数学物理学报(英文版)

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BOUNDEDNESS OF FORELLI-RUDIN TYPE OPERATORS ON TUBE DOMAINS OVER THE FORWARD LIGHT CONES

Jiaxin LIU, Guantie DENG, Zhiqiang GAO

数学物理学报(英文版). 2025 (4): 1235-1246. DOI: 10.1007/s10473-025-0401-6

摘要 ( 22 )

We explore some necessary and sufficient conditions for the boundedness of the Forelli-Rudin type operator $T$ on the weighted Lebesgue space associated with tubular domains over the forward light cone. Our approach involves conducting precise computations for a series of complex integrals to identify appropriate test functions, and through a detailed analysis of these test functions, we derive the boundedness properties of the operator $T$. This work is significant in the study of the Bergman projection operators.

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A POSITIVE SOLUTION FOR QUASILINEAR SCHRÖODINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

Lanxin HUANG, Jiabao SU

数学物理学报(英文版). 2025 (4): 1247-1264. DOI: 10.1007/s10473-025-0402-5

摘要 ( 29 )

In this paper, we study the quasilinear Schrödinger-Poisson system with critical Sobolev exponent $$\begin{aligned} \begin{cases} -\Delta_{p} u+|u|^{p-2}u+l(x)\phi |u|^{p-2}u=|u|^{p^{*}-2}u+\mu h(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=l(x)|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{aligned}$$ where $\mu >0$, $\frac{3}{2}<p<3$, $p\leqslant q<p^{*}=\frac{3p}{3-p}$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. Under certain assumptions on the functions $l$ and $h$, we employ the mountain pass theorem to establish the existence of positive solutions for this system.

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ON THE CONCENTRATION OF STANDING WAVES FOR NLS EQUATION WITH POINT-DIPOLE POTENTIAL

Jun WANG, Xiaoguang LI

数学物理学报(英文版). 2025 (4): 1265-1283. DOI: 10.1007/s10473-025-0403-4

摘要 ( 10 )

We study the following minimization problem: $$d_{p}(M_{p}):=\inf\{E_{p}(u): \|u\|_{L^{2}}=M_{p}\},$$ where the Gross-Pitaevskii energy functional $$E_{p}(u)=\int_{\mathbb{R}^{N}}|\nabla u|^{2}-c\frac{|u|^{2}}{|x|^{2}}+V(x)|u|^{2}{\rm d}x-\frac{2}{p+2}\int_{\mathbb{R}^{N}}|u|^{p+2}{\rm d}x.$$ When $p=p^{*}:=\frac{4}{N}$, the precise concentration behavior of minimizers is analyzed as $M_{p^{*}}\nearrow \|Q_{p^{*}}\|_{L^{2}}$, where $Q_{p^{*}}$ is the unique radially positive solution of $-\Delta \varphi-c\frac{\varphi}{|x|^{2}}-|\varphi|^{p^{*}+1}\varphi=0$. When $0<p<p^{*}$, we prove that all minimizers must blow up if $\lim\limits_{p\to p^{*}}M_{p}\geq \|Q_{p^{*}}\|_{L^{2}}$. On this argument, the detailed concentration behavior of minimizers is established as $p\nearrow p^{*}$.

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DIFFUSION ASYMPTOTICS OF A STEADY COUPLED MODEL FOR RADIATIVE TRANSFER IN A UNIT BALL

Lei LI, Zhengce ZHANG

数学物理学报(英文版). 2025 (4): 1284-1306. DOI: 10.1007/s10473-025-0404-3

摘要 ( 11 )

We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit ball in $\mathbb{R}^{3}$ with one-speed velocity. The model consists of a steady kinetic equation satisfied by the specific intensity of radiation coupled with a nonhomogeneous elliptic equation satisfied by the material temperature. For the $O(\epsilon)$ boundary data of the intensity of the radiation and the suitable small boundary data of the temperature, we prove the existence, uniqueness and the nonequilibrium diffusion limit of solutions to the boundary value problem for the coupled model.

