Authors:Janusz BRZDEK; Krzysztof CIEPLIŃSKI Pages: 377 - 390 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Janusz BRZDEK, Krzysztof CIEPLIŃSKI The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

Authors:Huifang JIA; Gongbao LI Pages: 391 - 418 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huifang JIA, Gongbao LI In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrödinger-Kirchhoff type - ∈ p M ( ∈ p - N ∫ R N ∇ u p ) Δ p u + V ( x ) u p - 2 u = f ( u ) in R N , where Δp is the p-Laplacian operator, 1 < p < N, M: R + → R + and V: R N → R + are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik-Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

Authors:Huijuan SONG; Jingxue YIN; Zejia WANG Pages: 419 - 428 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huijuan SONG, Jingxue YIN, Zejia WANG In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system - div ( h 1 ( x ) ∇ u p - 2 ∇ u ) = d ( x ) u r - 2 u + G u ( x , u , υ ) - div ( h 2 ( x ) ∇ υ q - 2 ∇ υ ) = f ( x ) υ s - 2 υ + G υ ( x , u , υ ) u = υ = 0 in Ω in Ω on ∂ Ω where Ω is a bounded domain in R N with smooth boundary ∂Ω, N≥2, 1 < r < p < ∞, 1 < s < q < ∞; h1(x) and h2(x) are allowed to have “essential” zeroes at some points in Ω ; d ( x ) u r - 2 u and f ( x ) υ s - 2 υ are small sources with Gu(x,u,v), Gv(x,u,v) being their high-order perturbations with respect to (u,v) near the origin, respectively.

Authors:Jing FU; Daqing JIANG; Ningzhong SHI; Tasawar HAYAT; Ahmed ALSAEDI Pages: 429 - 440 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jing FU, Daqing JIANG, Ningzhong SHI, Tasawar HAYAT, Ahmed ALSAEDI This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.

Authors:Mingquan WEI; Dunyan YAN Pages: 441 - 449 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Mingquan WEI, Dunyan YAN In this article, we obtain the sharp bounds from L P ( G n ) to the space wL P ( G n ) for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L P ( G n ) to the space L PI ( G n ) are obtained.

Authors:Huoyuan DUAN; Junhua MA Pages: 450 - 470 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Huoyuan DUAN, Junhua MA On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.

Authors:Jianjun ZHANG; Liangwen LIAO Pages: 471 - 478 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jianjun ZHANG, Liangwen LIAO In this article, we give a simple proof of Malmquist-Yosida type theorem of higher order algebraic differential equations, which is different from the methods as that of Gackstatter and Laine [2], and Steinmetz [12].

Authors:Zhou SHENG; Gonglin YUAN; Zengru CUI Pages: 479 - 496 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Zhou SHENG, Gonglin YUAN, Zengru CUI It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving unconstrained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.

Authors:Jiafeng LIAO; Yang PU; Chunlei TANG Pages: 497 - 518 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Jiafeng LIAO, Yang PU, Chunlei TANG In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered, { - Δ u = g ( x ) u 2 * - 2 u + λ f ( x ) u q - 2 u , x ∈ Ω u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N (N ≥ 3) is an open bounded domain with smooth boundary, 1 < q < 2,λ > 0. 2 * = 2 N N - 2 is the critical Sobolev exponent, f ∈ L 2 * 2 * - q ( Ω ) is nonzero and nonnegative, and g ∈ C ( Ω ¯ ) is a positive function with k local maximum points. By the Nehari method and variational method, k + 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75(2012) 2660-2671].

Authors:Yirang YUAN; Aijie CHENG; Dangping YANG; Changfeng LI; Qing YANG Pages: 519 - 545 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yirang YUAN, Aijie CHENG, Dangping YANG, Changfeng LI, Qing YANG The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in l 2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

Authors:Kamel BRAHIM; Latifa RIAHI Pages: 546 - 560 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Kamel BRAHIM, Latifa RIAHI In this article, we introduce the two dimensional Mellin transform M q ˜ ( f ) ( s , t ) , give some properties, establish the Paley-Wiener theorem and Plancherel formula, present the Hausdorff-Young inequality, and find several applications for the two dimensional Mellin transform.

