Authors:Jia Mu; Shuibo Huang; Ling Guo Pages: 3 - 14 Abstract: We discuss the existence and regularity of solutions to some fractional evolution equations in the q-norm. The linear part generates a noncompact semigroup, and the nonlinear part satisfies some conditions with respect to the fractional power norm of the linear part. In the end, we apply the obtained results to a control system. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0337-3 Issue No:Vol. 26, No. 1-3 (2018)

Authors:P. Muthukumar; K. Thiagu Pages: 15 - 36 Abstract: In this paper, we investigate the existence of mild solutions and the approximate controllability of a class of nonlinear fractional stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps which satisfies the nonlocal conditions in Hilbert space. The existence of mild solutions is proved by using Sadovskii’s fixed point theorem. Also the approximate controllability of the nonlinear fractional nonlocal stochastic differential equations of order \(1<q\le 2\) with infinite delay and Poisson jumps is checked by using Lebesgue dominated convergence theorem. Finally an example is included to illustrate the results. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0340-8 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Boubacar Diao; Khalil Ezzinbi; Mamadou Sy Pages: 37 - 55 Abstract: The aim of this work, is to study the existence and regularity of mild solutions for a class of abstract partial functional integrodifferential equations with infinite delay under the alpha-norm. We assume that the linear part generates an analytic semigroup, the nonlinear part is assumed to be continuous with respect to the alpha norm associated to the linear part. The phase espace is axiomatically defined. In the end, an illustration is proveded for some reaction-diffusion equation with infinite delay. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0315-9 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Saroj Kumar Sahani; Yashi Pages: 57 - 80 Abstract: Delayed models are a better representation of the nature of HIV. In the present paper, a multi-delayed model of HIV with combination drug therapy has been analysed. Effect of the immune response in the form of effector cell response has also been included to make the model more justified. The threshold properties related with the basic reproduction number \(R_0\) have been discussed. The local and global properties of the model have been analysed. Extensive numerical simulations have been performed to show the impact of highly effective drug on the concentration of virus. The numerical simulations and the result proved has led to the conclusion that a highly effective drug when combined with a less effective drug, can very efficiently bring down the viral load to undetectable levels. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0341-7 Issue No:Vol. 26, No. 1-3 (2018)

Authors:V. Sree Hari Rao; P. Raja Sekhara Rao Pages: 81 - 104 Abstract: The influence of time varying inputs on a two layer neural network involving transmission delays is studied in this paper. The asymptotic behavior of solutions of the system is studied for a variety of input functions. Conditions for solutions of a system with time varying inputs to approach the solutions (including equilibrium solutions) of the corresponding system with constant inputs are derived. This explains how a system tolerates variations in external inputs. Most important contribution of the study is that conditions on the inputs of the system are derived so that solutions of the system converge to a pre-specified output. The usefulness of the results is highlighted through examples and simulations. This research provides a thorough mathematical support for an upcoming technology that enables one to build intelligent machines with more sophisticated learning capabilities. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0312-z Issue No:Vol. 26, No. 1-3 (2018)

Authors:Navnit Jha; Venu Gopal; Bhagat Singh Pages: 105 - 123 Abstract: In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in three-space dimensions has been developed. The discretization formula is developed on a non-uniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of \(l^\infty \) - and \(l^2\) -error of the exact and approximate solution values. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0314-x Issue No:Vol. 26, No. 1-3 (2018)

