In this paper, we consider the $d$-boundedness of the Bergman metric and a vanishing theorem of ${L}^2$-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function, invariant function, holomorphic automorphism group and so on.

This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system \begin{equation*} \left\{\begin{array}{ll} \operatorname{div}\mathcal{A} (x,D {\bf u} )-\nabla\pi = \operatorname{div}{\bf F},& \mbox{in}\ \Omega, \\ \operatorname{div}{\bf u}= 0, & \mbox{in}\ \Omega. \end{array}\right. \end{equation*} In terms of the difference quotient method, our first result reveals that if ${\bf F}\in\mathbf{B}_{p,q,{\rm loc}}^{\beta}(\Omega,\mathbb{R}^{n})$ for $p=2$ and $1\leq q\leq\frac{2n}{n-2\beta}$, then such extra Besov regularity can transfer to the symmetric gradient $D{\bf u}$ and its pressure $\pi$ with no losses under a suitable fractional differentiability assumption on $x\mapsto\mathcal{A}(x,\xi)$. Furthermore, when the vector field $\mathcal{A}(x,D{\bf u})$ is simplified to the full gradient $\nabla{\bf u}$, we improve the aforementioned Besov regularity for all integrability exponents $p$ and $q$ by establishing a new Campanato-type decay estimates for $(\nabla{\bf u},\pi)$.

Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by $\mathfrak{B}_1$ and $\mathfrak{B}_2$, respectively. In this paper, a characterisation of groups in $\mathfrak{B}_1$ or in $\mathfrak{B}_2$ is given. By applying this new characterisation, some new properties of $\mathfrak{B}_1$ ($\mathfrak{B}_2$) and new proofs of many known results about $\mathfrak{B}_1$ or $\mathfrak{B}_2$ are obtained. Further, closure properties of $\mathfrak{B}_1$ and $\mathfrak{B}_2$ are discussed.

We prove the following properties: (1) Let $a\in \Lambda_{1,0,k,k'}^{m_0}({\mathbb R}^{n}\times {\mathbb R}^{n})$ with $m_0=-1|\frac {1} {p}-\frac {1} {2}|(n-1),\ n\geq 2\, (1< p \leq 2,\ k> \frac {n} {p},\ k'> 0;\ 2\le p\le \infty,\ k> \frac {n} {2},\ k'> 0$ respectively). Suppose the phase function $S$ is positively homogeneous in $\xi$-variables, non-degenerate and satisfies certain conditions. Then the Fourier integral operator $T$ is $L^p$-bounded. Applying the method of (1), we can obtain the $L^p$-boundedness of the Fourier integral operator if (2) the symbol $a \in \Lambda_{1,δ,k,k'}^{m_0},\ 0\le δ < 1$, with $m_{0},\, k,\, k'$ and $S$ given as in (1). Also, the method of (1) gives: (3) $a\in \Lambda_{1,δ ,k,k'}^{0},\ 0\leq δ < 1$ and $k,\, k'$ given as in (1), then the $L^{p}$-boundedness of the pseudo-differential operators holds, $1

We study additional non-isospectral symmetries of multicomponent constrained $N=2$ supersymmetric Kadomtsev—Petviashvili (KP) hierarchies. These symmetries are shown to form an infinite-dimensional non-Abelian superloop superalgebra.

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get an upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.

In this paper, we show that the Boussinesq operator $\mathcal{B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{R})$ as $t\to 0$ provided $s\geq\frac{1}{4}$--more precisely--on one hand, by constructing a counterexample in $\mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=\frac14$; on the other hand, we find that the Hausdorff dimension of the divergence set for $\mathcal{B}_tf$ is ${\alpha _1},\beta \left( s \right) = \left\{ {\begin{array}{*{20}{c}} {1 -2s,}\\ {1,} \end{array}\begin{array}{*{20}{c}} {as\frac{1}{4} \le s \le \frac{1}{2};}\\ {as0 < s < \frac{1}{4}.} \end{array}} \right.$ Moreover, a higher dimensional lift was also obtained for $f$ being radial.

