数学物理学报(英文版)

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EXISTENCE UNIQUENESS AND DECAY OF SOLUTION FOR FRACTIONAL BOUSSINESQ APPROXIMATION

郭春晓, 张景军, 郭柏灵

数学物理学报(英文版). 2013 (4): 883-900. DOI: 10.1016/S0252-9602(13)60048-X

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The Boussinesq approximation finds more and more frequent use in geologi-cal practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of commu-tator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results
obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively.

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POWER VARIATION OF SUBFRACTIONAL BROWNIAN MOTION AND APPLICATION

申广君, 闫理坦, 刘俊峰

数学物理学报(英文版). 2013 (4): 901-912. DOI: 10.1016/S0252-9602(13)60049-1

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In this paper, we consider the power variation of subfractional Brownian mo-tion. As an application, we introduce a class of estimators for the index of a subfractional Brownian motion and show that they are strongly consistent.

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EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN RN WITH ZERO MASS

李工宝, 叶红雨

数学物理学报(英文版). 2013 (4): 913-928. DOI: 10.1016/S0252-9602(13)60050-8

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In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ D^1,2(R^N) × D^1,2(R^N) to the following semilinear elliptic system
{−△u = K(x)f(v), x ∈ R^N,
−△v = K(x)g(u), x ∈ R^N(0.1)
by using a linking theorem, where K(x) is a positive function in Ls(R^N) for some s > 1 and the nonnegative functions f, g ∈ C(R, R) are of quasicritical growth, superlinear at infinity. We do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.
Our main result can be viewed as a partial extension of a recent result of Alves, Souto and Montenegro in [1] concerning the existence of a positive solution to the following semilinear elliptic problem

−△u = K(x)f(u), x ∈ R^N,

and a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in R^N.

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GLOBAL WELL-POSEDNESS OF THE 2D INCOMPRESSIBLE MICROPOLAR FLUID FLOWS WITH PARTIAL VISCOSITY AND ANGULAR VISCOSITY

陈明涛

数学物理学报(英文版). 2013 (4): 929-935. DOI: 10.1016/S0252-9602(13)60051-X

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This paper is concerned with the two-dimensional equations of incompress-ible micropolar fluid flows with mixed partial viscosity and angular viscosity. The global existence and uniqueness of smooth solution to the Cauchy problem is established.

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A NOTE ON THE EXISTENCE OF STATIONARY SOLUTIONS OF THE COMPRESSIBLE EULER-POISSON EQUATIONS WITH 6/5 < γ <|2

向建林

数学物理学报(英文版). 2013 (4): 936-942. DOI: 10.1016/S0252-9602(13)60052-1

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This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When 6/5 < γ < 2, under the weak smoothness of entropy function, we find a sufficient condition to guarantee the existence of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy.

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ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

胡动刚, 胡学海

数学物理学报(英文版). 2013 (4): 943-949. DOI: 10.1016/S0252-9602(13)60053-3

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A famous theorem of Szemer’edi asserts that any subset of integers with posi-tive upper density contains arbitrarily arithmetic progressions. Let F_q be a finite field with q elements and F_q((X⁻¹)) be the power field of formal series with coefficients lying in F_q. In this paper, we concern with the analogous Szemer´edi problem for continued fractions of Laurent series: we will show that the set of points x ∈ F_q((X−1)) of whose sequence
of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.

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WEAK SOLUTIONS OF MONGE-AMPÈRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

蒋飞达, 杨孝平

数学物理学报(英文版). 2013 (4): 950-962. DOI: 10.1016/S0252-9602(13)60054-5

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This paper concerns the weak solutions of some Monge-Amp`ere type equa-tions in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Amp`ere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

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GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

王宇钊, 杨杰, 陈文艺

数学物理学报(英文版). 2013 (4): 963-974. DOI: 10.1016/S0252-9602(13)60055-7

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Let (M, g, e^−fdv) be a smooth metric measure space. In this paper, we con-sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equation

∂u/∂t= efdiv(e−f |∇u|p−2∇u)
on M × [0, ∞), where 1 < p < ∞ and f is a smooth function on M under the assumption that the m-dimensional nonnegative Bakry-Émery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Émery Ricci curva-ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

