction model. The proposed model has been tested on ZAlizadeh Sani dataset from UCI and the results show that the DNN classifier improves prediction accuracy significantly. The performance improvement of 75.7% using DNN architecture has been achieved when compared to KNearest Neighbour (KNN). Keywords: Coronary Artery Disease; Heart Disease; Data Mining; Machine Learning; Deep learning; Deep Neural Network; KNN; Classification.
 Existence and Boundary Behavior of Positive Solutions for a Coupled Fractional System
by Imen Ben Saad, Sameh Turki, Zagharide Zine El Abidine Abstract: We consider the following semilinear fractional system
begin{equation*}
label{eq1} left{
begin{array}{ll}
displaystyle D^{alpha} u=p(t)displaystyle u^{a}displaystyle v^{r}textrm{ in }(0,1) ,
displaystyle D^{beta} v=q(t)displaystyle u^{s }displaystyle v^{b}textrm{ in }(0,1) ,
displaystylelim_{t rightarrow 0^{+}}displaystyle t^{1alpha}u(t)
=displaystylelim_{t rightarrow 0^{+}} displaystyle t^{1beta}v(t)=0,
end{array}
right.
end{equation*}%
where $ alpha, ; betain (0,1)$, $a,; b in(1,1)$, $r, ;sinmathbb{R}$ such that $(1'a')(1'b')'rs'>0$, $D^{alpha}$, $D^{beta}$ are the RiemannLiouville fractional derivatives of orders $alpha, ; beta$ and the nonlinearities $p, ; q$ are positive measurable functions on $(0, 1)$. Applying the Sch"{a}uder fixed point theorem, we establish the existence and the boundary behavior of positive solutions in the space of weighted continuous functions. Keywords: System of fractional differential equations; Boundary behavior; Karamata class; Schauder's fixed point theorem.
 Analysis on product graphs along with the utilization of Restrained step triple connected domination parameter
by Mahadevan G, Vimala Suganthi M, Iravithul Basira A Abstract: Recently, author's introduced the concept of restrained step triple connected domination number. In this paper we analysis general results for the strong product of paths and cycles along with application of restrained step triple connected domination number of a graph with reality. Keywords: Restrained domination number; triple connected domination number; Restrained step domination; rstcnumber.
 A constructive approach to degenerate center problem
by Mahdieh Molaei Derakhtenjani, Omid Rabiei Motlagh, Haji Mohammad Mohammadi Nejad Abstract: We give a constructive approach to the degenerate center problem?. ?First?, ?we consider homogeneous polynomial systems and provide various conditions for which the origin is a center?. ?Then?, ?by using the Poincare coefficients in polar coordinate?, ?we complete a rigorous computation such that the nonhomogeneous system perturbed by lower terms has an annular region surrounding the origin?. ?This enables us to show that a degenerate center may be the limit of a linear center?, ?a nilpotent singularity?, ?and even a hyperbolic saddle point?. ?Finally?, ?we provide sufficient conditions such that the origin is a degenerate center for a nonhomogeneous system?. ?The system may be of even degree?, ?so we have degenerate centers of even degree?, ?which are rare?. Keywords: Center Problem; Degenerate Center; Perturbation of Poincare Map.