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THE FREE INTERFACE PROBLEM OF PLASMA-VACUUM WITH SURFACE TENSION IN A TUBE DOMAIN

Biran ZHANG

数学物理学报(英文版). 2025 (4): 1307-1342. DOI: 10.1007/s10473-025-0405-2

摘要 ( 12 )

In this paper, we consider the plasma-vacuum interface problem in a cylindrical tube region impressed by a special background magnetic field. The interior region is occupied with plasma, which is governed by the incompressible inviscid and resistive MHD system without damping term. The exterior vacuum region is governed by the so-called the ``pre-Maxwell equations". And on the free interface, additionally, the effect of surface tension is taken into account. The original region can be transformed into a horizontally periodic slab through the cylindrical coordinate transformation, which will be impressed by a uniform non-horizontal magnetic field. Appending with the appropriate physical boundary conditions, the global well-posedness of the problem is established by the energy method.

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ON BLOW-UP TO THE ONE-DIMENSIONAL NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITY AND VACUUM

Yue CAO, Yachun LI, Shaojun YU

数学物理学报(英文版). 2025 (4): 1343-1354. DOI: 10.1007/s10473-025-0406-1

摘要 ( 12 )

In this paper, we consider the Cauchy problem of the isentropic compressible Navier-Stokes equations with degenerate viscosity and vacuum in $\mathbb{R}$, where the viscosity depends on the density in a super-linear power law (i.e., $\mu(\rho)=\rho^\delta, \delta>1$). We first obtain the local existence of the regular solution, then show that the regular solution will blow up in finite time if initial data have an isolated mass group, no matter how small and smooth the initial data are. It is worth mentioning that based on the transport structure of some intrinsic variables, we obtain the $L^\infty$ bound of the density, which helps to remove the restriction $\delta\leq \gamma$ in Li-Pan-Zhu [21] and Huang-Wang-Zhu [13].

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ASYMPTOTIC STABILIZATION IN A TWO-DIMENSIONAL SINGULAR CHEMOTAXIS-NAVIER-STOKES SYSTEM WITH INDIRECT SIGNAL CONSUMPTION

Feng DAI, Bin LIU

数学物理学报(英文版). 2025 (4): 1355-1383. DOI: 10.1007/s10473-025-0407-0

摘要 ( 12 )

This paper deals with the singular chemotaxis-Navier-Stokes system with indirect signal consumption $n_{t}+u\cdot\nabla n=\Delta n-\chi\nabla\cdot(\frac{n}{v}\nabla v)$; $v_{t}+u\cdot\nabla v=\Delta v-vw$; $w_{t}+u\cdot\nabla w=\Delta w-w+n$; $u_t+(u\cdot\nabla)u=\Delta u-\nabla P+n\nabla\Phi$; $\nabla\cdot u=0$, $x\in \Omega$, $t>0$ in a bounded and smooth domain $\Omega\subset\mathbb{R}^2$ with no-flux/no-flux/no-flux/no-slip boundary conditions, where $\Phi\in W^{2,\infty}(\Omega)$. A recent literature [Dai F, Liu B. J Differential Equations, 2023, 369: 115--155] has proved that for all reasonably regular initial data, the associated initial-boundary value problem possesses a global classical solution, but qualitative information on the behavior of solution has never been touched so far. In stark contrast to the positive effect of indirect signal consumption mechanism on the global solvability of system, the analysis of asymptotic behavior of solution to the system with indirect signal consumption is essentially complicated than that with direct signal consumption because the favorable coupled structure between cells and signal is broken down by the indirect signal consumption mechanism. The present study shows that the global classical solution exponentially stabilizes toward the corresponding spatially homogeneous equilibria under a smallness condition on the initial cell mass. In comparison to the previously known result concerning the uniform convergence of solution to the system with direct signal consumption, our result inter alia provides a more in-depth understanding on the asymptotic behavior of solution.