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OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DYNAMIC EQUATIONS WITH p-LAPLACIAN AND DAMPING

Taher S. HASSAN, Qingkai KONG

数学物理学报(英文版). 2013 (4): 975-988. DOI: 10.1016/S0252-9602(13)60056-9

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This paper concerns the oscillation of solutions of the second order nonlinear dynamic equation with p-Laplacian and damping
(r(t)φ_α(x^Δ(t))^Δ+ p (t)φ_α(x^Δα(t)+ q(t)f (x^α(t)) = 0
on a time scale T which is unbounded above. Sign changes are allowed for the coefficient functions r, p and q. Several examples are given to illustrate the main results.

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AREA INTEGRAL FUNCTIONS AND H^∞ FUNCTIONAL CALCULUS FOR SECTORIAL OPERATORS ON HILBERT SPACES

陈泽乾, 孙牧

数学物理学报(英文版). 2013 (4): 989-997. DOI: 10.1016/S0252-9602(13)60057-0

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Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi´s results on H^∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.

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RUIN PROBABILITY IN A SEMI-MARKOV RISK MODEL WITH CONSTANT INTEREST FORCE AND HEAVY-TAILED CLAIMS

杨虎, 薛凯

数学物理学报(英文版). 2013 (4): 998-1006. DOI: 10.1016/S0252-9602(13)60058-2

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In the present paper, we consider a kind of semi-Markov risk model (SMRM) with constant interest force and heavy-tailed claims, in which the claim rates and sizes are conditionally independent, both fluctuating according to the state of the risk business. First, we derive a matrix integro-differential equation satisfied by the survival probabilities. Second, we analyze the asymptotic behaviors of ruin probabilities in a two-state SMRM with special claim amounts. It is shown that the asymptotic behaviors of ruin probabilities depend only on the state 2 with heavy-tailed claim amounts, not on the state 1 with exponential claim sizes.

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GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

耿世锋, 陈国旺

数学物理学报(英文版). 2013 (4): 1007-1023. DOI: 10.1016/S0252-9602(13)60059-4

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In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equation
v_t − αΔv_t− βΔv + γΔ²v +∑ⁿ_j₌₁f_j (v)x_j = Δg(v) + G(v), x ∈ Rⁿ, t > 0 (1)
are proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation (1) are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.

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BLOW-UP OF THE SOLUTION FOR A CLASS OF POROUS MEDIUM EQUATION WITH POSITIVE INITIAL ENERGY

吴秀兰, 高文杰

数学物理学报(英文版). 2013 (4): 1024-1030. DOI: 10.1016/S0252-9602(13)60060-0

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This paper deals with a class of porous medium equation
u_t= Δu^m + f(u)
with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).

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ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

周军

数学物理学报(英文版). 2013 (4): 1031-1048. DOI: 10.1016/S0252-9602(13)60061-2

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We consider the growth rate and quenching rate of the following problem with singular nonlinearity
u_t = △u − v^−λ, v_t = △v − u^−μ, (x, t) ∈ Rⁿ × (0,∞),
u(x, 0) = u₀(x), v(x, 0) = v₀(x), x ∈Rⁿfor any n ≥ 1, where λ, μ > 0 are constants. More precisely, for any u₀(x), v₀(x) satisfying A₁₁(1+|x|²) ^α11 ≤ u₀ ≤ A₁₂(1+|x|²) ^α12 , A₂₁(1+|x|²) ^α21 ≤ v₀ ≤ A₂₂(1+|x|²) ^α22 for some constants α₁₂ ≥ α₁₁, α₂₂≥α₂₁, A₁₂ ≥ A₁₁, A₂₂ ≥ A₂₁, the global solution (u, v) exists and satisfies A₁₁(1+|x|²+b₁t) ^α11≤ u ≤ A₁₂(1+|x|²+b₂t) ^α12 , A₂₁(1+|x|²+b₁t) ^α21 ≤ v ≤ A₂₂(1+|x|²+b₂t) ^α22 for some positive constants b₁, b₂ (see Theorem 3.3 for the parameters A_ij , α_ij , b_i, i, j = 1, 2). When (1 − λ)(1 − λμ) > 0, (1 − λ)(1 − λμ) > 0 and 0 < u₀ ≤A₁(b₁T +|x|²)^{1−λ/1−λμ} , 0 < v₀ ≤ A₂(b₂T +|x|²)^{1−μ/1−λμ} in Rⁿ for some constants A_i, b_i(i = 1, 2)satisfying A^−λ₂ > 2nA₁1−λ/1−λμ , A^−μ₁ > 2nA₂1−μ/1−λμ and 0 < b₁ ≤ (1−λμ)A^−λ₂−(1−λ)2nA₁/(1−λ)A₁, 0 < b₂ ≤ (1−λμ)A^−μ₁−(1−μ)2nA₂/(1−μ)A₂, we prove that u(x, t) ≤ A₁(b₁(T −t)+|x|²)^{1−λ/1−λμ} , v(x, t) ≤A₂(b₂(T − t) + |x|²)^{1−μ/1−λμ} in Rⁿ × (0, T). Hence, the solution (u, v) quenches at the origin x = 0 at the same time T (see Theorem 4.3). We also find various other conditions for the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.

