This paper concerned with the orbital stability of solitary waves for the mKdV-Schrödinger system with cubic-quintic nonlinear terms through detailed spectral analysis and abstract stability theorem. First, we derived the explicit solitary wave solutions by assuming the solution expression. Then, through using the orbital stability theory developed by Grillakis et al., we established a general criteria for assessing the orbital stability for solitary waves of this system.

The initial boundary value problem of a class of coupled hyperbolic systems with logarithmic source terms is considered. In this article, we classify the initial data for the global existence, finite time blow-up and long time decay of the solution. By using potential well method combined with Sobolev embedding theorem, the sufficient initial conditions of global existence, asymptotic behavior, the upper and lower bounds of blow-up time are derived at low energy level $E(0) < d$. These results are extended in parallel to the critical case $E(0) = d$. Besides, with additional assumptions on initial data, the finite time blow up result is given with arbitrary positive initial energy $E(0) > 0$.

The Cauchy problem of the fifth-order nonlinear Schrödinger (foNLS) equation is investigated with nonzero boundary conditions in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are introduced to transform the original spectral parameter to a new spectral parameter in order to avoid the multi-valued problem. Based on Lax pair of the foNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions via solving a suitable Riemann-Hilbert problem. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which are helpful to study the propagation behavior of these solutions.

The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks contain at most two cut-vertices of the clique tree. Along with studying the structural properties of a clique tree in $\mathcal C \mathcal T$, we prove its eccentricity matrix to be irreducible, and then determine its inertia showing that every graph in $\mathcal C \mathcal T$ with more than four vertices and odd diameter has two positive and two negative $\mathcal E$-eigenvalues. Positive $\mathcal E$-eigenvalues and negative $\mathcal E$-eigenvalues turn out to be equal in number even for graphs in $\mathcal C \mathcal T$ with even diameter; that shared cardinality also counts the `diametrally distinguished' vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree $G$ in $\mathcal C \mathcal T$ is symmetric with respect to the origin if and only if $G$ has an odd diameter and exactly two adjacent central vertices. Our results generalize those achieved on trees by I. Mahato and M. R. Kannan in 2022.

We propose two nonmonotone retraction-based proximal gradient methods for solving a class of nonconvex nonsmooth optimization problems over the Stiefel manifold. The proposed methods are equipped with the descent direction obtained by a proximal mapping restricted in tangent space of the manifold and the Barzilai-Borwein stepsizes determined by two recent iteration points and the corresponding descent directions. By employing, respectively, the Grippo-Lampariello-Lucidi nonmonotone line search strategy and the Dai-Fletcher nonmonotone line search strategy, our proposed methods are proved to be globally convergent. Analysis on the iteration complexity for obtaining an $\epsilon$-stationary solution is provided. Numerical results on the sparse principle component analysis problems demonstrate the efficiency of our methods.

In this paper, an HIV-1 infection model with intracellular delay, humoral immunity and immune impairment is investigated, in which both virus-to-cell infection and cell-to-cell transmission are considered. The basic reproduction ratio is calculated and the existence of feasible equilibria is established. By analyzing the distributions of roots of the corresponding characteristic equations, the local asymptotic stability of each of feasible equilibria is established. With the help of appropriate Lyapunov functionals and LaSalle's invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable, and the virus is eventually eliminated; if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable. Finally, numerical simulations are carried out to illustrate the effects of some parameters on HIV-1 infection dynamics.

Let $\tau\in\mathbb{C}\setminus\{0\},$ let $p$ and $q$ be distinct positive integers, and let $a,$ $b,$ $c$ be meromorphic functions such that at least one of $b$ and $c$ is not identically equal to zero. The main purpose of this paper is to study the logistic delay differential equations of the Lotka-Volterra type $$w'(z)=w(z)[a(z)+b(z)w^{p}(z-\tau)+c(z)w^{q}(z-\tau)].$$ We prove that any admissible meromorphic solution $w$ of the equation satisfies that the counting function $N(r, w)$ of poles and the characteristic function $T(r, w)$ have the same growth category. Furthermore, we obtain that ``most" of admissible meromorphic solutions of a more general delay differential equation \begin{eqnarray*} w'(z)=w(z)\Big[a(z)+\sum_{j=1}^{k}b_{j}(z)w^{j}(z-\tau)\Big], \qquad k\in \mathbb{N}, \end{eqnarray*} have a pole at least.