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A DECREASING PROPERTY OF THE 3D MAGNETO-HYDRODYNAMIC FLOWS ON A TORUS

Zhaoxia LIU

数学物理学报(英文版). 2025 (4): 1384-1390. DOI: 10.1007/s10473-025-0408-z

摘要 ( 13 )

Let $(u, B)$ be a strong solution of the magneto-hydrodynamic system on three dimensional torus $\mathbb{T}^3$. In this note, using the properties of the curl operator, we show that $\|(\nabla\times(u-B), \nabla\times(u+B))(\cdot, t)\|_{L^1}+\frac{1}{2\nu}\|(u-B, u+B)(\cdot, t)\|^2_{L^2}$ is decreasing in time $t$ as long as the solution $(u, B)(\cdot,t)$ exists, where $\nabla\times w$ means the curl of the vector function $w$, and $\nu>0$ is the viscosity coefficient.

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GLOBAL STABILITY OF TRAVELING WAVES FOR SOME MULTIDIMENSIONAL SEMILINEAR HYPERBOLIC SYSTEMS

Dongbing ZHA, Yue ZHAO

数学物理学报(英文版). 2025 (4): 1391-1404. DOI: 10.1007/s10473-025-0409-y

摘要 ( 11 )

For multidimensional _rst order semilinear hyperbolic systems of diagonal form without self-interaction, we show the global nonlinear stability of traveling wave solutions.

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NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

Haolin WANG, Shoufu TIAN, Tiantian ZHANG

数学物理学报(英文版). 2025 (4): 1405-1437. DOI: 10.1007/s10473-025-0410-5

摘要 ( 11 )

In this work, we study wave state transitions of the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (2KdVSKR) equation by analyzing the characteristic line and phase shift. By converting the wave parameters of the N-soliton solution into complex numbers, the breath wave solution is constructed. The lump wave solution is derived through the long wave limit method. Then, by choosing appropriate parameter values, we acquire a number of transformed nonlinear waves whose gradient relation is discussed according to the ratio of the wave parameters. Furthermore, we reveal transition mechanisms of the waves by exploring the nonlinear superposition of the solitary and periodic wave components. Subsequently, locality, oscillation properties and evolutionary phenomenon of the transformed waves are presented. And we also prove the change in the geometrical properties of the characteristic lines leads to the phenomena of wave evolution. Finally, for higher-order waves, a range of interaction models are depicted along with their evolutionary phenomena. And we demonstrate that their diversity is due to the fact that the solitary and periodic wave components produce different phase shifts caused by time evolution and collisions.

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WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION

Siyan GUO, Jiangbo HAN, Runzhang XU

数学物理学报(英文版). 2025 (4): 1438-1470. DOI: 10.1007/s10473-025-0411-4

摘要 ( 12 )

In this paper, we consider the initial boundary value problem for the 2-D hyperbolic viscous Cahn-Hilliard equation. Firstly, we prove the existence and uniqueness of the local solution by the Galerkin method and contraction mapping principle. Then, using the potential well theory, we study the global well-posedness of the solution with initial data at different levels of initial energy, i.e., subcritical initial energy, critical initial energy and arbitrary positive initial energy. For subcritical initial energy, we prove the global existence, asymptotic behavior and finite time blowup of the solution. Moreover, we extend these results to the critical initial energy using the scaling technique. For arbitrary positive initial energy, including the sup-critical initial energy, we obtain the sufficient conditions for finite time blow-up of the solution. As a further study for estimating the blowup time, we give a unified expression of the lower bound of blowup time for all three initial energy levels and estimate the upper bound of blowup time for subcritical and critical initial energy.