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SOME NEW EXTENSIONS OF EDELSTEIN-SUZUKI-TYPE FIXED POINT THEOREM TO G-METRIC AND G-CONE METRIC SPACES

Fridoun MORADLOU, Peyman SALIMI, Pasquale VETRO

数学物理学报(英文版). 2013 (4): 1049-1058. DOI: 10.1016/S0252-9602(13)60062-4

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In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.

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LIE IDEALS, MORITA CONTEXT AND GENERALIZED (α, β)-DERIVATIONS

S. Khalid NAUMAN, Nadeem ur REHMAN, R. M. AL-OMARY

数学物理学报(英文版). 2013 (4): 1059-1070. DOI: 10.1016/S0252-9602(13)60063-6

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A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson´s famous result, several tech-niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

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A CHARACTERIZATION OF M_π-GROUPS

海进科, 李正兴

数学物理学报(英文版). 2013 (4): 1071-1075. DOI: 10.1016/S0252-9602(13)60064-8

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Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs´s work, we introduce the definition of M_π-groups and provide a characterization of M_π-groups.

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A NEW REDUCED-ORDER FVE ALGORITHM BASED ON POD METHOD FOR VISCOELASTIC EQUATIONS

李宏, 罗振东, 高骏强

数学物理学报(英文版). 2013 (4): 1076-1098. DOI: 10.1016/S0252-9602(13)60065-X

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A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illus-trate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.

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p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL

孙小妹

数学物理学报(英文版). 2013 (4): 1099-1112. DOI: 10.1016/S0252-9602(13)60066-1

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In this paper, we deal with the following problem:
{−Δ_pu − λ|y|^−p|u|^p−2u = |y|^−s|u|p*^(s)−2u + |u|^p*−2u in R^N, y ≠ 0,
u ≥ 0,
where u(x) = u(y, z) : R^m × R^N−m −→ R, N ≥ 3, 2 < m < N, 1 < p < m, λ <( (m−p)/p )^pand 0 < s < p, p*(s) = p(N−s)/N−p , p* = pN/N−p . By variational method, we prove the existence of a nontrivial weak solution when 0 < λ <( (m−p/ p ))^p and the existence of a cylindrical weak solution when λ < 0.

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FIXED POINTS AND STABILITY FOR QUARTIC MAPPINGS IN -NORMED LEFT BANACH MODULES ON BANACH ALGEBRAS

H. Azadi KENARY, A.R. ZOHDI, M. Eshaghi GORDJI

数学物理学报(英文版). 2013 (4): 1113-1118. DOI: 10.1016/S0252-9602(13)60067-3

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The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation
∑ⁿ_k₌₂(∑^k_i₁₌₂∑^k+1_i2=i1+1…∑ⁿ_in−k_+1=in−k+1)f (∑ⁿ_i_{=1, i6=i1}, …, _in−k+1x_i−∑^n−k+1_r=1x_ir)+ f (∑ⁿ_i₌₁x_i)= 2ⁿ⁻²∑_1≤i<j≤_n(f (x_i+ x_j)+f (x_i− x_j)−2ⁿ⁻⁵(n − 2)∑ⁿ_i₌₁f (2x_i)
(n ∈N, n ≥ 3) in β-Banach modules on Banach algebras.
The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

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WEIGHTED COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

刘小松, 娄增建

数学物理学报(英文版). 2013 (4): 1119-1126. DOI: 10.1016/S0252-9602(13)60068-5

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We characterize the boundedness and compactness of weighted composition operators on weighted Dirichlet spaces in terms of Nevanlinna counting functions and Carleson measure.