This study delves into the Hénon-type weighted elliptic equation, given by $-\Delta_g u = (\sinh r)^\alpha e^u$, within the context of hyperbolic space $\mathbb{H}^n$, where $\alpha > 0$ and $n > 2$. Our research reveals notable distinctions in the stability of solutions when compared to the Euclidean case.

In this paper, we investigate the stationary distribution of a novel stochastic hybrid predator-prey model with fear effect and Beddington-DeAngelis functional response. Based on Markov semigroup theory, the existence of asymptotically stable stationary distribution of stochastic hybrid system is established, which can converge in $L^1$ to an invariant density under appropriate conditions. Moreover, we derive sufficient conditions for extinction and persistence of prey and predator populations in the stochastic hybrid model.

A time-domain elastic scattering problem is considered in three dimensions. In problem setting, a rigid obstacle is immersed in an unbounded domain filled with homogeneous and isotropic elastic medium. In order to analyze the well-posedness of the target problem, we reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval by using a compressed coordinate transformation. The Galerkin method is adopted to prove the uniqueness results, and the energy method is used to prove the stability of the scattering problem. In addition, we derive a priori estimate with explicit time dependence.

In this work, we investigate a class of nonlinear boundary value problems within the framework of Sobolev spaces with variable exponents. After establishing precise formulations of these problems, we reformulate them as nonlinear hyperbolic-type equations. For each of the six problems considered, we establish both existence and uniqueness results. Furthermore, we address the associated stationary problems and prove the existence of solutions by applying a suitable variant of Brouwer's fixed-point theorem.

In this paper, we present some new regularity criteria for suitable weak solutions to the 3D corotational Beris-Edwards system. First, we prove that suitable weak solutions are regular if the scaled $L^{p ; q}$-norm of the velocity field or gradient of velocity is small with $\frac{2}{p}+\frac{3}{q}=2,1＜p \leq \infty$. Next, we give $\varepsilon$-regularity criteria in terms of velocity field $\mathbf{u}$ and director field $\mathbf{Q}$ in Lorentz spaces, which extends the results obtained by Wang et al (J. Evol. Equ. 21: 1627-1650, 2021) for Navier-Stokes equations.

This paper is concerned with the energy decay rate of the total energy to a wave equation with $p(x)$-Laplacian damping (nonlinear strong damping) and nonlinear source. With some suitable restrictions on variable growth exponents $r(x)$ and $p(x)$, first, we prove that the local solution can be extended to exist globally. Second, by using suitable and weighted multiplier techniques, it is proved that the total energy decays logarithmically. The key and main difficulty is to give a prior estimate for the wighted integral $\int_{\Omega}\chi^{p(x)-1}(\tau)|\nabla u(\tau)|^{p(x)}dx$ by some differential inequality techniques. In the proof of energy decay, the traditional method to eliminate the lower-order terms by exploiting the unique continuation and compactness arguments is not needed in our energy decay estimate.

This paper studies a model of $n$ insiders with perfect information trading on a risky asset with value normally distributed and disclosed at a random deadline. We propose a concept of information protocol equilibrium under semi-strong efficient pricing, consisting of an $n-$profile of insider trading strategies with terminal residual information protocols, and find that if a common protocol on terminal residual information before trading is obeyed by all insiders, then, in the market with more than two insiders there exists a uique equilibrium only when it requires to release common partial information eventually, or it does not exist if it requires to release all or not any; but in the market with a single insider, the insider may release all private information eventually to make a maximal profit. Thereby, the existence and uniqueness of information protocol equilibrium among $n$ insiders are deduced. Finally, numerical results illustrate some market characteristics of equilibria with different information protocols required before trading.