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EXISTENCE RESULT FOR FRACTIONAL STATE-DEPENDENT SWEEPING PROCESSES

Shengda ZENG, Abderrahim BOUACH, Tahar HADDAD

数学物理学报(英文版). 2025 (4): 1471-1481. DOI: 10.1007/s10473-025-0412-3

摘要 ( 10 )

This paper addresses the evolution problem governed by the fractional sweeping process with prox-regular nonconvex constraints. The values of the moving set are time and state-dependent. The aim is to illustrate how a fixed point method can establish an existence theorem for this fractional nonlinear evolution problem. By combining Schauder's fixed point theorem with a well-posedness theorem when the set $ C $ is independent of the state $u$ (i.e. $C := C(t)$, as presented in [22, 23]), we prove the existence of a solution to our quasi-variational fractional sweeping process in infinite-dimensional Hilbert spaces. Similar to the conventional state-dependent sweeping process, achieving this result requires a condition on the size of the Lipschitz constant of the moving set relative to the state.

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PROOFS OF CONJECTURES ON RAMANUJAN-TYPE SERIES OF LEVEL 3

John M. CAMPBELL

数学物理学报(英文版). 2025 (4): 1482-1496. DOI: 10.1007/s10473-025-0413-2

摘要 ( 9 )

The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging, out of all possible levels for Ramanujan-type series. This motivates the development of new techniques for constructing Ramanujan-type series of level 3. Chan and Liaw introduced an alternating analogue of the Borwein brothers' identity for Ramanujan-type series of level 3; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan--Liaw identity, via the use of Ramanujan's class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, via a limiting case of the Kummer--Goursat transformation, a new identity for evaluating the summands for alternating Ramanujan-type series of level 3, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.

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BERGMAN PROJECTION AND TOEPLITZ OPERATORS ON WEIGHTED HARMONIC BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS

Yongjiang DUAN, Sawlet JUNIS, Na ZHAN

数学物理学报(英文版). 2025 (4): 1497-1513. DOI: 10.1007/s10473-025-0414-1

摘要 ( 12 )

In this paper, it is shown that the harmonic Bergman projection $P^h_{\omega}$, induced by a radial weight $\omega,$ is bounded and onto from $L^{\infty}(\mathbb{D})$ to the harmonic Bloch space $\mathcal{B}_{h}$ if and only if $\omega\in \mathcal{D}$, which is a class of radial weights satisfying the two-sided doubling conditions. As an application, the bounded and compact positive Toeplitz operators $T^h_{\mu,\omega}$ on the endpoint case weighted harmonic Bergman space $L^1_{h,\omega}(\mathbb{D})$ are characterized.

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MEROMORPHIC SOLUTIONS OF DELAY SCHWARZIAN DIFFERENTIAL EQUATIONS

Xiaotang NIE, Jiaxing HUANG, Yuefei WANG, Chengfa WU

数学物理学报(英文版). 2025 (4): 1514-1528. DOI: 10.1007/s10473-025-0415-0

摘要 ( 10 )

In this paper, we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients $$f(z+1)-f(z-1)+a(z)S(f,z)=\frac{P(z, f(z))}{Q(z, f(z))}.$$ We obtain the necessary conditions on the degree of $R(z,f)$ for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of $Q(z, f)$ on $f$. Finally, we provide some examples to illustrate that all cases occur.

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BOUNDED, COMPACT AND SCHATTEN CLASSES HANKEL OPERATORS ON WEIGHTED FOCK SPACES

Chunxu XU

数学物理学报(英文版). 2025 (4): 1529-1554. DOI: 10.1007/s10473-025-0416-z

摘要 ( 10 )

Let $\phi$ be a smooth radial weight that decays faster than the class Gaussian ones. We obtain certain estimates for the reproducing kernels and the $L^p$-estimates for solutions of the $\overline{\partial}$-equation on the weighted Fock spaces $F_{\phi}^p~(1\leq p\leq\infty)$, which extends the classical Hörmander Theorem. Furthermore, for a suitable $f$, we completely characterize the boundedness and compactness of the Hankel operator $H_f:F_{\phi}^p\rightarrow L^q(\mathbb{C},{\rm e}^{-q\phi(\cdot)}{\rm d}m)$ for all possible $1\leq p,q<\infty$ and also characterize the Schatten-$p$ class Hankel operator $H_f$ from $F_{\phi}^2$ to $L^2(\mathbb{C},{\rm e}^{-2\phi}{\rm d}m)$ for all $0<p<\infty$. As an application, we give a complete characterization of the simultaneously bounded, compact and Schatten-$p$ classes Hankel operators $H_f$ and $H_{\overline{f}}$ on $F_\phi^2$.

