Vol. 12, No. 3 (2023): Regular Issue

In this article, we present several types of generalized $\alpha$-admissible almost type $\mathcal{Z}$-contractions, which can be considered as the generalizations of $\alpha$-admissible $\mathcal{Z}$-contractions and almost $\mathcal{Z}$-contractions, and obtain the fixed point results of these contractions in complete metric spaces. Moreover, we utilize some examples to verify the validity of main results. Finally, we give some fixed point results related to our results.

In this work, we aim to obtain the existence of weak solutions of a $p(x)$ -Laplacian pseudo-parabolic equation with memory term and logarithmic nonlinearity. Moreover, a lower bound for blow up time is also established using the differential inequality technique when the solution blows up at a finite time $T^\star$.

In this paper, a predator-prey model with Monod-Haldane type functional response is developed. Here, it is assumed that prey grows logistically in the absence of predator. Also, predator population is divided into two subpopulations such as juvenile predator and mature predator respectively. To incorporate the group defense behavior in the model, Monod-Haldane type functional response is considered. It is considered that a portion of juvenile predator becomes mature predator. It is assumed that the growth rate of predator at an instant is not depends only the density of prey at the present time, but also depends on the density of the prey on the previous instant of time. Different possible equilibrium points are determined. Also, the stability of the model around these equilibrium points is studied. Hopf bifurcation analysis of the model is done with respect to some important parameters. It is observed that exponential fading memory has a big role in the stability of the model. Finally, some numerical simulation results have been presented for better understanding of the dynamics of the model.

This work considers a new stochastic mathematical model for the transmission dynamics of the coronavirus COVID-19 by providing the healthy compartment together with the quarantine/isolation compartment. In the deterministic model, global stability conditions of the disease-free equilibrium $E_0$ and the endemic equilibrium $E_\star$ are derived in terms of the threshold quantity $R_0^d$. Based on the chaotic behavior, we develop and analyze a four-dimensional stochastic COVID-19 epidemic model. Uniqueness, boundedness, and positiveness of the proposed stochastic model are investigated in a biologically feasible region. In terms of the stochastic basic reproduction number $R_0^s$ of the stochastic model, extinction and persistence of the COVID-19 disease are derived. Our theoretical findings are supported by some numerical simulations. The sensitivity of the model with respect to the parameters involved in the system is studied to investigate the most sensitive parameter towards the highest number of infected individuals. We confirm the stability analysis by showing the elasticity of $R_0^s$ with respect to the variation of each parameter. We present real data of a case study with the first wave of the COVID-19 epidemic in the United Kingdom. We compare our numerical results with the real data.

Quasi-rational solutions to the cylindrical nonlinear Schrödinger equation (CNLS) in terms of wronskians and Fredholm determinants of order $2N$ depending on $2N-2$ real parameters are given. We get multi-parametric families of quasi-rational solutions to the CNLS equation and we construct explicitly solutions of order $1$ to $5$.

This work focuses on existence results of fractional order stochastic non-instantaneous impulsive neutral functional integro-differential equation with infinite delay. The results are obtained by using the Hausdorff measure of non-compactness, fractional calculus, stochastic analysis techniques and fixed point theorems. Finally, examples are given to illustrate the obtained theory.

In this work, we consider the two-term linear nabla fractional difference equation $$ {(}\nabla^{\nu}_{-1}u{)}(t) = \lambda u(t - 1), \quad t \in \mathbb{N}_{1}, $$ where $0 < \nu < 1$, $\lambda \in \mathbb{R}$, $\nabla^{\nu}_{-1}u$ denotes the $\nu$-th Riemann--Liouville nabla fractional difference of $u$ based at $-1$, and $\mathbb{N}_{1} = \{1, 2, 3, \cdots\}$. First we transform this nabla fractional difference equation into a Volterra difference equation of convolution-type. Using the well established qualitative theory of Volterra difference equations, we obtain sufficient conditions on asymptotic stability of solutions of the nabla fractional difference equation.