Authors:P. BASKAR; S. PADMANABHAN; M. Syed ALI Pages: 561 - 579 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): P. BASKAR, S. PADMANABHAN, M. Syed ALI In this article, we investigates finite-time H ∞ control problem of Markovian jumping neural networks of neutral type with distributed time varying delays. The mathematical model of the Markovian jumping neural networks with distributed delays is established in which a set of neural networks are used as individual subsystems. Finite time stability analysis for such neural networks is addressed based on the linear matrix inequality approach. Numerical examples are given to illustrate the usefulness of our proposed method. The results obtained are compared with the results in the literature to show the conservativeness.

Authors:A.M. NAGY; N.H. SWEILAM Pages: 580 - 590 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): A.M. NAGY, N.H. SWEILAM In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann-Liouville and the Grünwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.

Authors:Yangrong LI; Lianbing SHE; Jinyan YIN Pages: 591 - 609 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yangrong LI, Lianbing SHE, Jinyan YIN A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.

Authors:Yekini SHEHU; Olaniyi. S. IYIOLA Pages: 610 - 626 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yekini SHEHU, Olaniyi. S. IYIOLA In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.

Authors:Zhonglin WU; Shu WANG Pages: 627 - 642 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Zhonglin WU, Shu WANG We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L ∞(L 2) and L ∞(H1) norm.

Authors:Chunyan LIU; Zihou ZHANG; Yu ZHOU Pages: 643 - 650 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Chunyan LIU, Zihou ZHANG, Yu ZHOU In this article, we prove the following results: (1) A Banach space X is weak midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively weakly compact k-Chebyshev set; (2) A Banach space X is midpoint locally k-uniformly rotund if and only if every closed ball of X is an approximatively compact k-Chebyshev set.

Authors:Lei ZHANG; Bin LIU Pages: 651 - 672 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Lei ZHANG, Bin LIU This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton (gμHS2) system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative (gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative (gμHS2) system, we show that the semi-group of the solution operator { S ( t ) } t ≥ 0 has a bounded absorbing set. Moreover, we prove that the dynamical system { S ( t ) } t ≥ 0 possesses a global attractor in the Sobolev space H 2 ( S ) × H 2 ( S ) .

Authors:Xiaojun ZHAO Pages: 673 - 680 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Xiaojun ZHAO In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation - Δ u = ∫ R N u ( y ) p x - y α d y u ( x ) p - 2 u ( x ) in R N , where N ≥ 3, 0 < α < {4,N}. Suppose that 2 < p < 2 N - α N - 2 , we will show that this problem does not possess nontrivial solution with finite Morse index. While for p = 2 N - α N - 2 , if i(u) < ∞ then we have ∫ R N ∫ R N u ( x ) p u ( y ) p x - y α d x d y < ∞ and ∫ R N ∇ u 2 d x = ∫ R N ∫ R N u ( x ) p u ( y ) p x - y α d x d y .

Authors:Yuecai HAN; Yifang SUN Pages: 681 - 694 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Yuecai HAN, Yifang SUN The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H∈(1/2,1) and the underlying standard Brownian motions are studied. The generalization of the Itô formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.

Authors:Hongmei ZHU Pages: 695 - 708 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Hongmei ZHU In this article, we study a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric α and a 1-form β. We determine all of Douglas general (α, β)-metrics on a manifold of dimension n > 2.