In this paper, we establish the exponential stability of the one-dimensional Euler-Bernoulli beam equation with local viscosity. On the one hand, we demonstrate that the Euler-Bernoulli beam equation system is exponentially stable by employing the multiplier method, which relies on an appropriately constructed Lyapunov function. On the other hand, we discretize the Euler-Bernoulli beam equation system using the finite volume difference method. For the resulting semi-discrete system, we construct a discretized multiplier based on the discretized Lyapunov function. Finally, we prove that the semi-discrete Euler-Bernoulli beam equation system is also uniformly exponentially stable.

In this paper, we study bimagnetic monopoles which are topological solitons in three space dimensions. We prove the existence and uniqueness of solution of a static and radially symmetric Bogomol'nyi-Prasad-Sommerfield (BPS) bimagnetic monopoles formulated and presented in a recent study of Bazeia, Marques and Menezes. Our method is based on a dynamical shooting approach depending on two shooting parameters which provides an effective framework for constructing the BPS equations in magnetic core and magnetic shell. Furthermore, we obtain the relation between the BPS and non-BPS monopoles solutions, and properties of static BPS monopoles solutions.

The paper focuses on the ergodicity of a $\phi$-irreducible Markov chain $\{X_{n},n\geq0\}$ that is generated iteratively through the expression $X_{n+1}=f(X_{n})+\epsilon_{n+1}$. Here, $\{\epsilon_n,n\geq1\}$ is a sequence of independent identically distributed centered random variables, $f(\cdot)$ is an $\mathbb{R}$-valued continuous function, and $X_{0}$ is arbitrary but independent of $\{\epsilon_{n},n\geq1\}$. Our main contribution is to provide necessary and sufficient conditions for the ergodicity of this special class of Markov chains. We also present a generalized approach for $f(\cdot)$ in the end.

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. A linear (star) forest is a forest consisting of paths (stars). A path-star forest $F$ is a forest consisting of paths and stars. In this paper, we determine $ex(n,F)$ for sufficiently large $n$ and characterize the corresponding extremal graphs, and our result generalizes previous known results on the Turán numbers of linear forests and star forests.

In this paper, we investigate some asymptotic properties of the least squares estimator in nonlinear regression model with independent and identically distributed random errors under sub-linear expectations. The large deviation results for the estimator are established under some general conditions. As applications, the results on weak consistency and strong consistency are obtained under the meaning of capacity. A simulation study is also presented to verify the validity of the theoretical results.

This paper considers the existence of multiple normalized solutions of the following Schrödinger-Choquard equation \begin{align}\nonumber \Bigg\{ & -\Delta u=\lambda u+k(\varepsilon x)(I_{\alpha}\ast|u|^{q})|u|^{q-2}u+\mu(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, &x\in \mathbb{R}^N,\\ & \int_{\mathbb{R}^N}|u|^2 dx=c^2, &x\in \mathbb{R}^N, \end{align} where $c,\varepsilon,\mu>0,\ N\geq 3,\ \alpha\in(0,N),\ \frac{N+\alpha}{N}<q<1+\frac{\alpha+2}{N}<p\leq\frac{N+\alpha}{N-2},\ \lambda \in \mathbb{R}$ is a Lagrange multiplier which is unknown, $I_{\alpha}$ is the Riesz potential, $k: \mathbb{R}^N \rightarrow[0, \infty)$ is a continuous and positive function. When $\varepsilon$ is small enough, we prove that the numbers of normalized solutions are at least the numbers of global maximum points of $k$ by Ekeland's variational principle and truncated skill.

Let $w(z)$ be non-rational meromorphic solutions with hyper-order less than $1$ to a family of higher order nonlinear delay differential equations \begin{align*} w(z+1)w(z-1)+a(z)\frac{w^{(k)}(z)}{w(z)}=R(z,w(z)), \qquad k\in\mathbb{N^{+}}, \end{align*} where $a(z)$ is rational, $R(z,w(z))=\frac{P(z,w(z))}{Q(z,w(z))}$ is an irreducible rational function in $w$ with rational coefficients in $z$. This paper mainly show the relationships of the degree of $P(z,w(z))$ and $Q(z,w(z))$ when the above equations exist such solutions $w(z)$. There are also some examples to show that our results are sharp.

