Vol. 11, No. 3 (2022): Regular Issue

The model under discussion is an elaborated recurrent impulsive neural network. This is the first time in literature that the impacts are structured completely as the original neural network, such that physical sense of impacts has been explained. Moreover, the impact part comprises all types of impacts in neural networks, which were traditionally studied in conservative models. In the research, neuron membranes with negative as well as positive capacitance, are considered newly as parts of the neural networks. This was not studied before. The system is analyzed in matrix form to facilitate more transparent presentation. The existence and uniqueness of asymptotically stable discontinuous almost periodic solutions are investigated. An example with simulations is provided to illustrate the results.

We study existence,uniqueness and continuous dependence of a mild solution for an impulsive neutral functional differential nonlocal Cauchy problem with state- dependent delay in general Banach spaces are studied by using the fixed point technique and semigroup of operators.

In the present paper, we consider a class of reaction-diffusion systems based on the Lotka-Volterra differential equation model of a predator-prey interaction with the existence of memory terms. We show that every solution with initial values in $[0,l]$ and subject to homogeneous Neumann boundary conditions decays to a spatially homogeneous function of time.

In [J. Combinatorial Theory, Ser. B, 28 (1980), 305-359], Seymour introduced the binary matroid 3-sums and proved that if a 3-connected binary matroid $M$ is a 3-sum of matroids $M_1$ and $M_2$, then each of $M_1$ and $M_2$ is isomorphic to a proper minor of $M$. For a 3-connected binary matroid $M$ expressed as a 3-sum of $M_1$ and $M_2$, we show that in general, both $M_1$ and $M_2$ are 2-connected, and if $M_1$ and $M_2$ are simple matroids, then both $M_1$ and $M_2$ are also 3-connected.

Mathematically, Mandelbrot defined fractals as sets with non integer Hausdorff dimension which exceeds topological dimension. Later Hutchinson developed the theory of Iterated Function System (IFS) to explain self similarity mathematically. IFS theory and its generalisations were studied intensively from Barnsly onwards. Different forms of self similarities and their topological properties were discussed. They were carried out to higher dimensional spaces and corresponding results were established in the literature.

In this paper, the solutions of high-order nonlinear differential equations with delay and anticipation subject to mixed conditions are obtained by converting them into algebraic equations by using Bernstein polynomials and collocation points. Then the algebraic equations are solved by using Newton's method. Some examples are presented to illustrate the method proposed. In the problems involving delay and anticipation, the terms involving deviated arguments are converted into linear terms with the help of Taylor's series.

In this paper chaos control and synchronization techniques are applied on the introduced novel fractional chaotic systems. The system is extensively studied for its dynamical properties using various tools such as Lyapunov spectrum, bifurcation diagrams, phase portraits, equilibrium points, dissipative character, uniqueness of solution and so on. Besides the effect of changing fractional order on the dynamics are also studied in detail. The chaotic behavior of the novel system is controlled about any randomly chosen point. The systems are then synchronized in quad compound combination anti-synchronization with eight chaotic systems in presence of disturbances and uncertainties. The achieved synchronization is illustrated in secure communication with help of an example.

We consider the controllability problem for a class of fractional impulsive evolution systems of mixed type in an infinite dimensional Banach space. The existence of mild solutions and controllability results are discussed by a new estimation technique of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. However, the main results do not need any restrictive conditions on estimated parameters of the measure of noncompactness. Since we do not assume that the semigroup is compact and other conditions are more general, the outcomes we obtain here improve and generalize many known controllability results. An example is also given to demonstrate the applications of our main results.

This work deals with the existence of the system consists of coupling dynamics of the two types of tumor cells among the density of epithelial cells (ECs) and the mesenchymal cells (MCs) with the proteins matrix metalloproteinases (MMPs) and extra cellular matrix (ECM) which involved in the invasion and the intravasation processes. Along with square integrable mesenchymal epithelial transition function, the existence and uniqueness of mathematical model illustrated under Faedo-Galerkin approximation method which governed by the invasion model along with EMT and MET process which contains nonlinear terms due to acidification and interactions.

In this paper, we present the existence, uniqueness and asymptotic behaviour of mild solution for neutral stochastic impulsive integro-differential equations driven by fractional Brownian motion and Brownian motion with the Hurst index $H>\frac{1}{2}$ with nonlocal condition. The results are obtained by using Banach fixed point principle in a Hilbert space and the theory of resolvent operator.

A new 3-D chaotic system with a biscuit-like closed curve equilibrium is proposed in this paper. We analyze the qualitative properties of the new chaotic system in terms of phase plots, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We also establish that the new chaotic system has multistability with coexisting attractors. As a control application, we use integral sliding mode control for self-synchronization of the new chaotic system taken as master-slave systems. Finally, an electronic circuit realization of the new chaotic system is developed in MultiSIM, which confirms the feasibility of the system.

Two new stability results, Hyers-Ulam stability and Hyers-Ulam-Rassias stability, of a class Volterra-Fredholm integral equations are presented by using a fixed point theorem in a generalized complete metric space. In addition, for corresponding Volterra-Fredholm integral equations on infinite intervals the Hyers-Ulam-Rassias stability is also obtained.

This work highlights the influence of nonlinear mixed convective flow of a non-Newtonian fluid over a horizontal plate in the presence of heat generation/absorption. The Casson fluid model is employed to express the non-Newtonian behavior of the fluid. Also, the density of the Casson fluid is assumed to be a nonlinear function of temperature. The boundary layer analysis is adopted by introducing a set of non-dimensional transformations for deriving the non-dimensional form of flow governing equations. The proposed problem does not permit a similarity solution. Thus, local similarity and local non-similarity methods are adopted to convert the set of nonlinear PDEs to the set of nonlinear ODEs. On account of local similarity and non-similarity method, the consequential ODEs are solved numerically by the Runge-Kutta method together with the shooting technique. The control of pertinent parameters on the velocity and temperature fields, and on the non-dimensional heat transfer rate as well as on the skin friction coefficient, are analyzed through graphical representation and explored in detail. Prior knowledge about the effect of these parameters on the heat transfer rate and skin friction coefficient can be very useful in the perspective of industrial applications.

The degree of trigonometric approximation of periodic functions $f(x,y)$ belonging to generalized H\"older class by double Hausdorff matrix summability means of double Fourier series has been obtained in this paper. Some corollaries have also been established to find estimates of approximation using almost Euler means and $\left(C, \gamma, \delta \right)$ means.

The study of the longitudinal dispersion of a drug delivered through a microvessel is irreversibly absorbed or undergoes an exchange process at the boundary and has many applications in the field of chemical engineering, environmental dynamics, biomedical engineering and physiological fluid dynamics. The dispersion of nanoparticles plays a vital role in medical science during drug delivery through a microvessel. The current problem deals with the dispersion characteristic of blood during drug delivery through a microvessel. The nature of the blood flow is considered as a two phase fluid model, where clear region is considered as non-Newtonian Herschel-Bulkley fluid which followed a nonlinear relation between the shear stress and shear rate, while the peripheral region is considered as Newtonian fluid. The governing equations are solved analytically in the form of Bessel functions while others are solved in the form of numerical integration method. Several factors that influence the dispersion of nanoparticles during drug delivery in a microvessel, such as pressure distribution, nanoparticles volume fraction, the permeability of the blood vessel, yield stress and the radius of the nanoparticle were considered in the present problem. It is observed that the effective diffusion of the nanoparticles reduces with increase in nanoparticles volume fraction and the permeability of the blood vessels increases the effective dispersion at the inlet.