International Journal of Dynamical Systems and Differential Equations (45 papers in press)
Regular Issues
 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.
 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.
 Analysis of Brain Tumor Growth Model by Adomian Decomposition Method
by Archana Varsoliwala, Twinkle Singh Abstract: The current work involves the study of brain tumor growth (glioblastoma), which is a very aggressive brain tumor. The mathematical model is mainly based on two parameters  the diffusion and growth of tumor cells. Based on various medical studies conducted by researchers, which demonstrate that the combination of radiotherapy and chemotherapy can lead to negative tumor growth. This study uses the Adomian Decomposition Method and its convergence analysis to obtain an approximate solution of equation governing tumor growth. The result is consistent with the physical phenomenon of tumor growth, in which tumor concentration increases linearly after a patient is treated with combination therapy as opposed to rapid exponential growth. Keywords: Adomian Decomposition Method; Burgess equation; Adomian polynomials; Nonlinear partial differential equation.
 Optimal continuoussingular control of stochastic McKeanVlasov system in Wasserstein space of probability measures.
by Samira Boukaf, Lina Guenane, Mokhtar HAFAYED Abstract: In this paper, we study the local form of maximum principle for optimal stochastic continuoussingular control of nonlinear It Keywords: Derivative with respect to probability law; Optimal continuoussingular control; McKeanVlasov stochastic system; Wasserstein space of probability measures.
 Nonuniqueness of solution for initial value problem of impulsive fractional partial differential equations
by XianMin Zhang Abstract: This paper mainly researches the formulas of solution for the initial value problems (IVPs) of two impulsive fractional partial differential equations (IFrPDEs). For these IVPs of IFrPDEs, some properties of their solutions are found, which uncover that the formulas of solution given by some cited papers are inappropriate due to not meeting these properties. Next, by analyzing errors between the approximate solutions and exact solutions, two new formulas of solution of these IVPs of IFrPDEs are discovered that are the integral equations with some undetermined differentiable functions, which illustrate the nonuniqueness of solution of the IVPs of IFrPDEs to be expounded by two examples. Keywords: fractional partial differential equations; impulsive fractional partial differential equations; impulse;initial value problems; nonuniqueness of solution.
 ASYMPTOTICALLY POLYNOMIAL TYPE SOLUTIONS FOR 2DIMENSIONAL COUPLED NONLINEAR ODES WITH DERIVATIVE TERMS
by Bharadwaj BVK, Pallav Baruah Abstract: In this paper we have considered a generalized coupled system of nonlinear ordinary differential equations involving derivative terms. We have given sufficient conditions on the nonlinear functions such that the solutions pair asymptotically behaves like a pair of real polynomials. Keywords: Nonlinear Coupled Ordinary Differential Equations; Fixedpoint Theorem; Asymptotically Polynomial like solutions.
 Time optimal control for an epidemic system with isolation and quadratic treatment
by Soovoojeet Jana, Anupam Khatua, Tapan Kumar Kar, Manotosh Mandal Abstract: The main objective of this article is to explore the use of time optimal control problem to eradicate epidemic diseases. In the first part, an SIS type epidemic model with isolation and quadratic treatment control is proposed. Then the local and global dynamics of the system is studied. Next in the second part, a time optimal control problem is formulated in order to reach the disease free state in a minimum possible time. By means of Pontryagin's maximum principle, the optimal control problem is solved and the explicit expression of the optimal time is derived for the developed model system. Keywords: Epidemic model; isolation; quadratic treatment; global stability; time optimal control.
 Oscillation and Disconjugacy for Generalized HalfLinear Differential Equations with Bohr Almost Periodic Coefficients
by Mohammad MOALLA, Sami INJROU, Ramez KARROUM Abstract: This study aims to examine the disconjugacy and the oscillation domain of the generalized halflinear second order differential equations with almost periodic coefficients in the sense of Bohr. Some properties of topological D (the disconjugacy domain) and O (the oscillation domain) are identified, and the boundary of D is also studied. Keywords: halflinear; disconjugacy; oscillation; almost periodic coefficients; Riccati transformation.