Airplane refueling problem is a nonlinear unconstrained optimization problem. Given a fleet of $n$ airplanes with mid-air refueling technique, the question is to find the best refueling policy to make the last remaining airplane travel the farthest. At first, we proposed the definition of sequential feasible solution. We proved that if an airplane refueling instance has feasible solutions, it must have sequential feasible solutions; and the optimal feasible solution must be the optimal sequential feasible solution. Then we proved that any airplane refueling instance has $n!$ feasible solutions, and has at most $2^{n-2}$ sequential feasible solutions. So we proposed the sequential search algorithm which aims to seek out all of the sequential feasible solutions and to search for the maximal sequential feasible solution by bubble sorting all of the sequential feasible solutions. We observed that the number of the sequential feasible solutions will change to grow at a polynomial rate when $n$ is greater than an inflection point $N$. Moreover, we built an efficient computability scheme, according to which we could forecast within a polynomial time the computational complexity of the sequential search algorithm that runs on any given airplane refueling instance.

Based on the existence of smooth solution of smooth initial date for non-uniformly parabolic equation, we obtain the existence of smooth solution of the equation ${u_t} = {\left({a{{\left(u_x \right)}}} \right)_x} + b\left({x,u} \right),\quad \left({x,t} \right) \in \mathbb{R} \times \left({0,T} \right)$ under general initial date $u_0 \in L^p_{\rm loc}(\mathbb{R})$, where $p>1$. In the case $a^{\prime}=\left(1+s^2\right)^{-m / 2}(m>1)$, the same result is established for $u_0 \in W_{\mathrm{lo}}^{1, p}(\mathbb{R})$ and $p \geq m-1, m>2$. We also show the existence of weak solution of this equation under conditions $u_0 \in W^{1,p}_{\rm loc}(\mathbb{R})$ and $p>1$. The method depends on the prior estimate of the smooth solution and its derivative which allow us to use the standard parabolic theory to prove the existence of solutions. Moreover, an example is established to show that there may not be a classical solution of this equation for $p\in (1,m-1)$ under the condition $u_0 \in W^{1,p}_{\rm loc}(\mathbb{R})$.

The delayed fear of prey towards predator and delayed gestation of predator usually affect the system dynamical behaviors. On the other hand, additional food supplement is a commen measure for people to preserve species from extinction and keep natural balance for continuous development in system control. Thus in this paper, we construct a predator-prey model with predator induced fear, additional food, Holling-II type functional response and time delays, and then generalize it to the random environment. Next, we establish some criteria of the asymptotical stability of equilibrium state, and stationary distribution for stochastic case. Considering the impacts of fear, additional food and time delays, we analyze the Hopf bifurcation of them. Finally by some numerical examples and graphical analysis, we intuitively validate such properties as stability, instability and Hopf bifurcation. Our findings reveal the significant impacts of fear, additional food and time delays on the system stability.

We propose an exact explicit closed-form Laguerre series expansion formula to compute the $q$-scale function of a spectrally negative Lévy process (SNLP), and other functions associated to the scale function for the first time. The proposed closed-form formula for the scale function has many applications in applied probability and in particular in the Lévy insurance risk theory. We shall show that the new series expansion formulas can be used to express the expected discounted penalty functions, the moments of the present value of total dividend payments as well as the time value of Parisian ruin in the Lévy risk models.

In this paper, we consider a model which is derived from a class of the $2$-dimensional Kolmogorov systems. Our purpose is to investigate the continuity of periodic solutions for this model in coefficient functions with respect to weak topologies. Finally, we provide an example as an application to Lotka-Volterra systems.