 Existence of positive quasihomoclinic solutions for damped pLaplacian differential equations
by Monia Boujlida Abstract: In this paper we prove the existence of nontrivial quasihomoclinic solutions for the damped pLaplacian differential equation ('u′'^{p−2}u′)′ + c'u′'^{p−2}u′ − a(t)'u'^{p−2}u + f(t, u) = 0, t ∈ R, where p ≥ 2, c ≥ 0 is a constant and the functions a and f are continuous and not necessarily periodic in t. Using the MountainPass Theorem, we obtain the existence of positive homoclinic solution in both cases subquadratic and superquadratic. Keywords: quasihomoclinic solution; the (PS)condition; mountain pass theorem; damped pLaplacian differential equation. DOI: 10.1504/IJDSDE.2020.10035014
 Cheap controls for disturbances compensation in hyperbolic delayed systems
by Salma Souhaile, Larbi Afifi Abstract: This work applies to the remediability problem for a class of hyperbolic perturbed systems with constant or timevarying delays. With a convenient choice of input operator (control) and through the observation (output), we show how to remedy the effect of any disturbance f on the considered system. We give the main properties and characterisations of the concept according to the delay. Then, under the appropriate hypothesis, we prove how to find the optimal control ensures the compensation of a disturbance using the corresponding observation only. The usual case of actuators and sensors is examined. An application and numerical results for a onedimensional wave equation with delay are also presented. Keywords: hyperbolic systems; disturbance; control; observation; delay; remediability. DOI: 10.1504/IJDSDE.2020.10035015
 Oscillation of delay difference equations with finite nonmonotone arguments
by Limei Feng, Zhenlai Han Abstract: In this paper, the oscillation of difference equations with multiple nonmonotone delay arguments △x(t) + ^{m}Σ_{i=1} pi(t)x(τ_{i}(t)) = 0, t ∈ N is studied. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems. Keywords: difference equation; nonmonotone delay argument; oscillatory solution. DOI: 10.1504/IJDSDE.2020.10035016
 Delay feedback strategy for a fractionalorder chaotic financial system
by Changjin Xu Abstract: In this paper, we are concerned with a new fractional incommensurate order financial system which is a generalised version of the financial model investigated in earlier works. Designing a suitable timedelayed feedback controller, we have controlled the chaotic phenomenon of the fractional incommensurate order financial system. By analysing the characteristic equation of the involved financial system and regarding the delay as the bifurcation parameter, we establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation for fractional incommensurate order financial system. The study reveals that the delay and the fractional order have an important influence on the stability and Hopf bifurcation of considered financial system. Computer simulations are presented to illustrate the correctness of the theoretical results. The theoretical findings of this paper are new and have important meanings in dealing with the economic and financial problems. Keywords: chaos control; financial system; stability; Hopf bifurcation; fractional order; delay. DOI: 10.1504/IJDSDE.2020.10035017
Special Issue on: Computational Methods for Fuzzy, Delay, Impulsive and Stochastic Fractional Differential Equations
 Residual power series method for the time fractional FornbergWhitham equation
by Jianke Zhang, Luyang Yin Abstract: The purpose of this paper is to solve the time fractional FornbergWhitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalised fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional FornbergWhitham equation. Keywords: residual power series method; timefractional FornbergWhitham equation; Caputo derivative. DOI: 10.1504/IJDSDE.2020.10035018
Special Issue on: CDSM2CT2019 Advances in Qualitative Behaviours of Dynamical Systems
 Analysing of Complementary Perfect Hop Domination Numeral of Corona Products of Graphs
by Mahadevan G, Vijayalakshmi V Abstract: Recently, the authors introduced the concept of Complementary perfect hop domination number of a graph. A set S subset of V is a hop dominating set of G, if every vertex v belongs to VS there exists u belongs to S such that d(u,v) = 2. A set S subset of V is said to be complementary perfect hop dominating set of G, if S is a hop dominating set and has atleast one perfect matching. The minimum cardinality of complementary perfect hop dominating sets is called complementary perfect hop domination number of G and it is denoted by CPHD(G). In this paper we explore the CPHD number for the Corona product of two distinct paths and cycles. Keywords: complementary perfect hop dominating set; hop dominating set.
 Bifurcation Analysis of FractionalOrder VD Model
by Ramesh P Abstract: In this paper, we introduced the fractional order VD model. First, we established basic results such as existence, uniqueness, boundedness of the fractional order dynamical system. Next, we evaluate the local stability and Hopf bifurcation of the fractional order VD model. Finally, some numerical simulations evaluated with some examples. Keywords: Boundedness; Existence and uniqueness; Fractional dynamical system; Stability; Hopf bifurcation.
 2  Pebbling Property of Buttery Derived Graphs
by Sagaya Suganya Abstract: For a graph G, f(G) is the least configuration of p pebbles on the vertices of G, so that we can move a pebble to any vertex by a sequence of moves and each move is taking two pebbles of one vertex and placing one pebble on an adjacent vertex. A graph G is said to satisfy 2  pebbling property, if it is possible to move two pebbles to any arbitrarily chosen vertex with a possible configuration of 2f(G) q + 1 pebbles, where q is the number of vertices with at least one pebble. This paper determines the pebbling number and the 2  pebbling property of butterfly derived graphs. Keywords: pebbling; 2  pebbling; butterfly graph; Benes graph; augmented butterfly graph; enhanced butterfly graph.