In this paper, we consider Ramanujan's sums over arbitrary Dedekind domain with finite norm property. We define the Ramanujan's sums $\eta(a,A)$ and $\eta(B,A)$, where $a$ is an arbitrary element in a Dedekind domain, $B$ is an ideal and $A$ is a non-zero ideal. In particular, we discuss the Kluyver formula and Hölder formula for $\eta(a,A)$ and $\eta(B,A)$. We also prove the reciprocity formula enjoyed by $\eta(B,A)$ and the orthogonality relations for $\eta(a,A)$ in the last two parts.

Following the previous work, we shall study some inverse problems for the Dirac operator on an equilateral star graph. It is proven that the so-called Weyl function uniquely determines the potentials. Furthermore, we pay attention to the inverse problem of recovering the potentials from the spectral data, which consists of the eigenvalues and weight matrices, and present a constructive algorithm. The basic tool in this paper is the method of spectral mappings developed by Yurko.

In this article, we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the $p$-Laplacian equation $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=g(x,t)+f(t)$, where $g(x,t)$ and $f(t)$ are quasi-periodic in $t$ with Diophantine frequency. A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.

In this paper, we consider the 3-D Cauchy problem for the diffusion approximation model in radiation hydrodynamics. The existence and uniqueness of global solutions is proved in perturbation framework, for more general gases including ideal polytropic gas. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity, temperature, and radiation field.

We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients $M=\Gamma\backslash G$ of a simply-connected solvable Lie group $G$ by a lattice $\Gamma$, admitting a symplectic structure.

A Hom-group is the non-associative generalization of a group whose associativity and unitality are twisted by a compatible bijective map. In this paper, we give some new examples of Hom-groups, and show the first, second and third isomorphism theorems of Hom-groups. We also introduce the notion of Hom-group action, and as an application, we prove the first Sylow theorem for Hom-groups along the line of group actions.

In this article, we provide some sufficient conditions for the dynamical systems with the eventual shadowing property to have positive topological entropy and several equivalent conditions for the dynamical systems with the eventual shadowing property to be mixing.

We first study the Volterra operator $V$ acting on spaces of Dirichlet series. We prove that $V$ is bounded on the Hardy space $h^p_0$ for any $0< p \leq \infty$, and is compact on $h^p_0$ for $1< p \leq \infty$. Furthermore, we show that $V$ is cyclic but not supercyclic on $h^p_0$ for any $0 < p < \infty$. Corresponding results are also given for $V$ acting on Bergman spaces $h^p_{w,0}$. We then study the Volterra type integration operators $T_g$. We prove that if $T_g$ is bounded on the Hardy space $h^p$, then it is bounded on the Bergman space $h^p_w$.

In this paper, we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that ${\rm Sic}:={\rm Ric}-\alpha\nabla\phi\otimes\nabla\phi$ is bounded from below.

This paper deals with the following prescribed boundary mean curvature problem in $\mathbb{B}^N$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=0,u>0,~~& y\in\mathbb{B}^N,\\ \dfrac{\partial u}{\partial \nu}+\dfrac{N-2}{2}u=\dfrac{N-2}{2}\tilde{K}(y)u^{2^\sharp-1}, ~~&y\in\mathbb{S}^{N-1}, \end{array} \right. \end{equation*} where $\tilde{K}(y)=\tilde{K}(|y'|,\tilde{y})$ is a bounded nonnegative function with $y=(y',\tilde{y})\in \mathbb{R}^2\times\mathbb{R}^{N-3}$, $2^\sharp=\frac{2(N-1)}{N-2}$. Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if $N\geq 5$ and $\tilde{K}(r,\tilde{y})$ has a stable critical point $(r_0,\tilde{y}_0)$ with $r_0>0$ and $\tilde{K}(r_0,\tilde{y}_0)>0$, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of $\tilde{K}(r,\tilde{y})$.

Let $\mathcal{F}=\{H_1, \ldots, H_k\}$ ($k\ge 1$) be a family of graphs. The Turán number of the family $\mathcal{F}$ is the maximum number of edges in an $n$-vertex $\{H_1, \ldots, H_k\}$-free graph, denoted by ex$(n, \mathcal{F})$ or ex$(n, \{H_1,H_2,\ldots,H_k\})$. The blow-up of a graph $H$ is the graph obtained from $H$ by replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. In this paper we determine the Turán number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Turán number of the family consisting of a cycle, a star and linear forests with $k$ edges.