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ESTIMATES OF ALL TERMS OF HOMOGENEOUS POLYNOMIAL EXPANSIONS FOR THE SUBCLASSES OF G-PARAMETRIC STARLIKE MAPPINGS OF COMPLEX ORDER IN SEVERAL COMPLEX VARIABLES

Liangpeng XIONG, Qingchao WANG, Xiaoying SIMA

数学物理学报(英文版). 2025 (4): 1555-1566. DOI: 10.1007/s10473-025-0417-y

摘要 ( 11 )

In this paper, the class of starlike functions of complex order $\gamma\, (\gamma\in \mathbb{C}-\{0\})$ is extended from the case on unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ to the case on the unit ball $B$ in a complex Banach space or the unit polydisk $\mathbb{U}^n$ in $\mathbb{C}^n$. Let $g$ be a convex function in $\mathbb{U}$. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of $g$-parametric starlike mappings of complex order $\gamma$ on $B$ (resp. $\mathbb{U}^n$) when the mappings $f$ are $k$-fold symmetric, $k\in \mathbb{N}.$ Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.

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EXPONENTIAL STABILITY OF ATTRACTOR FOR RANDOM REACTION-DIFFUSION EQUATION DRIVEN BY NONLINEAR COLORED NOISE ON ℝ^N

Huijuan ZHU, Xiaojun LI, Yanjiao LI

数学物理学报(英文版). 2025 (4): 1567-1596. DOI: 10.1007/s10473-025-0418-x

摘要 ( 10 )

In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined on $\mathbb{R}^n$. The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on $\mathbb{R}^n$.

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GÂTEAUX DIRECTIONAL DIFFERENTIABILITY OF THE GENERALIZED METRIC PROJECTION IN BANACH SPACES

Jinlu LI

数学物理学报(英文版). 2025 (4): 1597-1618. DOI: 10.1007/s10473-025-0419-9

摘要 ( 14 )

Let $X$ be a real uniformly convex and uniformly smooth Banach space and $C$ a nonempty closed and convex subset of $X$. Let $\Pi_C{:}~X\to C$ denote the generalized metric projection operator introduced by Alber in [1]. In this paper, we define the Gâteaux directional differentiability of $\Pi_{C.}$ We investigate some properties of the Gâteaux directional differentiability of $\Pi_C$. In particular, if $C$ is a closed ball, or a closed and convex cone (including proper closed subspaces), or a closed and convex cylinder, then, we give the exact representations of the directional derivatives of $\Pi_{C.}$ By comparing the results in [12] and this paper, we see the significant difference between the directional derivatives of the generalized metric projection operator $\Pi_{C}$ and the Gâteaux directional derivatives of the standard metric projection operator $P_{C.}$

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A FORMULA OF CONDITIONAL ENTROPY FOR METRICS INDUCED BY PROBABILITY BI-SEQUENCES

M. RAHIMI, N. BIDABADI

数学物理学报(英文版). 2025 (4): 1619-1639. DOI: 10.1007/s10473-025-0420-3

摘要 ( 10 )

Westudytheconditional entropy of topological dynamical systems using a family of metrics induced by probability bi-sequences. We present a Brin-Katok formula by replacing the mean metric by a family of metrics induced by a probability bi-sequence. We also establish the Katoks entropy formula for conditional entropy for ergodic measures in the case of the new family of metrics.