 Optimal control of fractional stochastic systems with delay
by Sathiyaraj T Abstract: In this paper, the optimal control of timedelayed fractional stochastic dynamical systems wit Poisson jumps (FSDSP) are investigated in the finite dimensional space. Firstly, by applying Kranoselskiis fixed point theorem, some suitable sufficient conditions are established to guarantee the existence of solutions for the considered system. Then, the general conditions are used to extend the existence of optimal control for the considered Lagrange Problem (P). Concrete example is provided. Keywords: Optimal control; fractional integrals; stochastic systems; timedelays.
 Cartesian Product of the Extensions of Fuzzy Soft Ideals over Nearrings
by T. Manikantan, S. Ramkumar Abstract: In this paper, the notions of fuzzy translation, fuzzy multiplication and fuzzy magnified translation of the cartesian product of fuzzy soft sets are introduced. The cartesian product of the extensions of fuzzy soft sets over a nearring is defined. Using these notions, the concepts of fuzzy soft nearring and fuzzy soft ideal over a nearring are studied. Finally, the fuzzy translation, fuzzy multiplication and fuzzy magnified translation of the cartesian product of the extensions of two fuzzy soft sets over a regular commutative nearring are equivalent for a fuzzy soft nearring (resp. ideal) is proved. Keywords: Fuzzy magnified translation; Extension of fuzzy soft set; Cartesian product of the extensions of fuzzy soft sets; Fuzzy soft nearring; Fuzzy soft ideal.
 Convergence results of K iteration process for nonexpansive mappings with an application
by Sankara Narayanan M, Anbukkarasi V, Marudai M Abstract: This paper deals with the convergence theorems that approximate the fixed points of nonexpansive mappings via K iteration process under the framework of uniformly convex Banach space. One numerical example is provided to illustrate the derived result. Further, based on the proposed result, the existence of the mild solution for wave equation is discussed. In addition to that one new iterative scheme is proposed for finding the fixed points of nonexpansive and quasinonexpansive mappings.
Keywords: K iteration process; uniformly convex Banach space; nonexpansive mapping.
 Dengue Outbreaks Prediction Model for Urban Colombo using Meteorological Data
by KKWH Erandi, S.S.N. Perera, A.C. Mahasinghe Abstract: Dengue is a viral born disease with complex transmission dynamics. Disease outbreak can exert an increasing pressure on the health system with high mortality. Understanding and predicting the outbreaks of dengue transmission is vital in controlling the spread. In this work we propose a generalised linear regression model to understand the dynamics of the disease. Further, to moderate the model we analyse the correlation with meteorological parameters. Then we define a threshold value in order to capture the outbreak. Finally, we compare the proposed model with the existing methods. Keywords: Dengue; Climate Factors; Generalized Linear Model; Disease Outbreak; Threshold.
 Inventory control techniques in a twoechelon supply chain model with fuzzy demand and learning effect
by S. Ganesan, R. Uthayakumar Abstract: The crucial part of decisionmaking in a twoechelon supply chain modelling is to decide the production quantity of the manufacturer to satisfy the demand of the retailers. In this paper, we develop a twoechelon supply chain model with one manufacturer and multiple retailers. The production quantity of the manufacturer and demand of each retailer are the uncertain components of the model, and they are quantified by fuzzy numbers. Wright's learning function is applied in the fuzzy limits to appertain the knowledge acquired through experience of supply chain leaders in decisionmaking. We determine the optimal order quantity of each retailer by calculus method. An approximate value of generalized harmonic numbers is applied for the derivation of optimal values in learning model. Numerical examples are supplied to demonstrate both fuzzy and learning models. The robustness of the learning model is explained using numerical examples and comparative study. Keywords: supply chain; inventory control; parabolic fuzzy number; Wright's learning curve; generalized harmonic number.
 Stability Result for Fractional Neutral Stochastic Differential System Driven by Mixed Fractional Brownian Motion
by Dhanalakshmi K, Balasubramaniam P. Abstract: In this manuscript, stability results for fractional neutral stochastic integrodifferential system is established subject to mixed fractional Brownian motion(fBm). Sufficient conditions for stability results are derived based on the pth mean square norm, fixed point theorem and help of new integral inequality. As, a final point an example is given to illustrate the effectiveness of the obtained theory. Keywords: Fractional differential equations; Mild solution; Neutral stochastic differential equation; Exponential stability.
