International Journal of Dynamical Systems and Differential Equations (47 papers in press)

Regular Issues

• Interval oscillation criteria for damped second-order delay differential equation with nonlinearities given by Riemann-Stieltjes integral  
  by MUTHULAKSHMI V, MANJURAM R 
  Abstract: The purpose of this paper is to investigate the oscillatory behavior of certain types of damped second-order forced delay differential equation with nonlinearities given by Riemann-Stieltjes integral. By using the Riccati transformation, some inequalitiess and integral averaging technique, interval oscillation criteria of both El-Sayed 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; Riemann-Stieltjes integral.
  
  
• Eventually periodicity of solutions for some discrete max-type system of third order  
  by Huili Ma, Haixia Wang 
  Abstract: This paper is concerned with the eventually periodicity of the following max-type difference equation systemrn$$ x_{n+1}=maxleft{frac{A}{x_{n}y_{n-1}},x_{n-2}right},$$rn$$ y_{n+1}=maxleft{frac{A}{y_{n}x_{n-1}},y_{n-2}right},$$rnwhere $nin N$, $Ain R$, and the initial values $x_{-2}, x_{-1}, x_{0}, y_{-2}, y_{-1}, y_{0}$ are arbitrary non-zero numbers.
  Keywords: Periodic solutions; Difference equations; Max-type 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 first-order 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 multi-rnvalued 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 conver-rngent to zero; generalized Banach space; Poisson jumps; fixed point; set-valued 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 third-order 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 third-order 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: third-order 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 blow-up 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 well-posedness. Some conditions for extinction of prey and predators are derived. Feasibility criteria and stability analysis of all the equilibrium points are discussed here. Hopf-bifurcation condition for interior equilibrium point is carried out analytically. Mathematical conditions for finite time blow-up of top predator are established. Numerical simulations are carried out to validate our analytical findings.
  Keywords: Square root functional response; generalist predator; sexual reproduction; Hopf-bifurcation; finite time blow-up.
  
  
• 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 within-host 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 well-posedness 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 Caputo-Katugampola fractional derivative of order q(1, 2)  
  by Xianmin Zhang 
  Abstract: This paper mainly focuses on the non-uniqueness of solution to the initial value problem (IVP) of impulsive fractional differential equation (IFrDE) with Caputo-Katugampola 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 non-unique. Next, a numerical example is used to show the non-uniqueness of solution for the IVP for IFrDE.
  Keywords: fractional differential equation; impulsive fractional differential equation; impulse; Caputo-Katugampola fractional derivative.
  
  
• On a coupled nonlinear fractional integro-differential equations with coupled non-local generalized fractional integral boundary conditions  
  by Subramanian Muthaiah 
  Abstract: We investigate a coupled Liouville-Caputo 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 non-local generalized Riemann-Liouville fractional integral (GRLFI) boundary conditions. The existence and uniqueness results have endorsed by Leray-Schauder 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; Liouville-Caputo derivatives; Coupled system; Generalized fractional integrals; Non-local; Existence; Fixedpoint.
  
  
• A new hybrid collocation method for solving nonlinear two-point 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 DNA-containing capsids, and the adaptive immunity mediated by cytotoxic T lymphocytes (CTL) cells and antibodies. Also, the incidence of infection is presented by Hattaf-Yousfi 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 RBF-RA 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 ill-conditioned. Therefore, many studies have been dedicated to overcome this ill-conditioning by using different techniques.\ Here, the radial basis function algorithm based on vector-valued 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; Caputo-fuzzy fractional derivative.
  
  
• A two-echelon supply chain model with deterioration and stock-dependent 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 equal-sized 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 non-invasive 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 Z-Alizadeh 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 K-Nearest 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^{1-alpha}u(t) =displaystylelim_{t rightarrow 0^{+}} displaystyle t^{1-beta}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 Riemann-Liouville 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; rstc-number.
  
  
• 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; Non-linear partial differential equation.
  