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EMERGENT BEHAVIOR OF CLOSEST NEIGHBORS MODEL WITH NONLINEAR INHERENT DYNAMICS

Yuan LIANG, Chen WU, Jiugang DONG

数学物理学报(英文版). 2025 (4): 1640-1658. DOI: 10.1007/s10473-025-0421-2

摘要 ( 14 )

This paper studies the emergent dynamics of a flock with nonlinear inherent dynamics under closest neighbors model. We establish sufficient frameworks for convergence to flocking in terms of initial state and system parameters. When the number of closest neighbors is at least half of the population, it is shown that convergence to flocking occurs regardless of the initial state provided that the Lipschitz constant of nonlinear dynamics is smaller than the coupling strength. In contrast, when this number of closest neighbors is less than half of the population, we need to impose some restrictive conditions on the initial state to ensure the emergence of flocking based on the disturbed graphs approach. Our results are applicable to both continuous and discrete time cases. Finally, the validity of our theoretical analysis is tested by numerical simulations.

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HAUSDORFF DIMENSION OF RECURRENCE SETS FOR MATRIX TRANSFORMATIONS OF TORI

Zhangnan HU, Bing LI

数学物理学报(英文版). 2025 (4): 1659-1673. DOI: 10.1007/s10473-025-0422-1

摘要 ( 11 )

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax$ (mod 1), where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\cdots\le|\lambda_d|$. We investigate the Hausdorff dimension of the recurrence set $$R(\psi):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,\psi(n)) for infinitely many n\}$$ for $\alpha\ge\log|\lambda_d/\lambda_1|$, where $\psi$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $\alpha=\liminf\limits_{n\to\infty}\frac{-\log \psi(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.

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THREE INERTIAL ACCELERATION ALGORITHMS FOR SOLVING NON-MONOTONE EQUILIBRIUM PROBLEMS IN HILBERT SPACES

Yonghong YAO, Olaniyi S. IYIOLA, Yekini SHEHU

数学物理学报(英文版). 2025 (4): 1674-1700. DOI: 10.1007/s10473-025-0423-0

摘要 ( 19 )

Several results on iterative methods for equilibrium problems have been proposed and studied in the literature. Most of these results are obtained when the associated bifunction of the equilibrium problem is either a monotone or pseudomonotone operator. Results on iterative methods for equilibrium problems without monotonicity conditions on the bifunction are still few in the literature. In this paper, we study equilibrium problems for which the underlined bifunction is not assumed any form of monotonicity. We propose two weakly convergent iterative algorithms and one strongly convergent algorithm. We obtain our convergence results without assuming either monotonicity or pseudomonotonicity condition on the bifunction. Our proposed algorithms are tested numerically to be more efficient and faster than some few available algorithms for equilibrium problems without monotonicity in the literature.

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AN INEXACT SYMMETRIC PROXIMAL ADMM WITH CONVEX COMBINATION PROXIMAL CENTERS FOR SEPARABLE CONVEX PROGRAMMING

Jinbao JIAN, Xianke TANG, Jianghua YIN, Xianzhen JIANG

数学物理学报(英文版). 2025 (4): 1701-1722. DOI: 10.1007/s10473-025-0424-z

摘要 ( 24 )

In this paper, we develop an inexact symmetric proximal alternating direction method of multipliers (ISPADMM) with two convex combinations (ISPADMM-tcc) for solving two-block separable convex optimization problems with linear equality constraints. Specifically, the convex combination technique is incorporated into the proximal centers of both subproblems. We then approximately solve these two subproblems based on relative error criteria. The global convergence, and $O(\frac{1}{N})$ ergodic sublinear convergence rate measured by the function value residual and constraint violation are established under some mild conditions, where $N$ denotes the number of iterations. Finally, numerical experiments on solving the $l_1$-regularized analysis sparse recovery and the elastic net regularization regression problems illustrate the feasibility and effectiveness of the proposed method.

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