Let $G$ be a simple connected graph with vertex set $V(G)$. For $S \subseteq V(G)$, let $\pi_G(S)$ denote the maximum cardinality of internally disjoint $S$-paths in $G$. For an integer $k$ with $k\ge 2$, the $k$-path-connectivity $\pi_k(G)$ is defined as the minimum $\pi_G(S)$ over all $k$-subsets $S$ of $V(G)$. It is proved that deciding whether $\pi_G(S) \ge r$ is NP-complete problem [Graphs Combin. 37 (2021) 2521-2533]. The hypercube $Q_n$ is the famous Cayley graph, which is widely studied in the research of developing multiprocessor systems. The hierarchical cubic network $HCN_n$ is given in [IEEE TPDS 6 (1995) 427-435] which takes $Q_n$ as building clusters and emulates the desirable properties very efficiently. In this paper, we consider the $3$-path-connectivity of $HCN_n$ and prove that $\pi_3(HCN_n)=\lfloor \frac{3n+2}{4} \rfloor$ for $n \ge 2$ by constructing multiple internally disjoint $S$-paths. This result improves the $3$-tree-connectivity [Discrete Appl. Math. 322 (2022) 203-209] from trees to paths.

By means of the Whitham modulation theory, this paper studies the discontinuous initial value problem of the bad Jaulent-Miodek (JM) equation, which is compared with the good JM equation. The linear stability of the good and bad JM equations is analyzed to show the difference between the two equations. Also the physical significance of the JM equations is discussed by considering the reduction of the Euler's equation. Then the zero-, one-, two-phase periodic solutions and the corresponding Whitham equations in the framework of the bad JM equation are derived by finite-gap integration approach. The degeneration of the one-phase periodic solution along with the genus-one Whitham equation are analyzed by taking the two sides limits of the modulus $m$ of the Jacobi elliptic functions. The basic rarefaction wave patterns and dispersive shock wave patterns are proposed analytically and graphically, and then the complete classification of solutions and the all possible waveforms evolving from discontinuous initial values in the bad JM equation are established. In the results, various patterns of emergent nonlinear waves that appear to be new and are being detected for the first time

This paper concerns the existence and stability of solitary waves for the nonlinear Schrödinger equation with a partial confinement, which describes the limit case of the cigar-shaped model in Bose-Einstein condensate of dipolar quantum gases. More precisely, we applied a compactness argument, which comes from the confining potential $x_1^2+x_2^2$, to overcome the lack of compactness caused by the translation invariance with respect to $x_3$. Then, since the mass supercritical character of this equation, we construct orbitally stable solutions by adapting a suitable localized minimization problem. Finally, the stability of solitary waves is obtained.

In this article, using the method of Euler integral transforms, we obtain a new fundamental theorem for the Nikiforov-Uvarov-Suslov difference equation of hypergeometric type, which are essentially new results and their expressions are different from the Suslov Theorem. Meanwhile, we give two important examples to illustrate the applications of the new fundamental theorem. These indicate that the new fundamental theorem gives more general special functions which include the well-known Askey-Wilson polynomial as their particular case.

In this paper, we consider the free boundary value problem for a model of inviscid liquid-gas two-phase flow with cylindrical symmetry. For simplicity, we assume that the gas velocity is always equal to the liquid one and the gas and liquid are both connected continuously to the outer vacuum through the same free boundary. Furthermore, the free boundary is assumed to move in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. We construct two classes of global analytical solutions by using some ansatzs and show that the free boundary will spread outward linearly in time by using some new averaged quantities.

In this paper, we address exponential stabilization of transmission problem of the wave equation with dynamical boundary conditions. We consider waves passing from a medium in which the speed is $a_1$ into a medium in which the speed $a_2$ in the case of total internal reflection and show that such a system can be controlled by introducing both dynamical boundary control along the exterior boundary and distributed control near the transmission boundary. We obtain that the system is exponential stabilization without any restriction on $a_1$, $a_2$ and the transmission boundary.

Portmanteau tests have drawn much interest in economics and finance because of their strong relationship to model specification. The majority of current testing, however, concentrates on stationary time series. This article proposes an empirical likelihood-based portmanteau test for the autoregressive model, no matter if it is stationary, nearly integrated, or unit root, and with or without an intercept. It turns out that the final statistic is always asymptotically chi-squared distributed. A simulation study confirms the good finite sample performance of the proposed test before illustrating its practical merit in analyzing real data sets.