 A Model for the population dynamics of banana weevil, Cosmopolites sordidus (Germar) with harvesting and time delay: implications for management
by Eliab Horub Kweyunga, Julius Tumwiine, Eldad B. Karamura Abstract: In this paper, we study a single species logistic delayed model to represent the dynamics of the banana weevil Cosmopolites sordidus (Germar) population. In particular, we consider a logistic equation that incorporates time delay in the recruitment term and constant effort trapping. We analyse the model for the existence and stability of the steady states. The effects of time delay and constant effort trapping on the dynamics of the banana weevil population are investigated. We validate our analytical results using numerical simulations for a given set of parameter values and find that a reduction in eggtoadult banana weevil survival rate coupled with early and constant effort trapping of the banana weevils are key in management of infestation of the banana plantation.
Keywords: asymptotic stability; banana weevil; transcritical bifurcation; logistic delayed differential equation; time delay; constant effort harvesting.
Special Issue on: CDSM2CT2019 Advances in Qualitative Behaviours of Dynamical Systems
 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.
 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.
 Qualitative Analysis of Stochastic Fish Farm Model with Mussel Population
by Gokila C, Sambath M Abstract: In this paper, we analyze the dynamics of the fish farm model with the mussel population. For the stochastic systems, we establish the existence of globally positive solutions and we find the conditions for species to be extinct and the appearance of species. Construct appropriate Lyapunov functions and discuss the global asymptotic stability of a positive equilibrium solution. Also, we illustrate the condition for the existence of stationary distribution. To check our theoretical findings, some numerical simulations are worked out. Keywords: Fish Farm; Stochastically Permanent; Extinction; Stochastic AsymptoticrnStability; Stationary Distribution.
 H_{} Performance Analysis for Uncertain Systems with Actuator Fault Control via relaxed integral inequalities
by Karthik C, Nagmani G Abstract: This paper investigates the stability behavior of uncertain systems with timevarying delays under actuator fault control in the continuous case. The proposed H1 control problem is constructed such that the dynamics of the uncertain system under actuator fault is asymptotically stable. Based on LyapunovKrasovskii functional technique and using the relaxed integral inequality, the delaydependent Stability criterion is established for ensuring the stability behavior of the addressed time delay uncertain systems with regard to linear matrix inequality (LMI) with prescribed gain matrices. Lastly, two numerical examples with simulations are presented to illustrate the validity of the proposed theoretical results. Keywords: H_{∞} control; actuator faults; uncertainty; LynapunovKrasovskii functional(LKF); relaxed integral inequality; Linear matrix inequality(LMI).
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.
 ON THE GLOBAL BEHAVIOR OF A SYSTEM OF PIECEWISE LINEAR DIFFERENCE EQUATIONS
by Evelina Lapierre, Wirot Tikjha Abstract: In a previous paper we considered the system $x_{n+1} = ' x_n '  y_n  1$ and $y_{n+1} = x_n + ' y_n '  1$ and showed by mathematical induction that when the initial condition is an element of the closed second or fourth quadrant, the solution to the system is either a prime period3 solution or one of two prime period4 solutions. In this paper we complete the study of the global behavior of the system. We show that when the initial condition is an element of $mathbb{R}^2$ then the solution is the equilibrium point, one of two prime period3 solutions, or one of two prime period4 solutions. Keywords: Difference equation; Periodic solution; Stability.
 Mean Square Characterisation of a Stochastic Volterra Integrodifferential Equation with Delay
by John Appleby Abstract: In this paper the asymptotic behaviour of the mean square of the solution of a linear stochastic Volterra integrodifferential equation with delay is entirely characterised. In the case when the solution is meansquare asymptotically stable or unstable the exact rate ofrngrowth or decay can be determined by the real solution of a transcendental equation which is constructed as a byproduct of the proof. The proof of the mean square stability of an equation with fading memory is also sketched. Keywords: stochastic functional differential equations; stochastic Volterra equation; mean square stability; characteristic equation; characteristic exponent; renewal equation; exponential stability; variation of constants formula.
