The steady states of an antitone electric system are described by an antitone function with respect to the componentwise order. When this function is bounded from below by a positive vector, it has only one fixed point. This fixed point is attractive for the fixed point iteration method. In the general case, we find existence and uniqueness results of fixed points using the set of P-matrices.

In this work, we investigated the invariance analysis of fractional-order Hirota-Satsoma coupled Korteveg-de-Vries (HSC-KdV) system of equations based on Riemann-Liouville (RL) derivatives. The Lie Symmetry analysis is considered to obtain infinitesimal generators; we reduced the system of coupled equations into nonlinear fractional ordinary differential equations (FODEs) with the help of Erdelyi’s-Kober (EK) fractional differential and integral operators. The reduced system of FODEs solved by means of the power series technique with its convergence. The conservation laws of the system constructed by Noether’s theorem.

In this work we study the existence of positive solutions to the following fractional elliptic systems with Hardy-type singular potentials, and coupled by critical homogeneous nonlinearities \begin{equation*} \begin{cases} (-\Delta)^{s}u-\mu_{1}\frac{u}{|x|^{2s}}=|u|^{2^{\ast}_{s}-2}u+\frac{\eta\alpha}{2^{\ast}_{s}}|u|^{\alpha-2} |v|^{\beta}u+\frac{1}{2}Q_{u}(u,v) \ \ in \ \Omega, \\[2mm] (-\Delta)^{s}v-\mu_{2}\frac{v}{|x|^{2s}}=|v|^{2^{\ast}_{s}-2}v+\frac{\eta\beta}{2^{\ast}_{s}}|u|^{\alpha} |v|^{\beta-2}v+\frac{1}{2}Q_{v}(u,v) \ \ in \ \Omega, \\[2mm] \ \ u, \ v>0 \ \ \ \ \ in \ \ \Omega, \\[2mm] \ u=v=0 \ \ \ \ in \ \ \mathbb{R}^{N}\backslash\Omega, \end{cases} \end{equation*} where $(-\Delta)^{s}$ denotes the fractional Laplace operator, $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain such that $0\in\Omega$, $\mu_{1}, \mu_{2}\in [0,\Lambda_{N,s})$, $\Lambda_{N,s}=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$ is the best constant of the fractional Hardy inequality and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. In order to prove the main result, we establish some refined estimates on the extremal functions of the fractional Hardy-Sobolev type inequalities and we get the existence of positive solutions to the systems through variational methods.

In this paper, a backward problem for a nonlinear time-fractional diffusion equation in an axis-symmetric cylinder has been considered. Under some assumptions, we prove the existence and uniqueness of the solution to the nonlinear problem. The ill-posedness of the backward problem is established and we obtain the error estimates by a generalized quasi-boundary value regularization method.

The paper formulates the nonlinear problem of steady-state heat conduction at the constant electric potential difference on the surfaces of a plane dielectric layer with the temperature-dependent heat conduction coefficient and electrical resistivity. A fixed temperature value is set on one of the layer surfaces, and the convective heat exchange with the ambient medium occurs on the opposite surface. The formulation of the problem is transformed into integral ratios, which allows the calculation of the temperature distribution over the layer thickness, governed both by the monotonic and nonmonotonic function. The quantitative assay of the temperature state of a layer of a polymer dielectric made of amorphous polycarbonate is given as an example, as well as the analysis of nonuniformity of the absolute value of electric field intensity over the thickness of this layer.

In this paper we study the existence of a Nash-type equlibrium for a non-potential nonlinear system by combining variational methods with the monotonicity approach. The advance over existing research is that we can consider systems of Dirichlet problems in which the operator is not necessarily linear.

We consider a PDE/ODE system for two pairs of competing species and study the spatial segregation limit as the interspecific competition rate tends to infinity. We show that the limiting problem is a one-phase Stefan problem for nonlinear diffusion equations.

In this paper, we generalized E.C. Kim’ s estimates by taking in to account the trace of the divergencefree symmetric tensor non−zero. We have also shown that E.C. Kim’s estimates still valid in case of the trace of the divergencefree symmetric tensor vanished identically. In the equality case, we characterized eta−Killing spinor with Killing pair over the Sasakian spin manifolds.

We consider an age-size structured cell population model based on the cell cycle length. The model is described by a first order partial differential equation with initial-boundary conditions. Using the theory of semigroups of positive operators we establish new criteria for an asynchronous exponential growth of solutions to such equations. We discuss the question of exponential size growth of cells. We show how to incorporate into our description models with constant increase of size and with target size division. We also present versions of the model when the population is heterogeneous.

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and $f$-fixed point of Bregman relatively $f$-nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. As an application, we provide several applications of our method. Furthermore, we give a numerical example to demonstrate the behavior of the convergence of the algorithm.

In this paper, we define three types of 2-Ruled hypersurfaces in the Minkowski 4-space $\mathbb{E}^4_1$. We obtain Gaussian and mean curvatures of the 2-ruled hypersurfaces of type-1 and type-2, and some characterizations about its minimality. We also deal with the first Laplace-Beltrami operators of these types of 2-Ruled hypersurfaces in $\mathbb{E}^4_1$. Moreover, the importance of this paper is that the definition of these surfaces by using the octonions in $\mathbb{E}^4_1$. Thus, this is a new idea and make the paper original. We give an example of 2-ruled hypersurface constructed by octonion and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one-dimensional object embedded in the 4-dimensional Minkowski space $\mathbb{E}^4_1$. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2-ruled hypersurfaces in a four-dimensional Minkowski space

A malaria transmission disease model with host selectivity and Insecticide treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is much more expeditious than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two-slow for the host and one-fast for the vector. The basic reproduction number R0 is calculated and the local stability analysis is performed at equilibria of the model when the perturbation parameter ɛ > 0. The model is analyzed when ɛ → 0 using asymptotic expansions technique. Merging bed-net control, vector-bias, and singular perturbation have a notable effect on the model dynamics. It is shown that if over %30 of humans use ITNs, malaria disease burden can be reduced. The dynamics on the slow surface indicate that the infected vectors decays very fast when ɛ = 0.001 according to the numerical simulations.

The paper is concerned with the multiple solutions of a nonhomogeneous elliptic system with critical exponent over a non-contractible domain, precisely, a smooth bounded annular domain. We prove the existence of four solutions using variational methods and Lusternik-Schnirelmann theorey, when the inner hole of the annulus is sufficiently small.

We consider the equations of motion of a bar, with given density, infinite in both directions, subjected to longitudinal vibrations under the action of an external load, and a stress-strain relation represented by a fractional order operator.  Using three types of fractional operators, the initial-boundary value problems associated with the described phenomenon are posed and solved. Through the bivariate Mittag-Leffer function, which has been recently introduced, we find the fundamental solution of these problems and calculate their moments.

This paper is concerned with a West Nile virus (WNv) model on a growing domain, which accounts for habitat expansion of mosquitoes because of climate warming. We aim to understand the relationship of the growing rate and the transmission risk of WNv. The basic reproduction number, which is related to the growing rate and diffusion rate, is introduced through spectral theory. The conditions to determine whether the virus vanishes or spreads are deduced.The obtained results reveal that domain growth leads to increased risk of infection, and is detrimental to the control and prevention of WNv. To verify the feasibility of our analytical results on the long time behavior of WNv, some numerical simulations are given.

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether’s theorem.

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.