Sparse signal recovery is one of the key problems in the field of compressive sensing. Restricted isometry property (RIP) is an important metric for the recoverability of sparse signals. It can provide explicit and simple sufficient conditions for the convergences of many reconstruction methods such as orthogonal matching pursuit, basis pursuit (BP), and hard thresholding pursuit. However, RIP has several drawbacks. One drawback is that RIP can not be preserved for the scalar rescaling. In order to overcome this drawback, a new metric named combinatorial condition number is defined in this paper. It is invariant for the scalar rescaling. Subsequently, the Wielandt inequality and robust null space property are utilized to present a sufficient condition for the sparse signal recovery by BP in terms of the new metric.

In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some $0\leq \delta\leq 1 <L$ and $q=\frac {6(3-\delta)}{6-\delta}$, $\liminf _{R \rightarrow \infty} \frac{1}{R}\|(\mathbf{u}, \mathbf{h})\|_{R<|x|<L R}^{3-\delta}=0$. We also deduce sufficient conditions by allowing shrinking ratio $L=1+R^{-α}$. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions $\liminf$ rather than $\lim$.

This paper investigates the propagation dynamics of nonlocal dispersal cooperative systems within a shifting environment characterized by a contracting favorable region. We examine two distinct types of dispersal kernels. For thin-tailed kernels, we study the existence, uniqueness, and stability of forced waves using upper and lower solutions, the sliding method, and the dynamical systems approach. In the case of partially heavy-tailed kernels, considering compactly supported initial value functions, we demonstrate that for each species, the right side of the level sets exhibits accelerated rightward propagation, while transferability occurs. Conversely, the propagation on the left side does not move leftward but rather rightward, with a spreading speed equivalent to that of the shifting environment. Consequently, species with thin-tailed kernels inherently persist in a shifting habitat, provided they are part of a cooperative and irreducible system that includes at least one species with a heavy-tailed kernel, regardless of the magnitude of the shifting environment's speed. This behavior markedly diverges from the dynamics observed in scalar equations.

This paper mainly presents and studies some local and parallel finite element methods for the unsteady magnetohydrodynamic equations with low electromagnetic Reynolds number. Firstly, both the semi discrete and fully discrete local algorithms are provided and investigated, and the theoretical tool crucial to the analysis of the fully discrete local algorithm is obtained. Subsequently, we generalize the fully discrete local algorithm to the fully discrete parallel algorithm. At the end, some numerical experiments are provided to validate the effectiveness and efficiency of our algorithms.

This paper considers the customer's equilibrium joining-balking behavior in a retrial queue with two-phase service and priority-purchasing. If the arriving customer finds the server busy, the customer can make three choices (joining the orbit, purchasing the priority or balking) based on different levels of the system information. We consider two cases: the partially observable case and the fully unobservable case. The customer's equilibrium strategy for the partially observable case and the fully unobservable case are derived. Then, an algorithm is developed to solve the optimal fee that maximizes the service provider's profit. Finally, the influence of system parameters on performance measures, equilibrium joining strategies and the service provider's profits is illustrated by numerical examples.

In this paper, we study the existence of periodic solutions for Rayleigh type $p$-Laplacian equations $$ (\phi_{p}(x'))'+f(t, x')+g(x)=e(t). $$ By developing a continuation theorem, we prove that the given equation has at least one $T$-periodic solution provided that $G$ (a primitive function of $g$) satisfies $\liminf\limits_{x\to+\infty}\frac{pG(x)}{x^p}<(\frac{\pi_p}{T})^p$ and $f$ satisfies $p$-sublinear condition $\lim\limits_{|x|\to+\infty}\frac{f(t, x)}{\phi_p(x)}=0$.

In this paper, a nonmonotone smoothing Newton method is proposed for solving systems of nonlinear equalities and inequalities. By constructing a new smoothing function, the problem is approximated via a family of parameterized smooth equations. A smoothing Newton method is developed for solving the systems of nonlinear equalities and inequalities by adopting a modified nonmontone line search technique. And the global and local superlinear convergence of the algorithm are proved under mild assumptions. The preliminary numerical results are reported.