Authors:Rakesh KUMAR; Anuj Kumar SHARMA; Kulbhushan AGNIHOTRI Pages: 709 - 732 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Rakesh KUMAR, Anuj Kumar SHARMA, Kulbhushan AGNIHOTRI In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage (time delay) is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period (time delay, τ) passes through a critical value. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Authors:Xincai ZHU Pages: 733 - 744 Abstract: Publication date: March 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 2 Author(s): Xincai ZHU In this article, we study constrained minimizers of the following variational problem e ( ρ ) : = inf { u ∈ H 1 ( R 3 ) , ‖ u ‖ 2 2 = ρ } E ( u ) , ρ > 0 , where E(u) is the Schrödinger-Poisson-Slater (SPS) energy functional E ( u ) : = 1 2 ∫ R 3 ∇ u ( x ) 2 d x - 1 4 ∫ R 3 ∫ R 3 u 2 ( y ) u 2 ( x ) x - y d y d x - 1 p ∫ R 3 u ( x ) p d x in R 3 , and p∈ (2, 6). We prove the existence of minimizers for the cases 2 < p < 10 3 , ρ > 0, and p = 10 3 , 0 < ρ < ρ *, and show that e(ρ)= − ∞ for the other cases, where ρ* = ‖ ϕ ‖ 2 2 and ϕ(x) is the unique (up to translations) positive radially symmetric solution of - Δ u + u = u 7 3 in R 3 . Moreover, when e ( ρ * ) = - ∞ , the blow-up behavior of minimizers as ρ ↗ ρ * is also analyzed rigorously.

Authors:Jun LIU; Dachun YANG; Wen YUAN Pages: 1 - 33 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Jun LIU, Dachun YANG, Wen YUAN Let p∈ (0,1], q ∈(0,∞] and A be a general expansive matrix on ℝ n . Let H A p , q ( ℝ n ) be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize H A p , q ( ℝ n ) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley g λ *-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space L p,qℝ n . All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on ℝ n . Moreover, the range of λ in the g λ *-function characterization of H A p , q ( ℝ n ) coincides with the best known one in the classical Hardy space H p ( ℝ n ) or in the anisotropic Hardy space H A p ( ℝ n ) .

Authors:Jing ZHANG; Yongqian ZHANG Pages: 34 - 56 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Jing ZHANG, Yongqian ZHANG We study the initial-boundary value problem for the one dimensional Euler-Boltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions (UΔt,d, IΔt,d ) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.

Authors:Hangjin JIANG; Qiongli WU Pages: 57 - 72 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Hangjin JIANG, Qiongli WU In this paper, we proposed a new statistical dependency measure for two random vectors based on copula, called copula dependency coefficient (CDC). The CDC is proved to be robust to outliers and easy to be implemented. Especially, it is powerful and applicable to high-dimensional problems. All these properties make CDC practically important in related applications. Both experimental and application results show that CDC is a good robust dependence measure for association detecting.

Authors:Xian-jia WANG; Rui DONG; Lin CHEN Pages: 73 - 92 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Xian-jia WANG, Rui DONG, Lin CHEN Natural selection opposes the evolution of cooperation unless specific mechanisms are at work in Prisoner's Dilemma. By taking advantage of the modern control theory, the controller design is discussed and the optimal control is designed for promoting cooperation based on the recent advances in mechanisms for the evolution of cooperation. Two control strategies are proposed: compensation control strategy for the cooperator when playing against a defector and reward control strategy for cooperator when playing against a cooperator. The feasibility and effectiveness of these control strategies for promoting cooperation in different stages are analyzed. The reward for cooperation can't prevent defection from being evolutionary stable strategy (ESS). On the other hand, compensation for the cooperator can't prevent defection from emerging and sustaining. By considering the effect and the cost, an optimal control scheme with constraint on the admissible control set is put forward. By analyzing the special nonlinear system of replicator dynamics, the exact analytic solution of the optimal control scheme is obtained based on the maximum principle. Finally, the effectiveness of the proposed method is illustrated by examples.