In the setting of Fock–Sobolev spaces of positive orders over the complex plane, Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial, then the other must also be radial. In this paper, we extend this result to the Fock–Sobolev space of negative order using the Fock-type space with a confluent hypergeometric function.

In this paper, we consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators, \begin{eqnarray*} \begin{cases} \dot x=y+\epsilon f_1(\omega t,x,y),\\ \dot y=\lambda x^{l}+\epsilon f_2(\omega t,x,y), \end{cases} \end{eqnarray*} where $0\neq\lambda\in\mathbb{R}$, $l>1$ is an integer and the corresponding involution $G$ is $(-\theta,x,-y)\rightarrow (\theta,x,y)$. The existence of response solutions of the above reversible systems has already been proved in [22] if $[f_2(\omega t,0,0)]$ satisfies some non-zero average conditions (See the condition (H) in [22]), here $[\ \cdot\ ]$ denotes the average of a continuous function on $\mathbb{T}^d$. However, discussing the existence of response solutions for the above systems encounters difficulties when $[f_2(\omega t,0,0)]=0,$ due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where $[f_2(\omega t,0,0)]=0$. More precisely, with $2p < l$, if $f_2$ satisfies $[f_2(\omega t,0,0)]=[\frac{\partial f_2(\omega t,0,0)}{\partial x}]=[\frac{\partial^2 f_2(\omega t,0,0)}{\partial x^2}]=\cdots=[\frac{\partial^{p-1} f_2(\omega t,0,0)}{\partial x^{p-1}}]=0$, either $\lambda^{-1}[\frac{\partial^{p} f_2(\omega t,0,0)}{\partial x^{p}}]$ $< 0$ as $l-p$ is even or $\lambda^{-1}[\frac{\partial^{p} f_2(\omega t,0,0)}{\partial x^{p}}]\neq 0$ as $l-p$ is odd, we obtain the following results: (1) For $\tilde\lambda>0$ (see $\tilde\lambda$ in (2.2)) and $\epsilon$ sufficiently small, response solutions exist for each $\omega$ satisfying a weak non-resonant condition; (2) For $\tilde\lambda<0$ and $\epsilon_*$ sufficiently small, there exists a Cantor set $\mathcal{E}\in(0,\epsilon_*)$ with almost full Lebesgue measure such that response solutions exist for each $\epsilon\in\mathcal{E}$ if $\omega$ satisfies a Diophantine condition. In the remaining case where $\lambda^{-1}[\frac{\partial^{p} f_2(\omega t,0,0)}{\partial x^{p}}]> 0$ and $l-p$ is even, we prove the system admits no response solutions in most regions.

In applications involving, e.g., panel data, images, genomics microarrays, etc., trace regression models are useful tools. To address the high-dimensional issue of these applications, it is common to assume some sparsity property. For the case of the parameter matrix being simultaneously low rank and elements-wise sparse, we estimate the parameter matrix through the least-squares approach with the composite penalty combining the nuclear norm and the l¹ norm. We extend the existing analysis of the low-rank trace regression with i.i.d. errors to exponential β-mixing errors. The explicit convergence rate and the asymptotic properties of the proposed estimator are established. Simulations, as well as a real data application, are also carried out for illustration.

Let $\lambda=(\lambda_1, \ldots,\lambda_n)$ be $\beta$-Jacobi ensembles with parameters $p_1, p_2, n$ and $\beta$ while $\beta$ varying with $n.$ Set $\gamma=\lim_{n\to\infty}\frac{n}{p_1}$ and $\sigma=\lim_{n\to\infty}\frac{p_1}{p_2}.$ In this paper, supposing $\lim_{n\to\infty}\frac{\log n}{\beta n}=0,$ we prove that the empirical measures of different scaled $\lambda$ converge weakly to a Wachter distribution, a Marchenko—Pastur law and a semicircle law corresponding to $\sigma\gamma>0, \sigma=0$ or $\gamma=0,$ respectively. We also offer a full large deviation principle with speed $\beta n^2$ and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of $\beta$-Jacobi ensembles is obtained.