In this paper, we present and investigate a generalized stochastic AIDS/HIV epidemic model that includes both Brownian motions and Lévy jumps. Our proposed model is a staged progression compartmental one that takes the form of an Itô-Lévy stochastic differential equations system. First, we demonstrate its well-posedness in the sense that it admits one and only one solution which is positive and global in time. Then, and based on some assumptions and nonstandard analytical techniques, we prove two principal asymptotic properties: extinction and persistence in the mean. The theoretical results show that the dynamical behavior of our AIDS/HIV model is mainly determined by the parameters that are closely related to the perturbations intensities. In the end, we provide some numerical simulation examples to corroborate our theoretical results and exhibit the effect of the new adopted mathematical techniques on the findings.

The existence, uniqueness, and Ulam-Hyers stability of the Volterra-Fredholm integro-differential equation with non-instantaneous impulses and periodic boundary conditions over time scales are investigated in this paper using Banach fixed point theorems and Caputo delta fractional derivative. Finally, we present an example to confirm our main findings.

We analyze the existence results of fractional integro-differential equations via $\Psi$-Hilfer fractional derivative with nonlocal multi-point condition by using Schauder fixed point theorem. To establish the sufficient conditions for compactness of operators and an example is also discussed.

We study the variation along the normal direction of the deformation energy of a plane curve under the existence of an anisotropic term. The problem of variational character corresponds to a nonlinear nonhomogeneous differential equation of fourth order. This kind of problems arises from the elasticity theory, in particular from the deformation theory of elastic shells.

In this study, a Lotka-Volterra type prey-predator harvesting model with fuzzy biological parameters under some assumptions is presented. It is assumed that the parameters involved in biological model are vague/imprecise under consideration. The uncertainty of the said parameters is handled by triangular fuzzy numbers. First, the crisp harvesting model is formulated under some assumptions. Then the crisp model is converted to fuzzy model and then it is defuzzified by using utility function method. The existences of equilibrium points of the defuzzified model are identified and corresponding stabilities are checked.\ The economic features as well as the harvesting strategies at the optimal stage of our wished-for model is considered. Lastly, mathematical simulations of the defuzzified model with numerical data are carried out using MATLAB and MATHEMATICA to validate the theoretical results.

A reaction-diffusion model characterizing the dynamics of computer virus epidemic is considered in this paper. The propagation of viruses in computers is similar to the case of many infectious diseases so that the consideration of reaction-diffusion terms becomes necessary to look into the deep insights. The structure preserving analysis of virus propagation in the computers connected globally is performed through the extended reaction-diffusion mathematical model. A numerical scheme based on the collocation of cubic B-splines is proposed to investigate the computer virus epidemic model. The numerical results obtained are compared and validated by performing stability analysis and are found in good agreement with those already available in the literature. Due to the unavailability of the analytic solutions of these models, such a numerical simulation scheme can be of prime interest for biologists to interpret the results theoretically.

The present article studies chaos synchronization of a novel chaotic systems using nonlinear control method. Stability of system at equilibrium points are also discussed in brief for fractional order system. We use the nonlinear control method for synchronization between fractional order 3 scroll Dadras chaotic system with fractional order 2 scroll Lorenz and Chen chaotic systems. A nonlinear controller is designed for synchronization. Based on the design, the synchronization of considered chaotic systems is achieved only by using one controller. Nonlinear control method is a practicable method to synchronize chaotic systems. Adams-Boshforth-Moulton method is used for the computer simulation for integer order as well as fractional order in the Caputo sense. Graphical Results are also displayed to validate the effectiveness of the proposed method.

In the present study, our main work is focused on solving the fractional order nonlinear infinite system of Erd$\acute{\mbox{e}}$lyi-Kober type functional integral equations in sequence space $\ell_p,~ p>1$ by applying Hausdorff measure of non-compactness, and generalized Meir-Keeler (M-K) fixed point theorem. An example is presented to validate our existence theorem. We propose an iterative algorithm formed by homotopy perturbation along with the Adomian decomposition method to solve the considered problem with high accuracy. A numerical example is also used to show that our iterative algorithm converges strongly to the approximate solution of the proposed problem.