Authors:Meiman SUN; Guozheng YAN Pages: 110 - 124 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Meiman SUN, Guozheng YAN In this paper we consider a kind of exterior transmission problem in which the refractive index n(x) is a piecewise positive constant. Through establishing an equivalent boundary integral system, we obtain that the set of exterior transmission eigenvalues is a discrete set. Furthermore, we prove that there does not exist a transmission eigenvalue under some conditions.

Authors:Yingbo LIU; Ingo WITT Pages: 125 - 150 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Yingbo LIU, Ingo WITT For the 2-D quasilinear wave equation ( ∂ t 2 - Δ x ) u + ∑ i , j = 0 2 g i j ( ∂ u ) ∂ i j u = 0 satisfying null condition or both null conditions, a blowup or global existence result has been shown by Alinhac. In this paper, we consider a more general 2-D quasilinear wave equation ( ∂ t 2 - Δ x ) u + ∑ i , j = 0 2 g i j ( u , ∂ u ) ∂ i j u = 0 satisfying null conditions with small initial data and the coefficients depending simultaneously on u and ∂u. Through construction of an approximate solution, combined with weighted energy integral method, a quasi-global or global existence solution are established by continuous induction.

Authors:Lixin CHENG; Sijie LUO Pages: 151 - 156 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Lixin CHENG, Sijie LUO In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of norm-attaining functionals contains an infinite dimensional closed subspace of X * if and only if X * contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.

Authors:Zhiming WANG Pages: 157 - 168 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Zhiming WANG For a stochastic differential equation with non-Lipschitz coefficients, we construct, by Euler scheme, a measurable flow of the solution, and we prove the solution is a Markov process.

Authors:Yashan ZHANG Pages: 169 - 176 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Yashan ZHANG We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.

Authors:Wei SONG; Lin LI Pages: 177 - 186 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Wei SONG, Lin LI Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this paper, we prove the existence of decreasing solutions for a general iterative equation, which was proposed as an open problem in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995) 385–405] (or [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 (2000) 29–36]).

Authors:Chengzhi WANG; Mingshu ZHANG; Caidi ZHAO Pages: 187 - 202 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Chengzhi WANG, Mingshu ZHANG, Caidi ZHAO This paper studies the trajectory asymptotic behavior of a non-autonomous incompressible non-Newtonian fluid in 3D bounded domains. In appropriate topologies, the authors prove the existence of the uniform trajectory attractor for the translation semigroup acting on the united trajectory space.

Authors:Haiping NIU; Shu WANG Pages: 203 - 219 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Haiping NIU, Shu WANG We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t < T *. Then, when T *≤ t < T', we get a new shock wave between two rarefactions, and then, when t>T', another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t→∞. The technique does not involve dimensional reduction or coordinate transformation.

Authors:Kuo-Shou CHIU Pages: 220 - 236 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Kuo-Shou CHIU In this paper, we investigate the existence, uniqueness and the asymptotic equivalence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type (DEPCAG). This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.

Authors:Hairong LIU Pages: 237 - 247 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Hairong LIU In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.

Authors:Xiaojuan DUAN Pages: 269 - 288 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Xiaojuan DUAN In this paper, we explicitly construct some rotationally symmetric gradient pseudo-Kähler-Ricci solitons which depend on some parameters, on some line bundles and other bundles over projective spaces. We also discuss the “phase change” phenomenon caused by the variation of parameters.

Authors:Zhaoxing YANG; Guobao ZHANG Pages: 289 - 302 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Zhaoxing YANG, Guobao ZHANG This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts (waves with speeds c>c *, where c=c * is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x→ − ∞, but it can be allowed arbitrary large in other locations, which improves the results in [9, 18, 21].

Authors:Santhosh GEORGE; C.D. SREEDEEP Pages: 303 - 314 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Santhosh GEORGE, C.D. SREEDEEP In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the implementation of Lavrentiev regularization method. Using general Hölder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

Authors:Qiang TU; Wenyi CHEN Pages: 315 - 332 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Qiang TU, Wenyi CHEN In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on an n-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the structure theorem of the Lagrangian currents for semi-convex functions is given and the k-Hessian measures are calculated by a different method in terms of currents.