Multiple change-points estimation for functional time series is studied in this paper. The change-point problem is first transformed into a high-dimensional sparse estimation problem via basis functions. Group least absolute shrinkage and selection operator (LASSO) is then applied to estimate the number and the locations of possible change points. However, the group LASSO (GLASSO) always overestimate the true points. To circumvent this problem, a further Information Criterion (IC) is applied to eliminate the redundant estimated points. It is shown that the proposed two-step procedure estimates the number and the locations of the change-points consistently. Simulations and two temperature data examples are also provided to illustrate the finite sample performance of the proposed method.

We consider a Dirichlet nonlinear equation driven by the (p, 2)-Laplacian and with a reaction having the competing effects of a parametric asymmetric superlinear term and a resonant perturbation. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information.

Recently the quantum toroidal superalgebra ${\mathcal{E}}_{m|n}$ associated with ${\mathfrak{gl}}_{m|n}$ was introduced by L. Bezerra and E. Mukhin, which is not a quantum Kac-Moody algebra. The quantum toroidal superalgebra ${\mathcal{E}}_{m|n}$ exploits infinite sequences of generators and relations of the form, which are called Drinfeld realization. In this paper, we use only finite set of generators and relations to define an associative algebra ${\mathcal{E}}^\prime_{m|n}$ and show that there exists an epimorphism from ${\mathcal{E}}^\prime_{m|n}$ to the quantum toroidal superalgebra ${\mathcal{E}}_{m|n}$. In particular, the structure of ${\mathcal{E}}^\prime_{m|n}$ enjoys some properties like Drinfeld-Jimbo realization.

In this paper, we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3) L-functions for all k ≥ 1, based on our recent work on the twisted first moment of central values in this family of L-functions.

In this paper, we establish an improved Hardy-Littlewood-Sobolev inequality on $\mathbb S^{n}$ under higher-order moments constraint. Moreover, by constructing precise test functions, using improved Hardy-Littlewood-Sobolev inequality on $\mathbb S^{n}$, we show such inequality is almost optimal in critical case. As an application, we give a simpler proof of the existence of the maximizer for conformal Hardy-Littlewood-Sobolev inequality.

In this paper, we consider the following quadratic pencil of Schr$\ddot{\text{o}}$dinger operators $L(\lambda )$ generated in $L^{2}(\mathbb{R}^{+})$ by the equation \begin{equation*} -y^{\prime \prime} + [p(x) +2 \lambda q(x) ] y=\lambda^{2}y, \quad x \in \mathbb{R}^{+}=[0, +\infty ) \end{equation*} with the boundary condition \begin{equation*} \frac{y^{\prime}(0)}{y(0)}=\frac{\beta_{1} \lambda+\beta_{0}}{\alpha_{1} \lambda+\alpha_{0}}, \end{equation*} where $p(x)$ and $q(x)$ are complex valued functions and $\alpha_{0}$, $\alpha_{1}$, $\beta_{0}$, $\beta_{1}$ are complex numbers with $\alpha_{0}\beta_{1}-\alpha_{1}\beta_{0} \neq 0$. It is proved that $L(\lambda )$ has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions \begin{equation*} p(x ),q^{\prime}(x ) \in \mathrm{AC}(\mathbb{R^{+}} ),\quad \lim_{x \rightarrow \infty}[ |p(x)|+|q(x)|+|q^{\prime}(x)|] =0 \end{equation*} and \begin{equation*} \sup_{0\leq x< +\infty}\{{\rm e}^{\varepsilon \sqrt{x}}[ |p^{\prime}(x )|+ |q^{\prime \prime}(x)|] \}<+\infty \end{equation*} hold, where $\varepsilon >0.$

In this paper we study the existence of nontrivial solutions to the well-known Brezis-Nirenberg problem involving the fractional $p$-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue $\lambda_{p,s}(\widehat{\omega_{\delta}})$ with respect to the domain $\widehat{\omega_{\delta}}$. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi-Perera-Squassina-Yang [The Brezis-Nirenberg problem for the fractional $p$-Laplacian. Calc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis-Nirenberg type results of Ramos-Wang-Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192-199] to the fractional $p$-Laplacian setting.