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A MATHEMATICAL MODEL OF ENTERPRISE COMPETITIVE ABILITY AND PERFORMANCE THROUGH EMDEN-FOWLER EQUATION (II)

Meng-Rong LI, Yue-Loong CHANG, Yu-Tso LI

数学物理学报(英文版). 2013 (4): 1127-1140. DOI: 10.1016/S0252-9602(13)60069-7

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In this paper we work with the ordinary equation u" −u² (u + ¯u) = 0 and ob-tain some interesting phenomena concerning, blow-up, blow-up rate, life-span of solutions to those equations.

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ON PROPERTIES OF MEROMORPHIC SOLUTIONS FOR COMPLEX DIFFERENCE EQUATION OF MALMQUIST TYPE

黄志波, 陈宗煊, 李倩

数学物理学报(英文版). 2013 (4): 1141-1152. DOI: 10.1016/S0252-9602(13)60070-3

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In this paper, we present the properties on zeros, fixed points, poles, Borel exceptional value of finite order transcendental meromorphic solutions of complex difference
equation of Malmquist type
∑ⁿ_j₌₁f (z + c_j ) = R(f (z)) =P(f (z))Q(f (z))=a_pf(z)^p + a_p−1f (z)^p−1 + … + a₁f (z) + a₀/b_qf (z)^q+ b_q₋₁f (z)^q−1 + … + b₁f(z) + b₀,
where n(∈ N) ≥ 2, P(f(z)) and Q(f(z)) are relatively prime polynomials in f(z) with rational coefficients as (s = 0, 1, … , p) and bt (t = 0, 1, … , q) such that a0apbq 6≡ 0, and also consider the existence and the forms on rational solutions of this type of difference equations. Some examples are also listed to show that the assumptions of theorems, in certain senses, are the best possible.

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WELL-POSEDNESS IN CRITICAL SPACES FOR THE FULL COMPRESSIBLE MHD EQUATIONS

边东芬, 郭柏灵

数学物理学报(英文版). 2013 (4): 1153-1176. DOI: 10.1016/S0252-9602(13)60071-5

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In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N ≥2, under the assumptions that the initial density is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.

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SECTORIAL OSCILLATION THEORY OF LINEAR DIFFERENTIAL EQUATIONS

吴昭君, 陈裕先

数学物理学报(英文版). 2013 (4): 1177-1186. DOI: 10.1016/S0252-9602(13)60072-7

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In this paper, the sectorial oscillation of the solutions of higher order homo-geneous linear differential equations
f^(k) + A_n₋₂(z)f ^(k−2) +… + A₁(z)f ' + A₀(z)f = 0
with infinite order entire function coefficients is studied. Results are obtained to extend some results in [19] and [18].

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ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS

Souhir ELLEUCH, Maher MNIF

数学物理学报(英文版). 2013 (4): 1187-1201. DOI: 10.1016/S0252-9602(13)60073-9

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When A ∈ LR(H) and B ∈ LR(K) are given, for C ∈ LR(K, H) we denote by M_C the linear relation acting on the infinite dimensional separable Hilbert space HKof the form M_C =

(A C ).

0 B
In this paper, we give the necessary and sufficient conditions on A and B for which M_C is upper semi-Fredholm with negative index or Weyl for some C ∈ LR(K, H).

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HÖLDER ESTIMATES FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS

宋巧珍, 王立河, 李东升

数学物理学报(英文版). 2013 (4): 1202-1218. DOI: 10.1016/S0252-9602(13)60074-0

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In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the H¨older type estimates for the weak solutions.

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