Authors:Adel Ouannas; Abdulrahman Karouma Pages: 125 - 137 Abstract: This paper addresses the problem of generalized synchronization (GS) between different dimensional integer-order and fractional-order chaotic systems. Based on the stability theory of linear continuous time dynamical systems, stability results of linear fractional order systems and nonlinear controllers, different criterions are derived to achieve generalized synchronization. The effectiveness of the proposed control schemes are verified by considering two examples: fractional-order chaotic Lorenz and hyperchaotic Lorenz systems and hyperchaotic Chen and fractional-order chaotic Chen systems. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0317-7 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Mouffak Benchohra; Imene Medjadj Pages: 139 - 155 Abstract: Our aim in this work is to study the existence of solutions of first and second order functional differential equations with state-dependent delay. We use the Mönch’s fixed point theorem for the existence of solutions and the concept of measures of noncompactness. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0325-7 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Chandrima Banerjee; Pritha Das Pages: 157 - 176 Abstract: In this paper, dynamical effect of seasonal variations and impulsive perturbations on a hybrid tri-trophic food chain model has been investigated. In this model mixed functional responses (Holling type-II and modified Leslie–Gower type) are considered. Using Floquet theory and small amplitude perturbation techniques, it is found that a prey and middle predator free periodic solution is globally asymptotically stable if the impulsive period lies in between two critical values. Further, using comparison results of impulsive differential inequalities, it is derived that the model system is permanent under certain conditions. Moreover, the model system is simulated numerically to find the influence of the seasonal forcing and impulsive perturbation. Complex dynamical behavior has been observed for biologically feasible parametric values. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0328-4 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Moussa El-Khalil Kpoumiè; Khalil Ezzinbi; David Békollè Pages: 177 - 197 Abstract: The aim of this work is to study the existence of a periodic solution for some nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We use Massera’s approach (Duke Math 17:457–475, 1950), we prove that the existence of a bounded solution on \(\mathbb {R}^{+}\) implies the existence of an \(\omega \) -periodic solution. In nonlinear case, we use a fixed point for multivalued maps to show the existence of a periodic solution. Finally, we consider a reaction diffusion equation with delay to illustrate the main results of this work. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0331-9 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Boqiang Cao; Xining Li; Qiang Li; Ying Zhang Pages: 199 - 212 Abstract: Population systems often subject to both white noise and environment pollution, so a stochastic non-autonomous omnivory Lotka–Volterra model in a polluted environment is investigated in this paper. We establish the sufficient conditions for the existence of positive periodic solution and prove it by constructing Lyapunov function. According to Itô’s formula and the strong law of large numbers for martingales, we also discuss the extinction of population. The sufficient conditions for almost sure exponential stability of equilibrium point \(E^*(0,0,0,S^*,T^*)\) are obtained. Finally, we illustrate our results by some examples with the help of computer simulation. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0334-6 Issue No:Vol. 26, No. 1-3 (2018)

Authors:P. D. N. Srinivasu; D. K. K. Vamsi; I. Aditya Pages: 213 - 246 Abstract: Ecological and biological conservation of living systems has been an active area of research over the years by agriculturalists, biologists and mathematicians. One of the studies involves additional food supplement feeding (also called as diversionary feeding) for the purpose of biological (wildlife in some cases) conservation. The idea in this approach is to distract (thereby supplement) the wildlife from predating upon the other species with the end goal of wildlife conservation. On the other hand in agricultural entomology, insect control and optimization, additional food is supplemented as a tool for effective pest control thereby achieving the biological control. The study of these ecosystems is usually done using the predator–prey systems. In nature, we find situations wherein the group defense (toxicity) of the prey reduces the predator’s predation rate. This type of behaviour of the prey is also known as inhibitory effect of the prey. Biological conservation of such predator prey systems in the presence of additional food supplements is quite challenging and interesting. In this paper, we consider an additional food provided predator–prey system which is a variation of the standard predator–prey model in the presence of the inhibitory effect of the prey. The predators functional response is assumed to be of Holling type IV (considering the inhibitory effect). This model is analyzed to understand the inherent dynamics of the system. The findings suggest that the quality and quantity of additional food provided to the predators, play a very significant role in determining the eventual state of the ecosystem. The outcomes of the analysis suggests eco friendly strategies to eco-managers for biological conservation of living systems. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0344-4 Issue No:Vol. 26, No. 1-3 (2018)

Authors:Pan Mu; Lei Wang; Yi An; Yanping Ma Pages: 265 - 277 Abstract: This paper considers the microbial batch culture for producing 1,3-propanediol(1,3-PD) via glycerol disproportionation. Due to the nature of the fractional order operations, a novel fractional order model, which is based upon the original ordinary differential dynamic system, is introduced to describe the complex bioprocess in a more accurate manner. Existence and uniqueness of solutions to the novel fractional order system and the continuity of solutions with respect to the parameters are discussed respectively. In addition, a parameter identification problem of the system is presented, and a particle swarm optimization algorithm is constructed to solve it. Finally, the conclusion is drawn by numerical simulations. PubDate: 2018-01-01 DOI: 10.1007/s12591-017-0381-7 Issue No:Vol. 26, No. 1-3 (2018)