 A Deep Learning Approach for Brain Tumor Detection System using Convolutional Neural Networks
by Kalaiselvi T, Padmapriya S.T Abstract: The proposed work is aimed to develop convolution neural network (CNN) architecture based computer aided diagnostic system for human brain tumor detection process from magnetic resonance imaging (MRI) volumes. CNN is a class of Deep Learning networks that are commonly applied to analyze voluminous images. In the proposed method, a CNN model is constructed and trained using MRI volumes of BraTS2013 data. More than 4000 images of normal and tumor slices are used to train the proposed CNN system with 2layers. The system is tested with about 1000 slices from BraTS and got very accurate results about 9098% of accuracy. Further, the performance of proposed CNN system is tested by taking a set of clinical MRI volumes of popular hospital. The obtained results are discussed and focused for the future improvement of the proposed system. Keywords: Neural networks; MRI; brain tumor; deep learning; tumor detection;.
 Meansquare asymptotic stability of stochastic inertial neural networks with timedelay and Markovian jump parameters
by KRISHNASAMY RAMASAMY, Raju K. George Abstract: This article investigates the stability of inertial neural networks which incorporates the effects of both intrinsic and extrinsic noises along with timedelay. These intrinsic and extrinsic noises are taken to be in the form of Markovian jump parameters and Brownian motion respectively. Required sufficient stability conditions are established in the form of linear matrix inequalities from the construction of LyapunovKrasovskii functional. Derived conditions will be delaydependent which includes information about the bounds of the timedelay and also its derivatives. Theory of Lyapunov stability, Ito calculus and linear matrix inequality are used to derive the main results. Numerical example is given to demonstrate the validity of the derived theoretical results. Keywords: Inertial neural networks; Meansquare asymptotic stability; timedelay; Markovian jump; LyapunovKrasovskii functional.
Special Issue on: Differential, Difference and Dynamic Equations
 Analysis of the bilateral Laplace transform on time scales with applications
by Tom Cuchta, Svetlin Georgiev Abstract: The bilateral Laplace transform on time scales is investigated
analytically and its absolute convergence, uniform convergence, and inversion
integral are proven . Afterwards, a Fourier transform is defined and used to solve
partial dynamic equations. Keywords: time scales calculus; bilateral Laplace transform; Fourier transform; partial dynamic equations.
 A discrete SIS model of fractional order
by Tom Cuchta, Sabrina Streipert Abstract: In this work, we introduce two epidemic fractional difference equation models and derive their explicit solutions. The presented model is of the SusceptibleInfectedSusceptible class, which assumes that the disease is spread from susceptible to infected individuals who join the group of susceptible after recovery. The model is constructed using the fractional difference operators defined in [14], which sets it apart from the few existing discrete fractional epidemic model formulations. The unique solution of the presented fractional difference epidemic models is derived and relations to existing discrete SIS models are discussed. Keywords: Fractional Difference Equations; RiemannLouiville Derivative; Difference Equations; NablaDifference Equations; Epidemic Model; SIS; Explicit Solution; Unique Solution.
 FirstOrder Nonlinear Dynamic Initial Value Problems.
by Martin Bohner, Sanket Tikare, Iguer Luis Domini Dos Santos Abstract: We prove three existence theorems for solutions of firstorder dynamic initial value problems, including corresponding continuous and discrete cases. The main tools are fixed point theorems and dynamic inequalities. Two more results are given that discuss dependence of solutions on the initial conditions as well as convergence of sequences of solutions.rnrn Keywords: Time scales; dynamic equation; firstorder nonlinear; existence; continuous dependence; fixed point theorems; dynamic inequalities.
 A study on discrete Ponzi Scheme model through SturmLiouville theory
by Ferhan M. Atici, William Bennett Abstract: In this paper, we introduce a second order selfadjoint difference equation which describes the dynamics of Ponzi schemes: a type of investment fraud that promises more than it can deliver. We use the SturmLiouville theory to study the discrete equation with boundary conditions. The model is based on a promised, unrealistic interest rate $r_{p}$, a realized nominal interest rate $r_{n}$, a growth rate of the deposits $r_{i}$, and a withdrawal rate $r_{w}$. Giving some restrictions on the rates $r_{p}, r_{i}$, and $r_{w}$, we prove some theorems to when the fund will collapse or be solvent. Two examples are given to illustrate the applicability of the main results. Keywords: Ponzi scheme; difference equation; SturmLiouville boundary value problem; Green's function.
