We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

In this paper, variable exponent function spaces ${\mathbb{L}}^{p\left(·\right)}$, ${\mathbb{L}}_b^{p\left(·\right)}$, and ${\mathbb{L}}_c^{p\left(·\right)}$ are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.

In this paper, we investigate the existence of solution for differential systems involving a $\varphi -$Laplacian operator which incorporates as a special case the well-known $p-$Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.

The aim of this paper is to give fixed point theorems for $G$-monotone $\rho$-nonexpansive mappings over $\rho$-compact or $\rho$-a.e. compact sets in modular function spaces endowed with a reflexive digraph not necessarily transitive. Examples are given to support our work.

The main purpose of this article is to make use of the Horadam polynomials $h_n\left(x\right)$ and the generating function $\mathrm{\Pi }\left(x,z\right)$, in order to introduce three new subclasses of the bi-univalent function class $\sigma .$ For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete–Szegö inequalities. Some interesting observations of the results presented here are also discussed. We also provide relevant connections of our results with those considered in earlier investigations.

The Multivariate Geographically Weighted Regression (MGWR) model is a development of the Geographically Weighted Regression (GWR) model that takes into account spatial heterogeneity and autocorrelation error factors that are localized at each observation location. The MGWR model is assumed to be an error vector $\left(\mathbf{\epsilon }\right)$ that distributed as a multivariate normally with zero vector mean and variance-covariance matrix $\mathbf{\Sigma }$ at each location $\left(u_i,v_i\right)$, which $\mathbf{\Sigma }$ is sized $qxq$ for samples at the $i$-location. In this study, the estimated error variance-covariance parameters is obtained from the MGWR model using Maximum Likelihood Estimation (MLE) and Weighted Least Square (WLS) methods. The selection of the WLS method is based on the weighting function measured from the standard deviation of the distance vector between one observation location and another observation location. This test uses a statistical inference procedure by reducing the MGWR model equation so that the estimated error variance-covariance parameters meet the characteristics of unbiased. This study also provides researchers with an understanding of statistical inference procedures.

In this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the $3$-dimensional Nowak–May within-host HIV dynamics model. Taking into consideration that precise forms of functional responses are usually unknown, we introduce into this model a nonlinear incidence rate of a rather general form given by an unspecified function of the susceptible cells and free virus particles. We also add a term responsible to the loss of free virions due to infection of the target cells. To mirror the idea of highly active anti-HIV therapy, in this model we assume six controls that can act simultaneously. These six controls affecting different stage of virus life cycle comprise all controls possible for this model and account for all feasible actions of the existing anti-HIV drugs. With this control model, we consider an optimal control problem of minimizing the infection level at the end of a given time interval. Using an analytical mathematical technique, we prove that the optimal controls are bang-bang, find accurate estimates for the maximal possible number of switchings of these controls and establish qualitative types of the optimal controls as well as mutual relationships between them. Having the estimate for the number of switchings found, we can reduce the two-point boundary value problem for Pontryagin Maximum Principle to a considerably simpler problem of the finite-dimensional optimization, which can be solved numerically. Despite this possibility, the obtained theoretical results are illustrated by numerical calculations using BOCOP–2.0.5 software package, and the corresponding conclusions are made.

Cholera is an infectious intestinal disease which occurs as a result of poor sanitation and lack of basic education in its transmission. It is characterized by profuse vomiting and severe diarrhea when an individual eats food or drinks water contaminated with the Vibrio cholerae. A dynamic mathematical model that explicitly simulates the transmission mechanism of cholera by taking into account the role of control measures and the environment in the transmission of the disease is developed. The model comprises two populations: the human population and bacteria population. The next-generation method is used to compute the basic reproduction number, $R_0$. Both the disease-free and endemic equilibria are shown to be locally and globally stable for $R_0$ values less than unity and unstable otherwise. Necessary conditions of the optimal control problem were analyzed using Pontryagin’s maximum principle with control measures such as educational campaign and treatment of water bodies used to optimize the objective function. Numerical values of model parameters were estimated using the nonlinear least square method. The model simulations confirm the significant role played by control measures (education and treatment of water bodies) and the bacteria in the environment in the transmission dynamics as well as reducing the spread of cholera.

In this work, the Sumudu decomposition method (SDM) is utilized to obtain the approximate solution of two-dimensional nonlinear system of Burger’s differential equations. This method is considered to be an effective tool in solving many problems. Our results have shown that the SDM offers a much better approximation for solving several numbers of systems of two-dimensional nonlinear Burger’s differential equations. To clarify the facility and accuracy of the strategy, two examples are provided.

In this paper, we introduce the concept of a set-valued or multivalued quasi-contraction mapping in V-fuzzy metric spaces. Using this new concept, a fixed-point theorem is established. We also provide an example verifying and illustrating the fixed-point theorem in action.

In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.

In this article, we investigate spectrum estimation of law order moving average (MA) process. The main tool is the lag window which is one of the important components of the consistent form to estimate spectral density function (SDF). We show, based on a computer simulation, that the Blackman window is the best lag window to estimate the SDF of $\text{MA}\left(1\right)$ and $\text{MA}\left(2\right)$ at the most values of parameters ${\beta }_i$ and series sizes $n$, except for a special case when $\beta =-1$ and $n\ge 40$ in $\text{MA}\left(1\right)$. In addition, the Hanning–Poisson window appears as the best to estimate the SDF of$\hspace{0.17em}\text{MA}\left(2\right)$ when ${\beta }_1={\beta }_2=-0.5$ and $n\ge 40$.

In this paper, we introduce a new iterative method in a real Hilbert space for approximating a point in the solution set of a pseudomonotone equilibrium problem which is a common fixed point of a finite family of demicontractive mappings. Our result does not require that we impose the condition that the sum of the control sequences used in the finite convex combination is equal to 1. Furthermore, we state and prove a strong convergence result and give some numerical experiments to demonstrate the efficiency and applicability of our iterative method.

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation $\Delta \left(c_3\left(n\right)\Delta \left(c_2\left(n\right)\Delta \left(c_1\left(n\right)\Delta u\left(n\right)\right)\right)\right)+p\left(n\right)f\left(u\left(n-k\right)\right)=0$ under the conditions . New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.

The Bregman Proximal Gradient (BPG) algorithm is an algorithm for minimizing the sum of two convex functions, with one being nonsmooth. The supercoercivity of the objective function is necessary for the convergence of this algorithm precluding its use in many applications. In this paper, we give an inexact version of the BPG algorithm while circumventing the condition of supercoercivity by replacing it with a simple condition on the parameters of the problem. Our study covers the existing results, while giving other.

In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, unify, and generalize certain recent results in the literature.

In this work, the forced Duffing equation with secondary resonance will be considered our subject by assuming that the initial values has uncertainty in terms of a fuzzy number. The resulted fuzzy models will be studied by three fuzzy differential approaches, namely, Hukuhara differential and its generalization and fuzzy differential inclusion. Applications of fuzzy arithmetics to the models lead to a set of alpha-cut deterministic systems with some additional equations. These systems are then solved by the extended Runge-Kutta method. The extended Runge-Kutta method is chosen as our numerical approach in order to enhance the order of accuracy of the solutions by including both function and its first derivative values in calculations. Among our fuzzy approaches, our simulations show that the fuzzy differential inclusion is the most appropriate approach to capture oscillation behaviors of the model. Using the aforementioned fuzzy approach, we then demonstrate how to estimate parameters to our generated fuzzy simulation data.

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).