Authors:Hassan Eltayeb GADAIN Pages: 333 - 346 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Hassan Eltayeb GADAIN In this paper, the modification of double Laplace decomposition method is proposed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.

Authors:Yajuan XU; Guojing WANG Pages: 347 - 360 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Yajuan XU, Guojing WANG In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.

Authors:Hongwei LIU; Yijun HU; Linxiao WEI Pages: 361 - 376 Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Hongwei LIU, Yijun HU, Linxiao WEI In this paper, from the viewpoint of the time value of money, we study the risk measures for portfolio vectors with discount factor. Cash subadditive risk measures for portfolio vectors are proposed. Representation results are given by two different methods which are convex analysis and enlarging space. Especially, the method of convex analysis make the line of reasoning and the representation result be simpler. Meanwhile, spot and forward risk measures for portfolio vectors are also introduced, and the relationships between them are investigated.

Authors:Junfeng LIU; Ciprian A. TUDOR Pages: 1545 - 1566 Abstract: Publication date: November 2017 Source:Acta Mathematica Scientia, Volume 37, Issue 6 Author(s): Junfeng LIU, Ciprian A. TUDOR In this paper we study a fractional stochastic heat equation on R d ( d ≥ 1 ) with additive noise ∂ ∂ t u ( t , x ) = d δ _ α _ u ( t , x ) + b ( u ( t , x ) ) + W ˙ H ( t , x ) where d δ _ α _ is a nonlocal fractional differential operator and W ˙ H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

Authors:Samir KALLEL Pages: 1567 - 1593 Abstract: Publication date: November 2017 Source:Acta Mathematica Scientia, Volume 37, Issue 6 Author(s): Samir KALLEL The aim of this paper is to prove duality and reflexivity of generalized Lipschitz spaces Λ α , p , q k ( R ) , α ∈ R and 1 ≤ p , q ≤ ∞ , in the context of Dunkl harmonic analysis.

Authors:Yanli HAN; Yan GAO Pages: 1594 - 1606 Abstract: Publication date: November 2017 Source:Acta Mathematica Scientia, Volume 37, Issue 6 Author(s): Yanli HAN, Yan GAO This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a discriminating domain if the set is a discriminating domain. Secondly, in order to determine that a bounded polyhedron is a discriminating domain, we give a result that it only needs to verify that the extreme points of the polyhedron meet the viability conditions. The difference between our result and the existing ones is that our result just needs to verify the finite points (extreme points) and the existing ones need to verify all points in the bounded polyhedron.

Authors:Xiang GAO; Jihua MA Pages: 1607 - 1618 Abstract: Publication date: November 2017 Source:Acta Mathematica Scientia, Volume 37, Issue 6 Author(s): Xiang GAO, Jihua MA This paper is concerned with the Diophantine properties of the sequence { ξ θ n } , where 1 ≤ ξ < θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μ λ with λ = θ - 1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μ λ almost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.

Authors:Kwok-Pun Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Kwok-Pun HO We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob's inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob's inequality on martingale block spaces, we obtain the Davis' decompositions for martingale Morrey spaces.

Authors:ZHU Abstract: Publication date: January 2018 Source:Acta Mathematica Scientia, Volume 38, Issue 1 Author(s): Li ZHU In this paper we obtain the Plancherel formula for the spaces of L 2-sections of the line bundles over the pseudo-Riemannian symmetric space G/H where G = SL(n + 1, ℝ) and H = S(GL(1, ℝ) × GL(n 1, ℝ)). The Plancherel formula is given in an explicit form by means of spherical distributions associated with the character χλ of the subgroup H. We follow the method of Faraut, Kosters and van Dijk.