Authors:M. R. A. Moubarak; H. F. Ahmed; Omar Khorshi Pages: 279 - 291 Abstract: This paper presents a method to solve numerically the linear quadratic optimal control problems (LQOCPs) of the fractional singular control systems with Caputo fractional derivatives. Under certain conditions on the singular fractional control systems, the problems under consideration are transformed into LQOCPs of the standard normal fractional control systems. Based on the Grunwald–Letnikov approximation (GLA) of the fractional derivatives (FDs), a numerical technique is used to solve the LQOCPs of the standard normal fractional control systems. An illustrative example is introduced to demonstrate the applicability of the proposed technique. PubDate: 2018-01-01 DOI: 10.1007/s12591-016-0320-z Issue No:Vol. 26, No. 1-3 (2018)

Authors:Shruti Dubey; Sharad Dwivedi Abstract: In this article, we address the problem of stability and controllability of two-dimensional network of ferromagnetic particles of ellipsoidal shapes. The dynamics of magnetization inside the ferromagnetic material is governed by the Landau–Lifschitz equation of micromagnetism which is non-linear and parabolic in nature. The control is the magnetic field generated by a dipole whose position and amplitude can be selected. In the absence of control, first we prove the exponential stability of the relevant configurations of the network. Then, we investigate the controllability by the means of external magnetic field induced by the magnetic dipole. PubDate: 2018-01-13 DOI: 10.1007/s12591-018-0407-9

Authors:A. K. Misra; Arvind K. Singh; Pushkar Kumar Singh Abstract: In this paper, a nonlinear mathematical model to study the effect of skill development on the control of unemployment is proposed and analyzed. In the modeling process, three different classes; namely unemployed, self/temporary employed and regular employed are considered. It is also assumed that some academic institutions develop skills among the youths to get the employment; however the avenues to develop skills among the unemployed persons increases as the number of unemployed persons increases. It is further assumed that the rate of movement from unemployed class to self/temporary employed class increases as the avenues for skill development increases. The model is analyzed using stability theory of differential equations. It is found that the proposed model exhibits only one equilibrium, which is globally stable under certain conditions. Numerical simulation of the model has been carried out to confirm the analytical results. PubDate: 2017-12-21 DOI: 10.1007/s12591-017-0405-3

Authors:Jonathan Mitchell Abstract: The homotopy analysis method (HAM) has been shown to approximate solutions to nonlinear problems of various types. The method is useful because of the flexibility in defining the auxiliary parameter \(\hbar \) and that no assumption concerning relative size of parameters is required. We use the HAM to approximate frequency and solution of a conservative nonlinear oscillator which has many applications including lasers, epidemics, and microparasites. The oscillator is modeled by the simplified Liénard Equation \(\ddot{x}+\epsilon {\dot{x}}x+x=0\) . We compare our results for the weakly-nonlinear case to those gained from a more traditional perturbation technique, the Poincaré–Lindstedt method. We then show results from the HAM in the strongly-nonlinear regime and discuss some benefits and limitations of the method including conditions for convergence. PubDate: 2017-11-26 DOI: 10.1007/s12591-017-0404-4

Authors:G. M. Bahaa; Qing Tang Abstract: In this paper, fractional optimal control problem for two dimensional coupled evolution system is investigated. The fractional time derivative is considered in Caputo sense. Constraints on controls are imposed. First, the existence and uniqueness of the state for these systems is proved. Then, the necessary and sufficient optimality conditions for the fractional Dirichlet problems with the quadratic performance functional are derived. Finally we give some examples to illustrate the applicability of our results. PubDate: 2017-11-20 DOI: 10.1007/s12591-017-0403-5

Authors:A. Boudaoui; E. Lakhel Abstract: In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result. PubDate: 2017-11-05 DOI: 10.1007/s12591-017-0401-7