We study the following Schrödinger equation with variable exponent \begin{equation*} -\Delta u+u=u^{p+\epsilon a(x)},\;\; u>0\;\;\text{in}\;\mathbb{R}^N, \end{equation*} where $\epsilon>0$, $1 < p < \frac{N+2}{N-2}$, $a(x)\in C^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, $N\geq3$. Under certain assumptions on a vector field related to $a(x)$, we use the Lyapunov-Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^\alpha=f.$$ It is shown that when $5/3 \leq\alpha < 7/3$, the $\frac{3\alpha-5}{\alpha-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal., 51, 228-255 (2019)] for $\alpha=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes system.

A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycles in $G$. A graph $G$ is $acyclically$ $k$-$choosable$ if for any list assignment $L=\{L(v):v\in V(G)\}$ with $|L(v)|\geq k$ for each vertex $v\in V(G)$, there exists an acyclic proper vertex coloring $\phi$ of $G$ such that $\phi(v)\in L(v)$ for each vertex $v\in V(G)$. In this paper, we prove that every graph $G$ embedded on the surface with Euler characteristic number $\varepsilon=-1$ is acyclically $11$-choosable.

The Hartogs domain over homogeneous Siegel domain $D_{N,s}\ (s>0)$ is defined by the inequality $\|\zeta\|^2 < K_D(z,z)^{-s}$, where $D$ is a homogeneous Siegel domain of type II, $(z,\zeta)\in D\times\mathbb{C}^N$ and $K_D(z,z)$ is the Bergman kernel of $D$. Recently, Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains $D_{N,s}$ and $D'_{N',s'}$ are biholomorphisms for $N\geq 2$. In this article, we find a counter-example to show that the rigidity result is not true for $D_{1,s}$ and obtain a classification of proper holomorphic mappings between $D_{1,s}$ and $D'_{1,s'}$.

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ and $H$ are finite $p$-groups given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^m}\longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $m\geq 1$. Then $V(\mathbb{F}_pG) \cong V(\mathbb{F}_pH)$ if and only if $G\cong H$. Balogh and Bovdi only solved the isomorphism problem when $p$ is odd. In this paper, the case $p=2$ is determined.

In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on $\mathbb R^{\infty}$ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.

A multiplicative function $f$ is said to be resembling the Möbius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and Ω-results for the summatory function $\sum_{n\leq x} f(n)$ for a class of these $f$, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{1/3+ε})$ under the Riemann Hypothesis. On the other hand it is proved to be $\Omega(x^{1/4})$ unconditionally. It is interesting to compare these with the corresponding results for the Möbius function.

This paper is devoted to the conditions of the existence of CC-center for the generalized Abel equations. Using some new original methods, we obtain extended results of the main theorems in the paper by Llibre and Valls (2020) and the one by Zhou (2020), respectively. The proofs in this paper are much simpler than the previous ones.

Using Hartogs' fundamental theorem for analytic functions in several complex variables, we establish a multiple $q$-exponential differential operational identity for the analytic functions in several variables, which can be regarded as a multiple $q$-translation formula. This multiple $q$-translation formula is a fundamental result and play a pivotal role in $q$-mathematics. Using this $q$-translation formula, we can easily recover many classical conclusions in $q$-mathematics and derive some new $q$-formulas. Our work reveals some profound connections between the theory of complex functions in several variables and $q$-mathematics.

Faltings heights over function fields of complex projective curves are modular invariants of families of curves. The question on minimized Faltings heights was raised by Mazur. In this note, we consider this question for a simple class of families of hyperelliptic curves. We obtain a complete result of this question in this case.

This paper deals with a class of inertial gradient projection methods for solving a variational inequality problem involving pseudomonotone and non-Lipschitz mappings in Hilbert spaces. The proposed algorithm incorporates inertial techniques and the projection and contraction method. The weak convergence is proved without the condition of the Lipschitz continuity of the mappings. Meanwhile, the linear convergence of the algorithm is established under strong pseudomonotonicity and Lipschitz continuity assumptions. The main results obtained in this paper extend and improve some related works in the literature.