 Application of generalized Riccati equations to analysis of asymptotic forms of solutions of perturbed halflinear ordinary differential equations
by Sokea Luey, Hiroyuki Usami Abstract: Asymptotic forms of solutions of halflinear ordinary differential equations are investigated under several asymptotic conditions on the coefficient functions. The proof of the main results is based on analysis of solutions of generalized Riccati equations related to this halflinear equation. Keywords: halflinear ordinary differential equation; asymptotic form; Riccati equation.
 Linear Hilfer Nabla Fractional Difference Equations
by JAGAN MOHAN JONNALAGADDA, Gopal N. S. Abstract: In this article, we deal with the nabla analogue of Hilfer fractional derivative and obtain some of its salient properties such as composition and power rules. Further, we consider an initial value problem for a class of nonlinear Hilfer nabla fractional difference equations and obtain its equivalent Volterra summation equation, using these properties. Also, we derive expressions for general solutions of various classes of linear Hilfer nabla fractional difference equations by applying the discrete Laplace transform. Keywords: Hilfer nabla fractional difference; composition rule; power rule,; initial value problem; discrete Laplace transform.
 Effect of Pollution on PredatorPrey Systems
by Pinky Lawaniya, Soumya Sinha, Ravinder Kumar Abstract: In this paper a mathematical model is proposed to study the effect of environmental pollution on a predatorprey system. The conditions for the local and global stability of the equilibria are obtained. The possibility of occurrence of periodic solutions is analyzed and further existence of Hopf Bifurcation with respect to the appropriate parameter is examined. The conditions for uniform persistence of the model are obtained. The results of persistence and Hopf Bifurcation with respect to the appropriate parameter are verified through numerical simulations. Keywords: predatorprey system; biomagnification; Gause type model;global stability; persistence;periodic solutions; pollution .
 Pullback and forward attractors of contractive difference equations
by Abdullah Kalkan, Huy Huynh Abstract: The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. In this paper, we replace them with contractivity assumptions under which the pullback and forward attractors are identical. As a consequence, attractors degenerate to unique bounded entire solutions. As an application, we investigate attractors of integrodifference equations which are popular models in theoretical ecology. Keywords: Pullback attractor; Forward attractor; Contractive mapping; Dissipative difference equation; Semilinear difference equation; Contractive difference equation; Integrodifference equation.
 Oscillatory and stability of a mixed type difference equation with variable coefficients
by Sandra Pinelas, Nedjem Eddine Ramdani, Ali Fuat Yeniçerioglu, Yubin Yan Abstract: The goal of this paper is to study the oscillatory and stability of the mixed type difference equation with variable coefficientsrnThis paper generalize some known results and the examples illustrate the results. Keywords: Mixed type difference equation; Asymptotic behavior; Stability; Characteristic equation; Solution.rn.
 Modeling Analysis of Zika Virus with Saturated Incidence using Optimal Control Theory
by Naba Kumar Goswami Abstract: In this paper, a nonlinear mathematical model of the Zika virus is proposed and analyzed the impact of optimal control strategies with the saturated incident and bednet effect. The recent outbreak of the Zika virus in Brazil and other Latin American countries has posed a significant challenge in the domain of public health. The basic reproduction number $(R_0)$ is computed and performed sensitivity analysis to identify the key parameters that influence the basic reproduction number. To investigate the optimal control strategies, three types of timedependent control parameters are introduced in the system to reduce the transmission. Electronic devices, insecticidetreated bed nets, and mosquito repulsive lotions are used to reduce mosquito biting rates. Keeping this fact, found some suitable optimal control strategies to eradicate the transmission of the disease in the tropical area. Pontryagin's maximum principle is used to manifest the optimal control strategies. It is noticed that the optimal control model gives a better result than the model without optimal control. Finally, the results of the optimal controls are compared by using numerical simulation.
Keywords: Zika; Basic Reproduction Number and sensitivity analysis; Pontryagin's Maximum Principle; Optimal Control.