  
• Optimal continuous-singular control of stochastic McKean-Vlasov 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 continuous-singular control of nonlinear It
  Keywords: Derivative with respect to probability law; Optimal continuous-singular control; McKean-Vlasov stochastic system; Wasserstein space of probability measures.
  
  
• Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations  
  by Xian-Min 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 non-uniqueness 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; non-uniqueness of solution.
  
  
• ASYMPTOTICALLY POLYNOMIAL TYPE SOLUTIONS FOR 2-DIMENSIONAL COUPLED NONL-INEAR 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: Non-linear Coupled Ordinary Differential Equations; Fixed-point Theorem; Asymptotically Polynomial like solutions.
  
  

Special Issue on: CDSM2CT-2019 Advances in Qualitative Behaviours of Dynamical Systems

• Cartesian Product of the Extensions of Fuzzy Soft Ideals over Near-rings  
  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 near-ring is defined. Using these notions, the concepts of fuzzy soft near-ring and fuzzy soft ideal over a near-ring 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 near-ring are equivalent for a fuzzy soft near-ring (resp. ideal) is proved.
  Keywords: Fuzzy magnified translation; Extension of fuzzy soft set; Cartesian product of the extensions of fuzzy soft sets; Fuzzy soft near-ring; 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 two-echelon supply chain model with fuzzy demand and learning effect  
  by S. Ganesan, R. Uthayakumar 
  Abstract: The crucial part of decision-making in a two-echelon 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 two-echelon 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 decision-making. 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 integro-differential 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 2-layers. The system is tested with about 1000 slices from BraTS and got very accurate results about 90-98% 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;.
  
  
• Mean-square asymptotic stability of stochastic inertial neural networks with time-delay 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 time-delay. 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 Lyapunov-Krasovskii functional. Derived conditions will be delay-dependent which includes information about the bounds of the time-delay 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; Mean-square asymptotic stability; time-delay; Markovian jump; Lyapunov-Krasovskii 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 V-S 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 Fractional-Order 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 time-delayed 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; time-delays.
  
  
• 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 time-varying 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 Lyapunov-Krasovskii functional technique and using the relaxed integral inequality, the delay-dependent 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; Lynapunov-Krasovskii 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 Susceptible-Infected-Susceptible 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; Riemann-Louiville Derivative; Difference Equations; Nabla-Difference Equations; Epidemic Model; SIS; Explicit Solution; Unique Solution.
  
  
• First-Order Nonlinear Dynamic Initial Value Problems.  
  by Martin Bohner, Sanket Tikare, Iguer Luis Domini Dos Santos 
  Abstract: We prove three existence theorems for solutions of first-order 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; first-order nonlinear; existence; continuous dependence; fixed point theorems; dynamic inequalities.
  
  
• A study on discrete Ponzi Scheme model through Sturm-Liouville theory  
  by Ferhan M. Atici, William Bennett 
  Abstract: In this paper, we introduce a second order self-adjoint difference equation which describes the dynamics of Ponzi schemes: a type of investment fraud that promises more than it can deliver. We use the Sturm-Liouville 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; Sturm-Liouville boundary value problem; Green's function.
  
  
• Application of generalized Riccati equations to analysis of asymptotic forms of solutions of perturbed half-linear ordinary differential equations  
  by Sokea Luey, Hiroyuki Usami 
  Abstract: Asymptotic forms of solutions of half-linear 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 half-linear equation.
  Keywords: half-linear 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 Predator-Prey 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 predator-prey 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: predator-prey system; bio-magnification; 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 non-linear mathematical model of the Zika virus is proposed and analyzed the impact of optimal control strategies with the saturated incident and bed-net 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 time-dependent control parameters are introduced in the system to reduce the transmission. Electronic devices, insecticide-treated 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 period-3 solution or one of two prime period-4 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 period-3 solutions, or one of two prime period-4 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 integro--differential equation with delay is entirely characterised. In the case when the solution is mean--square 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 by--product 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.